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Math 586: Introduction to Algebraic Number Theory.

2013-04-25T20:10:26Z

Pmj5:

== Catalog Information ==

=== Title ===
Introduction to Algebraic Number Theory.

=== Credit Hours ===
3

=== Prerequisite ===
[[Math 372]] or equivalent; instructor's consent.

=== Description ===
Algebraic integers; different and discriminant; decomposition of primes; class group; Dirichlet unit theorem; Dedekind zeta function; cyclotomic fields; valuations; completions.

== Desired Learning Outcomes ==

=== Prerequisites ===
[[Math 372]] is a prerequisite for this course. In particular, students should be familiar with the concepts of groups and rings, and they should understand constructions of quotient groups and quotient rings. By this point in their mathematical career, students should be skilled at proving theorems by themselves.

=== Minimal learning outcomes ===
Students should achieve an advanced mastery of the topics listed below. This means that they should know all relevant definitions, correct statements and proofs of the major theorems (including their hypotheses and limitations), and examples and non-examples of the various concepts. The students should be able to demonstrate their mastery by solving difficult problems related to these concepts, and by proving theorems about the below concepts, even if the theorems go beyond the material in the text.
<div style="-moz-column-count:2; column-count:2;">
# Number Fields
#*Algebraic Numbers
#*Algebraic Integers
#*Cyclotomic Fields
#*Trace and Norm
#*Discriminants
#*Integral Bases
#*Computing Integral Bases
#Prime decomposition in rings of integers
#*Ideal theory of Dedekind domains
#*Splitting, ramification, inertia of primes
#*Computing prime decompositions
#*Decomposition and inertia groups
#*Frobenius maps
#*Functorial properties of the Frobenius
#Ideal Class Group
#*Finiteness of the class group
#*Minkowski bounds
#*Distribution of ideals in ideal classes
#*Class group computations in quadratic fields
#Dirichlet's unit theorem
#*Computation of fundamental units in quadratic fields
#Cebotarev Density Theorem (Statement)
#Dedekind zeta function
#* Class number formula

</div>
=== Textbooks ===

Possible textbooks for this course include (but are not limited to):

* Daniel Marcus, Number Fields
* Serge Lang, Algebraic Number Fields
* Pierre Samuel, Algebraic Theory of Numbers

=== Additional topics ===
As time permits, additional topics that might be considered include Galois representations, class field theory, module theory over Dedekind domains, etc.

=== Courses for which this course is prerequisite ===

[[Category:Courses|586]]
This course is not a prerequisite for any other courses.

Math 487: Intro to Number Theory

2011-08-04T22:26:36Z

Pmj5: Added missing punctuation.

== Catalog Information ==

=== Title ===
Number Theory.

=== (Credit Hours:Lecture Hours:Lab Hours) ===
(3:3:0)

=== Offered ===
F, Sp

=== Prerequisite ===
[[Math 371]].

=== Description ===
Foundations; congruences; quadratic reciprocity; unique factorization, prime distribution or Diophantine equations.

== Desired Learning Outcomes ==
This course is aimed at undergraduate mathematics majors. It is a first course in number theory, and is intended to introduce students to number theoretic problems and to different areas of number theory. Number theory has a very long history compared to some other areas of mathematics, and has many applications, especially to coding theory and cryptography.

=== Prerequisites ===
[[Math 371]] is a prerequisite for this course. In particular, students should be familiar with the concepts of groups and rings, and they should understand constructions of quotient groups and quotient rings. By this point in their mathematical career, students should be comfortable proving theorems by themselves.

=== Minimal learning outcomes ===
Students should achieve mastery of the topics listed below. This means that they should know all relevant definitions, correct statements of the major theorems (including their hypotheses and limitations), and examples and non-examples of the various concepts. The students should be able to demonstrate their mastery by solving non-trivial problems related to these concepts, and by proving simple (but non-trivial) theorems about the below concepts, related to, but not identical to, statements proven by the text or instructor.

<div style="-moz-column-count:2; column-count:2;">
# Divisibility in the integers
#* Prime numbers
#* Unique factorization
#* Euclid’s algorithm
#* GCD and LCM
# Congruence arithmetic
#* Complete and reduced residue systems
#* Linear congruences
#* Chinese remainder theorem
#* Polynomial congruences
#* Hensel’s lemma
#* Quadratic residues
#* Legendre and Jacobi symbols
#* Quadratic reciprocity
# Primitive roots
#* Existence of primitive roots
#* Structure of units modulo nonprimes
# Number Theoretic Functions
#* Moebius Function
#* Euler phi function
#* Sum of divisors function
#* Big Oh notation
#* Little Oh notation
#* Euler-Maclaurin Summation
#* Abel summation
# Distribution of Primes
#* Definition of Pi(''x'')
#* Estimates of Pi(''x'')
#* Primes in arithmetic progressions
#* Bertrand’s Hypothesis
# Sums of Squares
#* Representations of numbers as sums of two and four squares
#* Statement of Waring’s Problem

</div>
=== Textbooks ===

Possible textbooks for this course include (but are not limited to):

* Ivan Niven, Herbert S. Zuckerman, and Hugh L. Montgomery, <i>An Introduction to the Theory of Numbers (5th edition)</i>, Wiley, 1991.
* George Andrews, <i>Number Theory</i>, Dover, 1994.
* William LeVeque, <i>Fundamentals of Number Theory</i>, Dover, 1996.

=== Additional topics ===
Beyond the minimal learning outcomes, instructors are free to cover additional topics. These may include (but are certainly not limited to): diophantine approximation, continued fractions, elliptic curves, cryptography, partition theory. It is expected that some topics beyond the minimal learning objectives will typically be discussed.

=== Courses for which this course is prerequisite ===
This course is not a prerequisite for any other courses in the regular curriculum. Hence, this course may be the only opportunity for students to learn the topics listed here. As a result, it is important that all learning objectives be completed.

[[Category:Courses|487]]

Math 485: Mathematical Cryptography

2010-02-18T20:15:36Z

Pmj5:

== Catalog Information ==

=== Title ===
Mathematical Cryptography.

=== (Credit Hours:Lecture Hours:Lab Hours) ===
(3:3:0)

=== Offered ===
W


=== Prerequisite ===
[[Math 313]].

=== Recommended ===
[[Math 371]].

=== Description ===
A mathematical introduction to some of the high points of modern cryptography.

== Desired Learning Outcomes ==

This is a course in the mathematics and algorithms of modern cryptography. It complements, rather than being equivalent to, the current CS course on Computer Security (CS 465) and the current Information Technology course on Encryption and Compression (IT 531).

=== Prerequisites ===

The requirement for [[Math 313]] ensures both an appropriate level of mathematical maturity and a basic knowledge of linear algebra. The recommendation for [[Math 371]] encourages students to be familiar with the concepts of groups, rings, and fields.

=== Minimal learning outcomes ===

The student should gain a understanding of the following topics. In particular this includes knowing the definitions, being familiar with standard examples, and being able to solve mathematical and algorithmic problems by directly using the material taught in the course. This includes appropriate use of Maple, Mathematica, or another appropriate computing language.

<div style="-moz-column-count:2; column-count:2;">
# Classical systems, including:
#* Substitution theory
#* Block ciphers
#* Enigma
# Elementary number theory as follows:
#* Euclid's algorithm
#* Modular arithmetic and the algorithm for modular exponentiation
#* Chinese Remainder Theorem
#* Fermat and Euler Theorems
#* Primitive roots
#* Legendre and Jacobi symbols
#* Elementary continued fractions
#* Simple discussion of finite fields
# The DES and AES encryption standards
# RSA and its strengths and weaknesses; attacks on RSA
#* Wiener's continued fraction attack on low decryption exponent
# Primality testing algorithms
# Factorization techniques
#* The Quadratic Sieve
# Discrete logarithms
#* Diffie-Hellman key exchange
#* ElGamal

</div>

=== Additional topics ===
If time allows, additional topics may include, but are not limited to: Elliptic curve cryptography, birthday attacks and probability, quantum cryptography (key distribution, Shor's algorithm), hash functions, digital signatures, and lattices and lattice algorithms (LLL algorithm, NTRU system, lattice attacks on RSA).

=== Courses for which this course is prerequisite ===
None.

[[Category:Courses|485]]

Math 302: Math for Engineering 1

2010-02-17T20:47:40Z

Pmj5: Added learning outcomes from course materials

== Catalog Information ==

=== Title ===
Mathematics for Engineering 1.

=== (Credit Hours:Lecture Hours:Lab Hours) ===
(4:4:0)

=== Offered ===
F, W

=== Prerequisite ===
[[Math 113]] and passing grade on required preparatory exam taken during first week of class. (Practice exams available on class website.)

=== Description ===
Multivariable calculus, linear algebra, and numerical methods.

== Desired Learning Outcomes ==
This course is designed to give students from the College of Engineering and Technology the mathematics background necessary to succeed in their chosen field.

=== Prerequisites ===

Students are expected to have completed [[Math 113]].

=== Minimal learning outcomes ===

<div style="-moz-column-count:2; column-count:2;">

# Rectangular Space Coordinates; Vectors in Three-Dimensional Space
#* Define the following:
#** Cartesian coordinates of a point
#** sphere
#** symmetry about a point, a line, and a plane
#** vector
#** components of a vector
#** vector addition
#** scalar multiplication
#** zero vector
#** vector subtraction
#** vector norm (magnitude, length)
#** unit vector
#** coordinate unit vectors i, j, k
#** linear combination of unit vectors
#* Plot points in three-dimensional space.
#* Calculate the distance between two points in two-dimensional space and 3-dimensional space
#* Write the equation of a sphere centered about a given point with a given radius. Determine the center and radius of a sphere, given its equation.
#* Write the component equations of a line that passes through two given points.
#* Write the component equations of a line segment with given endpoints.
#* Find the midpoint of a given line segment.
#* Find the points of symmetry about a point, line, or plane.
#* Represent a vector by each of the following:
#** components
#** a linear combination of coordinate unit vectors
#* Carry out the vector operations:
#** addition
#** scalar multiplication
#** subtraction
#* Represent the operations of vector addition, scalar multiplication and norm geometrically.
#* Find the norm (magnitude, length) of a vector. Determine whether two vectors are parallel.
#* Recall, apply and verify the basic properties of vector addition, scalar multiplication and norm.
#* Model and solve application problems using vectors.
# The Dot Product
#* Define the following:
#** dot product.
#** perpendicular vectors.
#** unit vector in the direction of a vector a, denoted u_a.
#** the projection of a on b, denoted proj_b a.
#** the b-component of a, denoted comp_b a.
#** the direction cosines of a vector.
#** the direction angles of a vector.
#** the Schwarz Inequality.
#** the work done by a constant force on an object.
#** the dot product test for perpendicular vectors.
#** the dot product test for parallel vectors.
#** geometric interpretation of the dot product
#* Evaluate a dot product from the coordinate formula or the angle formula.
#* Interpret the dot product geometrically.
#* Evaluate the following using the dot product:
#** the length of a vector.
#** the angle between two vectors.
#** u_a, the unit vector in the direction of a vector a.
#** proj_b a, the projection of a on b.
#** comp_b a, the b-component of a.
#** the direction cosines of a vector.
#** the direction angles of a vector.
#** the work done by a constant force on an object.
#* Prove and verify the Schwarz Inequality.
#* Prove and apply the dot product tests for perpendicular and parallel vectors.
#* Recall and apply the properties of the dot product.
#* Prove identities involving the dot product.
#* Solve application problems involving the dot product.
#* Extend the vector operations and related identities for addition, scalar multiplication, and dot product to higher dimensions.
# The Cross Product
#* Define the following:
#** the cross product of two vectors
#** scalar triple product
#* Evaluate a cross product from the the coordinate formula or angle formula.
#* Interpret the cross product geometrically.
#* Evaluate the following using the cross product:
#** a vector perpendicular to two given vectors.
#** the area of a parallelogram.
#** the area or a triangle.
#** moment of force or moment of torque.
#* Evaluate scalar triple products.
#* Use the scalar triple product to determine the following:
#** volume of a parallelepiped.
#** whether or not three vectors are coplanar.
#* Recall and apply the properties of the cross product and scalar triple product.
#* Prove identities involving the cross product and the scalar triple product.
#* Solve application problems involving the cross product and scalar triple product.
# Lines
#*Define the following:
#** direction vector for a line
#** vector equation of a line
#** scalar parametric equations of a line
#** Cartesian equations or symmetric form of a line
#* Represent a line in 3-space by:
#** a vector equation
#** scalar parametric equations
#** Cartesian equations
#* Find the equation(s) representing a line given information about
#** a point of the line and the direction of the line or
#** two points contained in the line.
#** a point and a parallel line.
#** a point and perpendicular to a plane.
#** two planes intersecting in the line.
#* Find the distance from a point to a line.
#* Solve application problems involving lines.
# Planes
#* Define the following:
#** normal vector to a plane
#** cartesian equation of a plane
#** parametric equation of a plane
#* Find the equation of a plane in 3-space given a point and a normal vector, three points, or a geometric description of the plane.
#* Determine a normal vector and the intercepts of a given plane.
#* Represent a plane by parametric equations.
#* Find the distance from a point to a plane.
#* Find the angle between a line and a plane.
#* Determine a point of intersection between a line and a surface.
#* Sketch planes given their equations.
#* Solve application problems involving planes.
# Systems of Linear Equations
#* Define the following:
#** linear system of m equations in n unknowns
#** consistent and inconsistent
#** solution set
#** coefficient matrix
#** elementary row operations
#* Identify linear systems.
#* Represent a system of linear equations as an augmented matrix and vice versa.
#* Relate the following types of solution sets of a system of two or three variables to the intersections of lines in a plane or the intersection of planes in three space:
#** a unique solution.
#** infinitely many solutions.
#** no solution.
# Gaussian elimination
#* Define the following:
#** reduced row echelon form
#** leading variables or pivots
#** free variables
#** row echelon form
#** back substitution
#** Gaussian elimination
#** Gauss-Jordan elimination
#** homogeneous
#** trivial solution
#** nontrivial solutions
#* Identify matrices that are in row echelon form and reduced row echelon form.
#* Determine whether a linear system is consistent or inconsistent from its reduced row echelon form. If the system is consistent, write the solution.
#* Identify the lead variables and free variables of a system represented by an augmented matrix in reduced row echelon form.
#* Solve systems of linear equations using Gaussian elimination and back substitution.
#* Solve systems of linear equations using Gauss-Jordan elimination.
#* Model and solve application problems using linear systems.
# Matrices and Matrix Operations
#* Define the following:
#** vector, row vector, and column vector
#** equal matrices
#** scalar multiplication
#** sum of matrices
#** zero matrix
#** scalar product
#** linear combination
#** matrix multiplication
#** transpose
#** trace
#** identity matrix
#* Perform the operations of matrix addition, scalar multiplication, transposition, trace, and matrix multiplication.
#* Represent matrices in terms of double subscript notation.
# Inverses; Rules of Matrix Arithmetic
#* Define the following:
#** commutative property
#** singular
#** nonsingular or invertible
#** multiplicative inverse
#* Recall, demonstrate, and apply algebraic properties for matrices.
#* Recall that matrix multiplication is not commutative in general. Determine conditions under which matrices do commute.
#* Recall and prove properties and identities involving the transpose operator.
#* Recall and prove properties and identities involving matrix inverses.
#* Recall and prove properties and identities involving matrix powers.
#* Recall, demonstrate, and apply that the cancelation laws for scalar multiplication do not hold for matrix multiplication.
#* Recall and apply the formula for the inverse of 2x2 matrices.
# Elementary Matrices
#* Define the following:
#** elementary matrix
#** row equivalent matrices
#* Identify elementary matrices and find their inverses or show that their inverse does not exist.
#* Relate elementary matrices to row operations.
#* Factor matrices using elementary matrices.
#* Find the inverse of a matrix, if possible, using elementary matrices.
#* Prove theorems about matrix products and matrix inverses.
#* Solve a linear equation using matrix inverses.
# Further Results on Systems of Equations and Invertibility
#* Solve matrix equations using matrix algebra.
#* Recall and prove properties and identities involving matrix inverses.
#* Recall equivalent conditions for invertibility.
# Further Results on Systems of Equations and Invertibility
#* Define the following:
#** diagonal matrix
#** upper and lower triangular matrices
#** symmetric matrix
#** skew-symmetric matrix
#* Determine powers of diagonal matrices.
#* Recall and prove properties and identities involving the transpose operator.
#* Prove basic facts involving symmetric and skew-symmetric matrices.
# Determinants
#* Define the following:
#** minor
#** cofactor
#** cofactor expansion
#** determinant
#** adjoint
#** Cramer's Rule
#* Apply cofactor expansion to evaluate determinants of nxn matrices.
#* Recall and apply the properties of determinants to evaluate determinants.
#* Evaluate the adjoint of a matrix.
#* Determine whether or not a matrix has an inverse based on its determinant.
#* Evaluate the inverse of a matrix using the adjoint method.
#* Use Cramer's rule to solve a linear system.
# Properties of Determinants
#* Recall the effects that row operations have on the determinants of matrices. Relate to the determinants of elementary matrices.
#* Recall, apply and verify the properties of determinants to evaluate determinants, including:
#** det(AB) = det(A) det(B)
#** det(kA) = k^n det(A)
#** det(A^-1)= 1/det(A)
#** det(A^T) = det(A)
#** det(A) = 0 if and only if A is singular
#* Evaluate the determinant of a matrix using row operations.
#* Apply determinants to determine invertibility of matrix products.
# Linear Transformations: Definitions and Examples
#* Define the following:
#** linear transformation
#** image
#** range
#* Describe geometrically the effects of a linear operator.
#* Determine whether or not a given transformation is linear.
#* Prove theorems and solve application problems involving linear transformations.
# Matrix Representations of Linear Transformations
#* Define the following:
#** standard matrix representation
#** eigenvalues and eigenvectors
#* Determine the matrix that represents a given linear transformation of vectors given an algebraic description.
#* Determine the matrix that represents a given linear transformation of vectors given a geometric description.
#* Prove theorems and solve application problems involving linear transformations.
# Vector Spaces: Definitions and Examples
#* Define the following:
#** vector space
#** vector space axioms
#** vector space R^n
#** vector space R^(mxn)
#** vector space of real-valued functions
#** additional properties of vector spaces
#* Prove or disprove that a given set of vectors together with an addition and a scalar multiplication is a vector space.
#* Prove and verify properties of a vector space.
# Subspaces
#* Define the following:
#** subspace
#** closure under addition
#** closure under scalar multiplication
#** zero subspace
#** linear combination
#** span (or subspace spanned by a set of vectors)
#** spanning set
#* Prove or disprove that a set of vectors forms a subspace.
#* Prove or disprove a set of vectors is a spanning set for R^n.
#* Prove or disprove a given vector is in the span of a set of vectors. Determine the span of a set of vectors.
#* Prove theorems about vector spaces and spans.
# Linear Independence
#* Define the following:
#** linearly independent
#** linearly dependent
#** Wronskian
#* Determine whether a set of vectors is linearly dependent or linearly independent.
#* Geometrically describe the span of a set of vectors. For sets that are linearly dependent, determine a dependence relation.
#* Prove theorems about linear independence.
# Basis and Dimension
#* Define the following:
#** basis
#** dimension
#** finite and infinite dimensional
#** standard basis
#* Prove or disprove a set of vectors forms a basis.
#* Find a basis for a vector space.
#* Determine the dimension of a vector space.
#* Geometrically interpret the ideas of span, linear dependance, basis, and dimension.
# Row Space, Column Space, and Null Space
#* Define the following:
#** row space
#** column space
#** null space
#** particular solution
#** general solution
#* Express a product Ax as a linear combination of column vectors.
#* Find a basis for a the column space, the row space, and the null space of a matrix.
#* Find the basis for a span of vectors.
# Rank and Nullity
#* Define the following:
#** rank
#** nullity
#** The Consistency Theorem
#** equivalent statements of invertibility
#* Find the rank and nullity of a matrix.
#* Recall and prove identities involving rank and nullity
#* Recall and apply the Consistency Theorm
#* Recall and apply the equivalent statements of invertibility.
# Eigenvalues and Eigenvectors
#* Define the following:
#** eigenvalue or characteristic value
#** eigenvector or characteristic vector
#** characteristic polynomial or characteristic polynomial
#** equivalent statements of invertibility
#* Find the eigenvalues and eigenvectors of an nxn matrix.
#* Prove theorems and solve application problems involving eigenvalues and eigenvectors.
# Diagonalization
#* Define the following:
#** diagonalizable
#** algebraic multiplicity
#** geometric multiplicity
#* Determine whether or not a matrix is diagonalizable.
#* Find the diagonalization of a matrix, if possible.
#* Find powers of a matrix using the diagonalization of a matrix.
#* Prove theorems and solve application problems involving the diagonalization of matrices.
# Limit, Continuity, Vector Derivative; The Rules of Differentiation
#* Define the following:
#** scalar functions
#** vector functions
#** components of a vector function
#** plane curve or space curve
#** parametrization of a curve
#** limit of a vector function
#** a vector function continuous at a point
#** derivative of a vector function
#** a differentiable vector function
#** integral of a vector function
#* Graph a parametric curve.
#* Identify a curve given its parametrization.
#* Determine combinations of vector functions such as sums, vector products and scalar products.
#* Evaluate limits, derivatives, and integrals of vector functions.
#* Recall, derive and apply rules to combinations of vector functions for the following:
#** limits
#** differentiation
#** integration
#* Determine continuity of a vector-valued function.
#* Prove theorems involving limits and derivatives of vector-valued functions.
#* Solve application problems involving vector-valued functions.
# Curves; Vector Calculus in Mechanics
#* Define the following:
#** directed path
#** differentiable parameterized curve
#** tangent vector
#** tangent line
#** unit tangent vector
#** principal normal vector
#** normal line
#** osculation plane
#** force vector
#** momentum vector
#** angular momentum vector
#** torque
#* Find the tangent vector and tangent line to a curve at a given point.
#* Find the principle normal and normal line to a curve at a given point.
#* Determine the osculating plane for a space curve at a given point.
#* Reverse the direction of a curve.
#* Solve application problems involving curves.
#* Solve application problems involving force, momentum, angular momentum, and torque.
# Arc Length
#* Define the following:
#** arc length
#** arc length parametrization
#* Evaluate the arc length of a curve.
#* Determine whether a curve is arc length parameterized.
#* Find the arc length parametrization of a curve.
# Curvilinear Motion; Curvature
#* Define the following:
#** velocity vector function
#** speed
#** acceleration vector function
#** uniform circular motion
#** curvature
#** tangential component of acceleration
#** normal component of acceleration
#* Given the position vector function of a moving object, calculate the velocity vector function, speed, and acceleration vector function, and vice versa.
#* Calculate the curvature of a space curve.
#* Recall the formulas for the curvature of a parameterized planar curve or a planar curve that is the graph of a function. Apply these formulas to calculate the curvature of a planar curve.
#* Determine the tangential and normal components of acceleration for a given parameterized curve.
#* Solve application problems involving curvilinear motion and curvature.
# Functions of Several Variables; A Brief Catalogue of the Quadric Surfaces; Projections
#* Define the following:
#** real-valued function of several variables
#** domain
#** range
#** bounded functions
#** quadric surface
#** intercepts
#** traces
#** sections
#** center
#** symmetry
#** boundedness
#** cylinder
#** ellipsiod
#** elliptic cone
#** elliptic paraboloid
#** hyperboloid of one sheet
#** hyperboloid of two sheets
#** hyperbolic paraboloid
#** parabolic cylinder
#** elliptic cylinder
#** projection of a curve onto a coordinate plane
#* Describe the domain and range of a function of several variables.
#* Write a function of several variables given a description.
#* Identify standard quadratic surfaces given their functions or graphs.
#* Sketch the graph of a quadratic surface by sketching intercepts, traces, sections, centers, symmetry, boundedness.
#* Find the projection of a curve, that is the intersection of two surfaces, to a coordinate plane.
# Graphs; Level Curves and Level Surfaces
#* Define the following:
#** level curve
#** level surface
#* Describe the level sets of a function of several variables.
#* Graphically represent a function of two variables by level curves or a function of three variables by level surfaces.
#* Identify the characteristics of a function from its graph or from a graph of its level curves (or level surfaces).
#* Solve application problems involving level sets. functions.
# Partial Derivatives
#* Define the following:
#** partial derivative of a function of several variables
#** second partial derivative
#** mixed partial derivative
#* Interpret the definition of a partial derivative of a function of two variables graphically.
#* Evaluate the partial derivatives of a function of several variables.
#* Evaluate the higher order partial derivatives of a function of several variables.
#* Verify equations involving partial derivatives.
#* Apply partial derivatives to solve application problems.
# Open and Closed Sets; Limits and Continuity; Equity of Mixed Partials
#* Define the following:
#** neighborhood of a point
#** deleted neighborhood of a point
#** interior of a set
#** boundary of a set
#** open set
#** closed set
#** limit of a function of several variables at a point
#** continuity of a function of several variables at a point
#* Determine the boundary and interior of a set.
#* Determine whether a set is open, closed, neither, or both.
#* Evaluate the limit of a function of several variables or show that it does not exists.
#* Determine whether or not a function is continuous at a given point.
#* Recall and apply the conditions under which mixed partial derivatives are equal.
# Differentiability and Gradient
#* Define the following:
#** differentiable multivariable function
#** gradient of a multivariable function
#* Evaluate the gradient of a function.
#* Find a function with a given gradient.
# Gradient and Directional Derivative
#* Define the following:
#** directional derivative
#** isothermals
#* Recall and prove identities involving gradients.
#* Give a graphical interpretation of the gradient.
#* Evaluate the directional derivative of a function.
#* Give a graphical interpretation of directional derivative.
#* Recall, prove, and apply the theorem that states that a differential function f increases most rapidly in the direction of the gradient (the rate of change is then ||f(x)||) and it decreases most rapidly in the opposite direction (the rate of change is then -||f(x)||).
#* Find the path of a heat seeking or a heat repelling particle.
#* Solve application problems involving gradient and directional derivatives.
# The Mean-Value Theorem; The Chain Rule
#* Define the following:
#** the Mean Value Theorem for functions of several variables
#** normal line
#** chain rules for functions of several variables
#** implicit differentiation
#* Recall and apply the Mean Value Theorem for functions of several variables and its corollaries.
#* Apply an appropriate chain rule to evaluate a rate of change.
#* Apply implicit differentiation to evaluate rates of change.
#* Solve application problems involving chain rules and implicit differentiation.
# The Gradient as a Normal; Tangent Lines and Tangent Planes
#* Define the following:
#** normal vector
#** tangent vector
#** tangent line
#** tangent plane
#** normal line
#* Use gradients to find the normal vector and normal line to a smooth planar curve at a given point.
#* Use gradients to find the tangent vector and tangent line to a smooth planar curve at a given point.
#* Use gradients to find the normal vector to a smooth surface at a given point.
#* Use gradients to find the tangent plane to a smooth surface at a given point.
#* Use gradients to find the normal line to a smooth surface at a given point.
#* Solve application problems involving normals and tangents to curves and surfaces.
# Local Extreme Values
#* Define the following:
#** local minimum and local maximum
#** critical points
#** stationary points
#** saddle points
#** discriminant
#** Second Derivative Test
#* Find the critical points of a function of two variables.
#* Apply the Second-Partials Test to determine whether each critical point is a local minimum, a local maximum, or a saddle point.
#* Solve word problems involving local extreme values.
# Absolute Extreme Values
#* Define the following:
#** absolute minimum and absolute maximum
#** bounded subset of a plane or three-space
#** the Extreme Value Theorem
#* Determine absolute extreme values of a function defined on a closed and bounded set.
#* Apply the Extreme Value Theorem to justify the method for finding extreme values of functions defined on certain sets.
#* Solve word problems involving absolute extreme values.
# Maxima and Minima with Side Conditions
#* Define the following:
#** side conditions or constraints
#** method of Lagrange
#** Lagrange multipliers
#** cross-product equation of the Lagrange condition
#* Graphically interpret the method of Lagrange.
#* Determine the extreme values of a function subject to a side conditions by applying the method of Lagrange.
#* Apply the cross-product equation of the Lagrange condition to solve extreme value problems subject to side conditions.
#* Apply the method of Lagrange to solve word problems.
# Differentials; Reconstructing a Function from its Gradient
#* Define the following:
#** differential
#** general solution
#** particular solution
#** connected open set
#** open region
#** simple closed curve
#** simply connected open region
#** partial derivative gradient test
#* Determine the differential for a given function of several variables.
#* Determine whether or not a vector function is a gradient.
#* Given a vector function that is a gradient, find the functions with that gradient.
# Multiple-Sigma Notation; The Double Integral over a Rectangle R; The Evaluation of Double Integrals by Repeated Integrals
#* Define the following:
#** double sigma notation
#** triple sigma notation
#** upper sum
#** lower sum
#** double integral
#** integral formula for the volume of a solid bounded between a region Omega in the xy-plane and the graph of a non-negative function z = f(x,y) defined on Omega.
#** integral formula for the area of region in a plane
#** integral formula for the average of a function defined on a region Omega.
#** projection of a region onto a coordinate axis
#** Type I and Type II regions
#** reduction formulas for double integrals
#** the geometric interpretation of the reduction formulas for double integrals
#* Evaluate double and triple sums given their sigma notation.
#* Recall and apply summation identities.
#* Approximate the integral of a function by a lower sum and an upper sum.
#* Evaluate the integral of a function using the definition.
#* Evaluate double integrals over a rectangle using the reduction formulas.
#* Sketch planar regions and determine if they are Type I, Type II, or both.
#* Evaluate double integrals over Type I and Type II regions.
#* Change the order of integration of an integral.
#* Apply double integrals to calculate volumes, areas, and averages.
# The Double Integral as the Limit of Riemann Sums; Polar Coordinates
#* Define the following:
#** diameter of a set
#** Riemann sum
#** double integral as a limit of Riemann sums
#** polar coordinates (r; theta)
#** transformation formulas between Cartesian and polar coordinates
#** double integral conversion formula between Cartesian and polar coordinates
#* Represent a region in both Cartesian and polar coordinates.
#* Evaluate double integrals in terms of polar coordinates.
#* Evaluate areas and volumes using polar coordinates.
#* Convert a double integral in Cartesian coordinates to a double integral in polar coordinates and then evaluate.
# Further Applications of the Double Integral
#* Define the following:
#** integral formula for the mass of a plate
#** integral formulas for the center of mass of a plate
#** integral formulas for the centroid of a plate
#** integral formulas for the moment of an inertia of a plate
#** radius of gyration
#** the Parallel Axis Theorem
#* Evaluate the mass and center or mass of a plate
#* Evaluate the centroid of a plate.
#* Evaluate the moments of inertia of a plate.
#* Calculate the radius of gyration of a plate.
#* Recall and apply the parallel axis theorem.
# Triple Integrals; Reduction to Repeated Integrals
#* Define the following:
#** triple integral
#** integral formula for the volume of a solid
#** integral formula for the mass of a solid
#** integral formulas for the center of mass of a solid
#* Evaluate physical quantities using triple integrals such as volume, mass, center of mass, and moments of intertia.
#* Recall and apply the properties of triple integrals, including: linearity, order, additivity, and the mean-value condition.
#* Sketch the domain of integration of an iterated integral.
#* Change the order of integration of a triple integral.
# Cylindrical Coordinates
#* Define the following:
#** cylindrical coordinates of a point
#** coordinate transformations between Cartesian and cylindrical coordinates
#** cylindrical element of volume
#* Convert between Cartesian and cylindrical coordinates.
#* Describe regions in cylindrical coordinates.
#* Evaluate triple integrals using cylindrical coordinates.
# Spherical Coordinates
#* Define the following:
#** spherical coordinates of a point
#** coordinate transformations between Cartesian and spherical coordinates
#** spherical element of volume
#* Convert between Cartesian and spherical coordinates.
#* Describe regions in spherical coordinates.
#* Evaluate triple integrals using spherical coordinates.
# Jacobians; Changing Variables in Multiple Integration
#* Define the following:
#** Jacobian
#** change of variable formula for double integration
#** change of variable formula for triple integration
#* Find the Jacobian of a coordinate transformation.
#* Use a coordinate transformation to evaluate double and triple integrals.
# Line Integrals
#* Define the following:
#** work along a curved path
#** smooth parametric curve
#** directed or oriented curve
#** path dependence
#** closed curve
#* Evaluate the work done by a varying force over a curved path.
#* Evaluate line integrals in general including line integrals with respect to arc length.
#* Evaluate the physical characteristics of a wire such as centroid, mass, and center of mass using line integrals.
#* Determine whether or not a vector field is a gradient.
#* Determine whether or not a differential form is exact.
# The Fundamental Theorem for Line Integrals; Work-Energy Formula; Conservation of Mechanical Energy
#* Define the following:
#** path-independent line integrals
#** closed vector field
#** simply connected
#* Recall, apply, and verify the Fundamental Theorem for Line Integrals (Theorem 2 in Section 15.3).
#* Determine whether or not a force field is closed on a given region, and if so, find its potential function.
#* Solve application problems involving work done by a conservative vector field
# Vector Fields
#* Define the following:
#** vector field
#** open
#** path connected
#** region
#** integral curve (field lines, flow lines, or streamlines)
#** gradient vector field (or conservative vector field)
#** potential function
#** continuously differentiable vector field
#* Sketch a vector field.
#* Write the formula for a vector field from a description.
#* Write the gradient vector field associated with a given scalar-valued function.
#* Recover a function from its gradient or show it is not possible.
#* Find the integral curves of a vector field.
# Green's Theorem
#* Define the following:
#** Jordan curve
#** Jordan region
#** Green's Theorem
#* Recall and verify Green's Theorem.
#* Apply Green's Theorem to evaluate line integrals.
#* Apply Green's Theorem to find the area of a region.
#* Derive identities involving Green's Theorem
# Parameterized Surfaces; Surface Area
#* Define the following:
#** parameterized surface
#** fundamental vector product
#** element of surface area for a parameterized surface
#** surface integral
#** integral formula for the surface area of a parameterized surface
#** integral formula for the surface area of a surface z = f(x; y)
#** upward unit normal
#* parameterize a surface.
#* evaluate the fundamental vector product for a parameterized surface.
#* Calculate the surface area of a parameterized surface.
#* Calculate the surface area of a surface z = f(x; y).
# Surface Integrals
#* Define the following:
#** surface integral
#** integral formulas for the surface area and centroid of a parameterized surface
#** integral formulas for the mass and center of mass of a parameterized surface
#** integral formulas for the moments of inertia of a parameterized surface
#** integral formula for flux through a surface
#* Calculate the surface area and centroid of a parameterized surface.
#* Calculate the mass and center of mass of a parameterized surface.
#* Calculate the moments of inertia of a parameterized surface.
#* Evaluate the flux of a vector field through a surface.
#* Solve application problems involving surface integrals.
# The Vector Differential Operator Del
#* Define the following:
#** the vector differential operator Del
#** divergence
#** curl
#** Laplacian
#* Evaluate the divergence of a vector field.
#* Evaluate the curl of a vector field
#* Evaluate the Laplacian of a function.
#* Recall, derive and apply formulas involving divergence, gradient and Laplacian.
#* Interpret that divergence and curl of a vector fields physically.
# The Divergence Theorem
#* Define the following:
#** outward unit normal
#** the divergence theorem
#** sink and source
#** solenoidal
#* Recall and verify the Divergence Theorem.
#* Apply the Divergence Theorem to evaluate the flux through a surface.
#* Solve application problems using the Divergence Theorem.
# Stokes' Theorem
#* Define the following:
#** oriented surface
#** outward, upward, and downward unit normal
#** the positive sense around the boundary of a surface
#** circulation
#** component of curl in the normal direction
#** irrotational
#** Stokes' theorem
#* Recall and verify Stoke's theorem.
#* Use Stokes' Theorem to calculate the flux of a curl vector field through a surface by a line integral.
#* Apply Stokes' theorem to calculate the work (or circulation) of a vector field around a simple closed curve.

</div>

=== Additional topics ===

=== Courses for which this course is prerequisite ===
[[Math 303]]

[[Category:Courses|302]]

Math 303: Math for Engineering 2

2010-02-16T21:45:33Z

Pmj5: Added detailed learning outcomes from course materials

== Catalog Information ==

=== Title ===
Mathematics for Engineering 2.

=== (Credit Hours:Lecture Hours:Lab Hours) ===
(4:4:0)

=== Offered ===
F, W

=== Prerequisite ===
[[Math 302]] or [[Math 314]].

=== Description ===
ODEs, Laplace transforms, Fourier series, PDEs.

== Desired Learning Outcomes ==
This course is designed to give students from the College of Engineering and Technology the mathematics background necessary to succeed in their chosen field.

=== Prerequisites ===
Students are expected to have completed [[Math 302]] or [[Math 314]].

=== Minimal learning outcomes ===
Students should achieve mastery of the topics below.
<div style="-moz-column-count:2; column-count:2;">
# Some Basic Mathematical Models; Direction Fields
#* Model physical processes using differential equations.
#* Sketch the direction field (or slope field) of a differential equation using a computer graphing program.
#* Describe the behavior of the solutions of a differential equation by analyzing its slope field. Identify any equilibrium solutions.
# Solutions of Some Differential Equations; Classification of Differential Equations
#* Solve basic initial value problems; obtain explicit solutions if possible.
#* Characterize the solutions of a differential equation with respect to initial values.
#* Use the solution of an initial value problem to answer questions about a physical system.
#* Determine the order of an ordinary differential equation. Classify an ordinary differential equation as linear or nonlinear.
#* Verify solutions to ordinary differential equations.
#* Determine the order of a partial differential equation. Classify a partial differential equation as linear or nonlinear.
#* Verify solutions to partial differential equations.
# Linear First Order Equations with Variable Coefficients
#* Identify and solve first order linear equations.
#* Analyze the behavior of solutions.
#* Solve initial value problems for first order linear equations.
# Separable First Order Equations
#* Identify and solve separable equations; obtain explicit solutions if possible.
#* Solve initial value problems for separable equations, and analyze their solutions.
#* Apply the transformation $y=xv(x)$ to obtain a separable equation, if possible.
# Modeling with First Order Equations
#* Construct models of tank problems using differential equations. Analyze the models to answer questions about the physical system modeled.
#* Construct growth and decay problems using differential equations. Analyze the models to answer questions about the physical system modeled.
#* Construct models of problems involving force and motion using differential equations. Analyze the models to answer questions about the physical system modeled.
#Differences Between Linear and Nonlinear Equations
#* Recall and apply the existence and uniqueness theorem for first order linear differential equations.
#* Recall and apply the existence and uniqueness theorem for first order differential equations (both linear and nonlinear).
#* Summarize the nice properties of linear equations. Contrast with nonlinear equations.
# Autonomous Equations and Population Dynamics
#* Determine and classify the equilibrium solutions of an autonomous equation as asymptotically stable, semistable or unstable by analyzing a graph of $\dfrac{dy}{dt}$ versus $y$. Sketch the phase line.
#* Analyze solutions of the logistic equation and other autonomous equations.
# Exact Equations and Integrating Factors
#* Identify whether or not a differential equation is exact.
#* Solve exact differential equations with or without initial conditions, and obtain explicit solutions if possible.
#* Use integrating factors to convert a differential equation to an exact equation and then solve.
#* Determine an integrating factor of the form $\mu(x)$ or $\mu(y)$ which will convert a non-exact differential equation to an exact equation, if possible.
# Introduction to Second Order Equations
#* Determine the characteristic equation of a second order linear differential equation with constant coefficients.
#* Solve second order linear differential equations with constant coefficients that have a characteristic equation with real and distinct roots.
#* Describe the behavior of solutions.
#* Convert a second order differential equation to a first order differential equation in the following cases: i) y"=f(t,y'), ii) y"=f(y,y').
# Fundamental Solutions of Linear Homogeneous Equations; the Wronskian
#* Recall and apply the existence and uniqueness theorem for second order linear differential equations.
#* Recall and verify the principal of superposition for solutions of second order linear differential equations.
#* Evaluate the Wronskian of two functions.
#* Determine whether or not a pair of solutions of a second order linear differential equations constitute a fundamental set of solutions.
#* Recall and apply Abel's theorem.
# Complex Roots of the Characteristic Equation
#* Use Euler's formula to rewrite complex expressions in different forms.
#* Solve second order linear differential equations with constant coefficients that have a characteristic equation with complex roots.
#* Solve initial value problems and analyze the solutions.
# Repeated Roots; Reduction of Order
#* Solve second order linear differential equations with constant coefficients that have a characteristic equation with repeated roots.
#* Solve initial value problems and analyze the solutions.
#* Apply the method of reduction of order to find a second solution to a given differential equation.
# Nonhomogeneous Equations; Method of Undetermined Coefficients
#* For a nonhomogeneous second order linear differential equation, determine a suitable form of a particular solution that can be used in the method of undetermined coefficients.
#* Apply the method of undetermined coefficients to solve nonhomogeneous second order linear differential equations.
#* Solve initial value problems and analyze the solutions.
# Variation of Parameters; Reduction of Order
#* Apply the method of variation of parameters to solve nonhomogeneous second order linear differential equations with or without initial conditions.
#* Apply the method of reduction of order to solve nonhomogeneous second order linear differential equations with or without initial conditions.
# Mechanical Vibrations
#* Model undamped mechanical vibrations with second order linear differential equations, and then solve. Analyze the solution. In particular, evaluate the frequency, period, amplitude, phase shift, and the position at a given time.
#* Model damped mechanical vibrations with second order linear differential equations, and then solve. Analyze the solution. In particular, evaluate the quasi frequency, quasi period, and the behavior at infinity.
#* Define critically damped and overdamped. Identify when these conditions exist in a system.
# Forced Vibrations
#* Model forced, undamped mechanical vibrations with second order linear differential equations, and then solve. Analyze the solution.
#* Describe the phenomena of beats and resonance. Determine the frequency at which resonance occurs.
#* Model forced, damped mechanical vibrations with second order linear differential equations, and then solve. Determine and analyze the solutions, including the steady state and transient parts.
# General Theory of nth Order Linear Equations
#* Recall and apply the existence and uniqueness theorem for higher order linear differential equations.
#* Recall the definition of linear independence for a finite set of functions. Determine whether a set of functions is linearly independent or linearly dependent.
#* Use the Wronskian to determine if a set of solutions form a fundamental set of solutions.
#* Recall the relationship between Wronskian and linear independence for a set of functions, and for a set of solutions.
#* Apply the method of reduction of order to solve higher order linear differential equations.
# Homogeneous Equations with Constant Coefficients
#* Apply Euler's formula to write a complex number in exponential form. Find the distinct complex roots of a number.
#* Determine the characteristic equation of higher order linear differential equations.
#* Solve higher order linear differential equations.
#* Solve initial value problems.
# The Method of Undetermined Coefficients
#* For a nonhomogeneous higher order linear differential equation, determine a suitable form of a generalized particular solution that can be applied in the method of undetermined coefficients.
#* Use the method of undetermined coefficients to solve nonhomogeneous higher order linear differential equations.
#* Solve initial value problems.
# The Method of Variation of Parameters
#* Use the method of variation of parameters to solve nonhomogeneous higher order linear differential equations.
#* Solve initial value problems.
# Review of Power Series
#* Determine the radius of convergence of a power series.
#* Find the power series expansion of a function.
#* Manipulate expressions involving summation notation. Change the index of summation.
# Series Solutions near an Ordinary Point, Part I
#* Find the general solution of a differential equation using power series.
#* Solve initial value problems. Analyze the solution.
# Series Solutions near an Ordinary Point, Part II
#* Given an initial value problem, use the differential equation to inductively determine the terms in the power series of the solution, expanded about the initial value.
#* Determine a lower bound for the radius of convergence of a series solution.
# Euler Equations
#* Find the general solution to an Euler equation in the cases of real distinct roots, equal roots, and complex roots.
#* Solve initial value problems for Euler equations and analyze their solutions.
# Definition of Laplace Transform
#* Sketch a piecewise defined function. Determine if it is continuous, piecewise continuous or neither.
#* Evaluate Laplace transforms from the definition.
#* Determine whether an infinite integral converges or diverges.
# Solution of Initial Value Problems
#* Evaluate inverse Laplace transforms.
#* Use Laplace transforms to solve initial value problems.
#* Evaluate Laplace transforms using the derivative identity given in Problem 28 (p. 322) of the textbook.
# Step Functions
#* Sketch the graph of a function that is defined in terms of step functions.
#* Convert piecewise defined functions to functions defined in terms of step functions and vice versa.
#* Find the Laplace transform of a piecewise defined function.
#* Apply the shifting theorems (Theorems 6.3.1 and 6.3.2) to evaluate Laplace transforms and inverse Laplace transforms.
# Differential Equations with Discontinuous Forcing Functions
#* Use Laplace transforms to solve differential equations with discontinuous forcing functions.
#* Analyze the solutions of differential equations with discontinuous forcing functions.
# Impulse Functions
#* Define an idealized unit impulse function.
#* Use Laplace transforms to solve differential equations that involve impulse functions.
#* Analyze the solutions of differential equations that involve impulse functions.
# The Convolution Integral
#* Evaluate the convolution of two functions from the definition.
#* Prove and disprove properties of the convolution operator.
#* Evaluate the Laplace transform of a convolution of functions.
#* Use the convolution theorem to evaluate inverse Laplace transforms.
#* Solve initial value problems using convolution.
# Introduction to Systems of First Order Equations
#* Transform a higher order linear differential equation into a system of first order linear equations.
#* Transform a system of first order linear equations to a single higher order linear equation.
#* Model physical systems that involve more than one unknown function with a system of differential equations.
#* Recall and apply methods of linear algebra.
# Basic Theory of Systems of First Order Linear Equations
#* Recall and verify the superposition principle for first order linear systems.
#* Relate the Wronskian to linear independence and a fundamental set of solutions.
# Homogeneous Linear Systems with Constant Coefficients
#* Sketch a direction field and a phase portrait for a homogeneous linear system with constant coefficients.
#* Find the general solution of a homogeneous linear system with constant coefficients in the case of real, distinct eigenvalues.
#* Determine if the origin is a saddle point or a node for a homogeneous linear system. Classify a node as asymptotically stable or unstable.
#* Find general solutions, solve initial value problems, and analyze their solutions.
# Complex Eigenvalues
#* Sketch a direction field and a phase portrait for a homogeneous linear system with constant coefficients.
#* Find the general solution of a homogeneous linear system with constant coefficients in the case of complex eigenvalues.
#* Classify the origin as a saddle point, a node, a spiral point or a center.
#* Solve and analyze physical problems modeled by systems of differential equations.
# Fundamental Matrices
#* Determine a fundamental matrix for a system of equations.
#* Solve initial value problems using a fundamental matrix.
#* Prove identities using fundamental matrices.
# Repeated Eigenvalues
#* Sketch a direction field and a phase portrait for a homogeneous linear system with constant coefficients.
#* Find the general solution of a homogeneous linear system with constant coefficients in the case of repeated eigenvalues.
#* Identify improper nodes. Classify them as asymptotically stable or unstable.
#* Solve initial value problems.
# Nonhomogeneous Linear Systems
#* Use diagonalization to solve nonhomogeneous linear systems.
#* Use the method of undetermined coefficients to solve nonhomogeneous linear systems.
#* Use the method of variation of parameters to solve nonhomogeneous linear systems.
#* Solve initial value problems.
# Two-Point Boundary Value Problems
#* Solve boundary value problems involving linear differential equations.
#* Find the eigenvalues and the corresponding eigenfunctions of a boundary value problem.
# Fourier Series
#* Identify functions that are periodic. Determine their periods.
#* Find the Fourier series for a function defined on a closed interval.
#* Determine the $m$th partial sum of the Fourier series of a function. Compare to the function.
# The Fourier Convergence Theorem
#* Find the Fourier series for a periodic function.
#* Recall and apply the convergence theorem for Fourier series.
# Even and Odd Functions
#* Determine whether a given function is even, odd or neither.
#* Sketch the even and odd extensions of a function defined on the interval [0,L].
#* Find the Fourier sine and cosine series for the function defined on [0,L].
#* Establish identities involving infinite sums from Fourier series.
# Separation of Variables; Heat Conduction in a Rod
#* Apply the method of separation of variables to solve partial differential equations, if possible.
#* Find the solutions of heat conduction problems in a rod using separation of variables.
# Other Heat Conduction Problems
#* Solve steady state heat conduction problems in a rod with various boundary conditions.
#* Analyze the solutions.
# The Wave Equation; Vibrations of an Elastic String
#* Solve the wave equation that models the vibration of a string with fixed ends.
#* Describe the motion of a vibrating string.
# Laplace's Equation
#* Solve Laplace's equation over a rectangular region for various boundary conditions.
#* Solve Laplace's equation over a circular region for various boundary conditions.

</div>

=== Additional topics ===

=== Courses for which this course is prerequisite ===

[[Category:Courses|303]]

Math 112 Calculus Learning Goals

2009-08-19T21:21:29Z

Pmj5: /* Appendix E (Paul) */

----

== Section 1.1 (Jessica) ==

Homework:

Written: 25, 31, 40, 43, 55, 56, 64, 65
Online: 3, 4, 5, 6, 8, 9, 11. Modify 12 (Given f like 66 or 67 in book, find f(-x). Is f even or odd or neither?) Add problem like 51, 57 in book.

Goals

1. Given a function described algebraically, find the
- domain 31, 07, 08,
- range,
- value at a given number 04,
- value when we plug in another function, e.g. difference quotient 04, 05, 25.
2. Convert one representation of a function to another.
- verbal to/from graphical 03
-algebraic to/from graphical 09
-verbal to/from algebraic 55, 56, 2 new online
3. Piecewise functions: Given an algebraic description of a piecewise function, find its domain and sketch its graph. 011, 43.
Be able to write the absolute value as a piecewise function, and use this to sketch its graph. 40, 09
4. Use the definition of an even/odd function to decide whether a function described algebraically is even or odd. 012, 65
Use symmetry to decide whether a graph is an even or odd function. 64

----

== Section 1.2 (Sam) ==

Homework:

Written: 4,5,7,9,12 and Appdx D: 24, 37*, 46
Online: O1, O2, O4, O5 (these are the numbers under the current numbering),
and Appdx D: 29, 60, 69
The * on problem 37 means a calculator will be necessary.

Goals

1. Writing down a linear function given a set of information 5,12,O4,O5
2. Recognizing what a function's graph should look like 4,7,O1
3. Based on a graph, write down a function O2
4. Writing down a polynomial based on information 9, O2
5. Trig review (definitions, proofs, values of trig functions at particular points) Appdx D:24, 37, 46, 29, 60, 69 (the latter three going online)

# Trig to know, including problems to skim for practice. (Jessica)

1. Given angles in degrees and radians, draw the angle, convert from radians to degrees and vice versa. (See appendix D: 1-12)
2. Write down sin, cos, tan, sec, scs, cot of any angle 0, pi/6, pi/4, pi/3, pi/2, and all angles obtained from these angles by adding a multiple of pi/2. (appendix D:23-28)
3. Given a right triangle with side measurements, write sin, cos, tan, sec, csc, cot of any of the angles of the triangle in terms of the side lengths. (Appendix D: 35-38)
Similarly, given one trig ratio, find the others. (Appendix D: 29-34)
4. Graph trig functions. (app D: 77-82)
5. Use trig identities to simplify expressions (appendix D: 43-57 odd, 59-63), and to solve equations and inequalities involving trig functions (app D: 65-76).
* Most important identities: sin^2(x) + cos^2(x) = 1
* sin(-x) = -sin(x)
* cos(-x) = cos(x)
* sin(x+y) = sin(x)cos(y) + cos(x)sin(y)
* cos(x+y) = cos(x)cos(y) - sin(x)sin(y)
Using the above, you can derive other identities:
* 1+tan^2(x)=sec^2(x); 1+cot^2(x) = csc^2(x)
* sin(x-y)
* cos(x-y)
* sin(2x)
* cos(2x)

----

== Section 1.3 (Savannah) ==

Homework

Written: 1,7,13, 14, 22, 39, 46, 60, 65
Online: Modify 1 to be y as a function of x. Replace 6 with a different word problem. Add 3 from section 1.2 online assignment.

Goals

1. Apply and recognize transformations of functions:
- Translating, Stretching, Reflecting, Absolute Value 1
- Graph --> Algebraic 7, O1, O5, 1.2-O3
- Algebraic --> Graph 13, 14, 22, O7, O8
2. Given functions f and g, find:
- f+g
- f-g
- fg
- f/g
- domains for above functions. O2, O9
3. Understand and apply compositions of functions:
- Given functions f and g, find g o f and f o g. 28, 39, 60, O3, O6
- Given f o g, find functions f and g. 46, 65, O4

----

== Section 1.5 (James) ==

Homework:

Written: 11, 15, 18, 19, 21, 25, 29
Online: o2, o3, o4, o5, one more word problem

Goals

1. Graph exponential functions a^x for a<1, a>1, as well as transformations of exponential functions from section 1.3. Find domains. 11, 15, 21, o2
2. Given two points that an exponential function passes through, find the equation of the exponential function. 18, o3.
3. Use laws of exponents to simplify exponential expressions, to show various facts about exponential functions. 19, 29.
4. Convert a word problem involving population change into an algebraic expression involving exponential functions. Use this to find sizes of populations at given times, and to graph. 25, o4, o5.

----

== Section 1.6 (McKay) ==

Homework

Written: 12,16,18,19,23,35,45,48,54,60,67,71,73
Online: 01,02,03,04,05,06,07,08,010,011,012,013,016,018

Goals

1) Be able to tell if a function is one-to-one
-algebraically 12
-graphically 01
2) Know the definition of an inverse function and be able to use the steps to find the inverse
-algebraically 16,19,23,54,02,03,04,05,06
-graphically 18,45,73,07
3)Know the laws of logarithms and be able to solve and simplify logarithmic and exponential expressions. 35,48,06,08,010,011,012
4)Know the inverses of the trigonometric functions
- domain and range 71
- values at a point 60, 013,016
- simplify a trig function composed with an inverse trig function 67, o18

----

== Section 2.1 (Tyler) ==

Homework

Written:4*, 5*? (require calculator)

Goals

1. Know definitions of tangent and secant line. Compute the slope of a secant line. 4
2. Know definition of average velocity and the idea of instantaneous velocity. Compute an average velocity. 5
3. Explain how average velocity is equivalent to slope of a secant line, same for instantaneous velocity and slope of tangent line.

----

== Section 2.2 (Tyler) ==

Homework

Written: 6, 7, 9, 16, 27, 32, 34a, 40
Online: kill O2, O5. Modify: O3 fix so different parts don't have same answer.
O6, O7, O8: Use the same input notation for infinity. O8 Delete irrelevant lecture about asymptotes.

Goals:

1. Idea of a limit: (Note: the "definition" of a limit in this section is sloppy and is not a real definition of a limit. See section 2.4)
* Find the limit of a function at a point from its graph, including when the limit does not exist. 6,7, O1, O3
* Recognize the difference between the limit of a function at a point and the value of the function at a point. 6, 7, O1
* Give examples of functions that have prescribed limits at certain points. 16
* Understand that calculators and tables can entice you to guess a wrong limit. Give examples of functions whose limit is not what you would guess from plugging in values to your calculator. (no problems)
2. Idea (and notation) for one-sided limit: Do the same things for one-sided limits as done before for regular limits. 6, 7, O1, O3, O4, O6, O7, O8
3. Explain how one-sided and two-sided limits are connected. Use this to compute limits. 6, 7, O1, O3, O4, O6, O7, O8
4. Infinite limits:
* Explain why infinity is not a number and how the definition of an infinite limit gets around this.
* Find all vertical asymptotes for a function. 9, 34a
* For rational functions, find when a limit is infinity and when it is negative infinity. 27, 32, O6, O7, O8, 40

----

== Section 2.3a (Rebecca) ==

Homework

Written: 10,15,19,20,21,22,28,29
Online: Get rid of 3

Goals

1. Be able to apply limit laws to simplify limits
- algebraically 01
- graphically 02
2. Recognize when you can directly substitute to compute a limit and when you cannot. 10, 03, 04, 05, 06
3. Use algebra to simply and find limits.
- for polynomials over polynomials. 15, 19, 20, o5, o6, o7, o8, o9, o10
- expressions with square roots. 21, 22, 29, o11
- expressions with multiple fractions. 28, 29, o12

----

== Section 2.3b (Drew) ==

Homework

Written: 36-39, 42, 55, 56, 58
Online: Add o2 from 2.2. Kill o3, o4. Modify o6 so that direct substitution doesn't work. Add two problems using substitution.

Goals

1. Use the Squeeze Theorem to find limits. 36, 37, 38, o5, o6, o7
2. Evaluate limits that involve absolute values and other piecewise functions. 39, 42, o1, o2, o2 from 2.2
3. Use limit laws to find lim f(x) given that lim g(f(x)) = L. 55, 56, 58
4. Use substitution to rewrite lim g(f(x)) as lim g(u). 2 new online problems.

----

== Section 2.4 (Jessica) ==
Homework:
Written: 2, 14, 15, 16, 19, 20
Online: kill o6 (done), modify problems to find the *largest* delta (done).

Goals:
1. Use a graph showing a function and lines y=L+epsilon, y=L-epsilon to find the largest value of delta so that if 0<|x-a|<delta, then |f(x)-L|<epsilon. 2
2. Know the epsilon-delta definition of a limit. Given a limit and a value for epsilon, be able to find the largest value of delta corresponding to that epsilon. 14, o1, o2, o3, o4, o5
3. Use the epsilon-delta definition of a limit to prove a statement about the limit of a linear function. 15, 16, 19, 20.

----

== Section 2.5a (McKay/Savannah) ==

Written:4, 6, 15, 18, 22, 23, 43ab, 58
Online - Kill O2-O4, O11-O16; Change O8, O9, O10 to ask, "Where is the following function continuous?"; Add a graph problem where the student must identify where the graph is discontinuous; Add a graph problem where the student must identify the types of discontinuities

1. Given the graph of a function, be able to tell:
a) where it is continuous. 4
b) where it is discontinuous, and the type of discontinuity. 6
2. Given a function described algebraically, be able to tell:
a) where it is continuous. 22, 23, O1, O8, O9, O10
b) where it is discontinuous and types of discontinuities. 15, 18, 43(a,b), O5, O6, O7
This includes functions from Theorem 7, piecewise functions, and combinations of functions.
3. Use limit laws and the definition of continuity to prove facts about continuity (e.g. Theorem 4). 58

----

== Section 2.5b (Mark) ==

Homework

Written: 28, 31, 32, 36, 37, 39, 41, 45, 47, 49, 65
Online: Kill 3, 5, 6, 7. Add 3 problems: Find intervals where f is positive, negative. One quadratic, easy factor; one rational with linear top and bottom; one rational with quadratic top, perfect square bottom.

Goals

1. Use continuity to evaluate limits 31,32
2. Continuity of piecewise functions
-determine if a piecewise function is continuous (or continuous from the right/left) 36,37,39
-determine parameters to make a piecewise function continuous 41,O1,O2,O4
3. Given a composition of functions, tell where it is continuous/discontinuous. 28
4. Use the Intermediate Value Theorem to prove the existence of solutions to equations. 45,47,49,65
5. Find intervals where a continuous function is positive/negative. (3 new online problems)

----

== Section 2.6 (Skyler) ==

Homework:
Written: 4, 5, 7, 13, 33, 43, 48
Online: Drop #15

Goals

1. Understand definition of limit at +/- inf
2. Find the limit of a rational function as x -> +/-inf and equations of horizontal asymptotes (when they exist)
3. Identify functions whose limit at +/- inf does not exist or whose limit is infinite
4. Compute limits at infinity by graphs and algebraic techniques

----

== Section 2.7 (Sam) ==
Homework:
Written: 5,9,19, 20, 43,44, 46
Online: eliminate o7,o9,o12

Goals:
1. Given an algebraic function, take the derivative (using the definition of the derivative)
Write down an equation for a tangent line through a point on a graph 3, 5, 05, o10, o11
2. Given a real life situation, interpret instantaneous change or average of change 43, 44, 46
Given a position function (or graph thereof), interpret the graph or find (average) velocity o1, o2, o3
3. Sketch the graph of a function such that certain criteria are met (values at certain points, derivatives, etc.) 19, 20
4. Given a limit expression, write down the point and function whose derivative it represents o6, o8
5. Write both formal definitions of a derivative
Explain how the derivative can be interpreted as slope of a line, velocity, instantaneous change

Changes:
Added: 43, 44
Eliminating: 2,25, o7, o9, o12

----

== Section 2.8 (Savannah) ==

Homework

Written: 1, 3, 5, 6, 7, 9, 11, 19, 20, 25, 35, 36, 37, 38, 44, Prove that if a function f is differentiable at a, then f is continuous at a.
Online: Kill O2, O3, Change O9 to f'(x)

Goals

1. Given a function f, find a formula for the derivative of f using f'(x) = lim (h->0) [f(x+h)-f(x)]/h, and be able to state the domain of f'(x). 19, 20, 25, O9, O10
2. Given a graph of a function f, be able to find the graph of f', f''. Given a graph of a function f', f'', be able to find th graph of f. 1, 3, 5, 6, 7, 9, 11, O1, O4, O5
3. Recognize when and why a function fails to be differentiable: 35, 36, 37, 38
a) corner
b) discontinuity (Theorem 4)
c) vertical tangent line
4. Be able to compute and apply higher derivatives: O6, O7, O8, 44
- (f')'=f''
- position [s(t)], velocity [v(t)=s'(t)], acceleration [a(t)=v'(t)=s''(t)]
5. Prove that if f is differentiable at a, then f is continuous at a. [Written homework]

----

== Section 3.1 (Drew) ==

Homework:
Written: 2, 13, 17, 20, 25, 28, 29, 33, 55, 61, 77
Online: Kill o6 last part, o14, o8

Goals:
1. Compute derivatives using:
a. The power rule. Especially, be able to rewrite functions so the power rule can be applied. 13, 17, 20, 25, 28, 29, o1, o3, o7, o8, o9, o10, o11, o12
b. The derivative of e^x. 2, 17, 28, o12
c. Sums, differences, and scalar multiples of functions you can differentiate with these rules. (almost all problems)
2. Use the definition of the derivative and limit laws to:
a. Prove simple rules, including the power rule for n=-2, -1, -1/2, 1/2, 1, 2, as well as sum and difference rules. 61
b. Compute certain limits. 77
3. Find a tangent line to the graph of a function at a given point. 33, 55, o2, o4, o5

----

== Section 3.2 (Rebecca) ==

Homework

Written: 2,11,13,23,24,32,33,47,49,55,57
Online: As listed.

Goals

1. Be able to apply the product rule to find the derivative of functions. 11,24,33,47,49,55,57, 02,04,05,06,07,09
2. Be able to apply the quotient rule to find the derivative of functions. 2,13,23,24,32,47,49, 01,03,08,010,011
3. Be able to prove the product and quotient rules and variations. 55,57*

----

== Section 3.3 (Skyler/James) ==

Homework

Written: 9, 10, 18, 20, 35, 42, 45, 49
Online: As listed. Change ordinal number suffixes on #3 (e.g. 83th makes no sense)

Goals

1. Know derivatives for the 6 basic trig functions. Be able to use these to compute derivatives of more complicated functions.
2. Know limits [2] and [3] in the book. Be able to use these to solve other limits involving trig functions
3. Know the "periodicity" of differentiation of trig functions--i.e. the fourth derivative of the sine function is again the sine function
4. Be able to use trig identities to find derivatives and limits
5. Use trig derivatives in various real world problems.

----

== Section 3.4 (Savannah) ==

Homework

Written: 7, 24, 25, 31, 37, 47, 63, 65, 71, 84, 89
Online: Kill O6, O7, O8; Reorder problems (see Dr. Purcell); Add #77 from the book to online (change "particle" to "point")

Goals

1. Fluency in chain rule: Given a composition of functions, use the chain rule and other rules to compute its derivative quickly. 7, 25, 37, 47, O1, O2, O3, O9, O10, O11, O12, O13, O14, O15, O16
2. Given values of two functions and their derivatives at certain points, use the formal statement of the chain rule to compute the derivative of compositions of these functions. 63, 65, 71, O4, O5
3. Given a word problem involving the composition of functions, compute rates of change. Interpret the meaning of the derivative as a particular rate of change. 84
4. Use the chain rule and definition of even/odd functions to prove properties of derivatives of even/odd functions. 89
5. Compute the derivative of a^x. 24, 31

----

== Section 3.5 (Rebecca) ==
Homework

Written: 3,11,12,15,18,21,23,34,41,57 (Add 43,67)
Online: Kill 03,014; Replace 08 with 35 from the book and make it online

Goals

1) For an implicitly defined equation find dy/dx, dx/dy, and d^2y/dx^2 using implicit differentiation.
2) Find the tangent line to an implicitly defined function.
3) Know the derivatives of the inverse trig functions.
----

== Section 3.6 (Sam) ==

----

== Section 3.8 (Jessica) ==

Homework:

Written: 14, 15
Online: kill o5 (pH scale)

Goals:

Use exponential functions to model each of the following:
1. Population growth. o1, o3, o4
2. Radioactive decay. o6, o7
Find half-life, or given half-life, find equation.
3. Cooling. 14, 15
4. Interest. o2, o8
- Derive the formula for compounding interest in n periods o2, o8 (see also section 1.3, number 60)
- Use expression of e as a limit to find formula for continuously compounded interest.
----

== Section 3.9 (Mark) ==

Homework

Written: 5, 22, 33, 35, 37, 42
Online: Kill O1, O4, O7, O8, O11, O12, O13; remove geometry formulas from online questions.

Goals

1. Solve related rates problems. (all homework problems)
a) Draw picture
b) Recognize variables from the problem, and rates of change as their derivatives.
c) Write equations relating variables (using geometry, trigonometry)
d) Differentiate, remembering to use the chain rule.
e) Plug in specific values to determine the requested answer.

----

== Section 3.10 (Tyler) ==

Homework
Written 1,2,3,5,23,28,43,44
Online Kill O4, O5, Renumber so that O3 is first and O1 is third. Possibly add one more. Change the hint in O1 to read "Try using linear approximation..."

Goals
1. Compute the linearization of a function at a point. 1,2,3,5, O2, O3
- Explain the relationship between the linearization and the tangent line. 5, O2
- Explain how linearization is useful. 23,28
2. Use the linearization at a point x=a to approximate the value of a function f(a+epsilon). 23,28, O1, O2

Suggestion: merge 3.10 and 3.11 for written as well as online.
----

== Section 3.11 (Savannah) ==

----

== Section 4.1 ==

----

== Section 4.2 ==

----

== Section 4.3 ==

Goals:

1. Know and apply the increasing/decreasing test o1, o5
2. Find local maxima and minima (First and Second derivative tests, endpoints) 21,23
3. Know the definition of concavity and how to determine concavity (The concavity test) o2, o5, 72
4. Know what an inflection point is and how to find it o2
5. Know how the above information affects the graph of a function
Draw a graph such that.... 25, 28
Given a graph, where is concavity positive, etc.? 1, 6, 7, 31 and a new online problem similar to problem 8 on page 295

Changes:

o3 and o4 are to be moved to section 4.5 and a problem is to be added online that is like problem 8.
----

== Section 4.4 ==

----

== Section 4.5 (Skyler) ==

Goals:

Use all of the following to draw graphs of functions:

* Algebraic Properties (Domain, intercepts, even/odd/periodic)

* Limit Properties (Asymptotes, singularities, discontinuities)

* Derivative Properties (Increasing/decreasing, extrema, concavity/inflection)

Written homework - as listed
Online homework - pulled from 4.3 (O3 and O4)

----

== Section 4.7 ==

----

== Section 4.8 (Mark) ==

Goals

1. Use Newton's Method to approximate roots/solutions of equations.
2. Derive the formula for Newton's Method from the tangent line equation.
3. Give an example of a function 'f' and a starting point 'x_1' that makes Newton's Method fail.

Homework

Written: 1, 2, 3, 4, 29.
Online: Kill O1, Add new online problem with graph.

----

== Section 4.9 (Tyler)==

Goals:
1. Use the Mean value Theorem to prove Theorem 1.
2. Memorize the table of anti-differentiation formulas on page 341 and
relate it to the table of derivatives on the inside back cover of the book. 12, 15, 20, 27, 33, 49, 51, O1-O12
3. Use the table and Theorem 1 to find *all* antiderivatives of various functions. 12, 15, 20, 27, 49, 51
4. Use Theorem 1 and known differentiation formulas to find a unique anti-derivative with given values at specified points. 33, 53, 67,
69, 70, 74,O2,O3, O4,O5,O10, O11, O13
5. Use anti-differentiation to solve physical and economic problems, especially problems of motion. 67, 69, 70, 74, O4, O5

Changes to problems:
Written: Add story problems: 67, 69, 70, 74
Online: too long, Delete O1, O6, O12.
Adjust numbers in 04 to give a clean answer.
O10--change so that F(0) not equal to 0 (else they will get right answer for wrong reason).
O8 and O9 problem that Arctan/Arcsin not accepted--just arctan/arcsin.
O13 should not say "the anti-derivative, but rather "an anti-derivative".

----

== Appendix E (Paul)==
Homework:

Written: 1, 3, 5, 10, 11, 15, 17, 20, 23, 41, 43
Online: o1, o2, o3, o4, o5, o6, o7, o8, o9, o10

Goals

1. Write expanded sums in sigma notation (1, 3, 5, 10)
and expand sums written in sigma notation (11, 15, 17, 20).
Numerically evaluate sums (23). Apply appropriate rules to
distribute constants or expand sums of sums (o2, o4, o5, o8, o9, o10).
2. Explicitly evaluate $\sigma_{i=1}^n f(i)$ in terms of 'n', where 'f(x)'
is a polynomial in x of degree at most 3. (Degree 0 o3, o7; degree 1
o1, o2, o10; degree 2 o4, o5.)
Numerically evaluate this sum when 'i' runs over a given range of integers (o6, o8, o9).
3. Evaluate telescoping sums (41) and limits of sums (43).
4. Change the index of summation.

Kill o3, o6, o8, o9, o10, add written 30, 35, add 3 online problems on changing index of summation and one limit of a sum problem like 44.
----

== Section 5.1 ==

----

== Section 5.2 ==

----

== Section 5.3 (Tyler a, Ryan b)==

Goals:
A. Explain the relationship between area and derivative described by the Fundamental Theorem.
A. Evaluate and describe properties of functions defined by integrals 3,
A. Differentiate functions defined by integrals 9,15, R44,R45,R47,R48,R68 , O1-O6
B. Given a function and limits of integration verify that the hypotheses
of the Fundamental Theorem hold
B. Use the Fundamental Theorem to evaluate the definite integral of a function.
B. Apply the Fundamental Theorem to interpret the meaning of a
definite integral and its derivative in physical, economic, and other applications.

Changes:
A. O5. Make the hint part of the problem or make it not show until after they try once.
A. All the online problems are essentially the same--differentiate an integral (and don't forget the chain rule).

B.

----

== Section 5.4 (Paul)==
Homework:

Written: 4, 9, 23, 32, 43, 61, 65
Online: o1, o2, o3, o4, o5, o6, o7, o8, o9, o10

Goals

1. Evaluate definite and indefinite integrals for all standard functions
(polynomials 9, o1a, o3, o6a, o6c, rational functions 4, o1b, o1c, o4, o6b, o7, o8,
trigonometric functions and inverses 32, o5, o8, o9abc, o10, exponentials 23, 32 and
logarithms o8, absolute value 43, etc.)
2. Solve problems about net change by setting up and evaluating appropriate
integrals of rates of change (density 61, marginal cost 65,
displacement/distance o2, etc.).
----

== Section 5.5 ==

----

Math 112 Calculus Learning Goals

2009-08-19T20:24:15Z

Pmj5: /* Section 5.4 */

----

== Section 1.1 (Jessica) ==

Homework:

Written: 25, 31, 40, 43, 55, 56, 64, 65
Online: 3, 4, 5, 6, 8, 9, 11. Modify 12 (Given f like 66 or 67 in book, find f(-x). Is f even or odd or neither?) Add problem like 51, 57 in book.

Goals

1. Given a function described algebraically, find the
- domain 31, 07, 08,
- range,
- value at a given number 04,
- value when we plug in another function, e.g. difference quotient 04, 05, 25.
2. Convert one representation of a function to another.
- verbal to/from graphical 03
-algebraic to/from graphical 09
-verbal to/from algebraic 55, 56, 2 new online
3. Piecewise functions: Given an algebraic description of a piecewise function, find its domain and sketch its graph. 011, 43.
Be able to write the absolute value as a piecewise function, and use this to sketch its graph. 40, 09
4. Use the definition of an even/odd function to decide whether a function described algebraically is even or odd. 012, 65
Use symmetry to decide whether a graph is an even or odd function. 64

----

== Section 1.2 (Sam) ==

Homework:

Written: 4,5,7,9,12 and Appdx D: 24, 37*, 46
Online: O1, O2, O4, O5 (these are the numbers under the current numbering),
and Appdx D: 29, 60, 69
The * on problem 37 means a calculator will be necessary.

Goals

1. Writing down a linear function given a set of information 5,12,O4,O5
2. Recognizing what a function's graph should look like 4,7,O1
3. Based on a graph, write down a function O2
4. Writing down a polynomial based on information 9, O2
5. Trig review (definitions, proofs, values of trig functions at particular points) Appdx D:24, 37, 46, 29, 60, 69 (the latter three going online)

# Trig to know, including problems to skim for practice. (Jessica)

1. Given angles in degrees and radians, draw the angle, convert from radians to degrees and vice versa. (See appendix D: 1-12)
2. Write down sin, cos, tan, sec, scs, cot of any angle 0, pi/6, pi/4, pi/3, pi/2, and all angles obtained from these angles by adding a multiple of pi/2. (appendix D:23-28)
3. Given a right triangle with side measurements, write sin, cos, tan, sec, csc, cot of any of the angles of the triangle in terms of the side lengths. (Appendix D: 35-38)
Similarly, given one trig ratio, find the others. (Appendix D: 29-34)
4. Graph trig functions. (app D: 77-82)
5. Use trig identities to simplify expressions (appendix D: 43-57 odd, 59-63), and to solve equations and inequalities involving trig functions (app D: 65-76).
* Most important identities: sin^2(x) + cos^2(x) = 1
* sin(-x) = -sin(x)
* cos(-x) = cos(x)
* sin(x+y) = sin(x)cos(y) + cos(x)sin(y)
* cos(x+y) = cos(x)cos(y) - sin(x)sin(y)
Using the above, you can derive other identities:
* 1+tan^2(x)=sec^2(x); 1+cot^2(x) = csc^2(x)
* sin(x-y)
* cos(x-y)
* sin(2x)
* cos(2x)

----

== Section 1.3 (Savannah) ==

Homework

Written: 1,7,13, 14, 22, 39, 46, 60, 65
Online: Modify 1 to be y as a function of x. Replace 6 with a different word problem. Add 3 from section 1.2 online assignment.

Goals

1. Apply and recognize transformations of functions:
- Translating, Stretching, Reflecting, Absolute Value 1
- Graph --> Algebraic 7, O1, O5, 1.2-O3
- Algebraic --> Graph 13, 14, 22, O7, O8
2. Given functions f and g, find:
- f+g
- f-g
- fg
- f/g
- domains for above functions. O2, O9
3. Understand and apply compositions of functions:
- Given functions f and g, find g o f and f o g. 28, 39, 60, O3, O6
- Given f o g, find functions f and g. 46, 65, O4

----

== Section 1.5 (James) ==

Homework:

Written: 11, 15, 18, 19, 21, 25, 29
Online: o2, o3, o4, o5, one more word problem

Goals

1. Graph exponential functions a^x for a<1, a>1, as well as transformations of exponential functions from section 1.3. Find domains. 11, 15, 21, o2
2. Given two points that an exponential function passes through, find the equation of the exponential function. 18, o3.
3. Use laws of exponents to simplify exponential expressions, to show various facts about exponential functions. 19, 29.
4. Convert a word problem involving population change into an algebraic expression involving exponential functions. Use this to find sizes of populations at given times, and to graph. 25, o4, o5.

----

== Section 1.6 (McKay) ==

Homework

Written: 12,16,18,19,23,35,45,48,54,60,67,71,73
Online: 01,02,03,04,05,06,07,08,010,011,012,013,016,018

Goals

1) Be able to tell if a function is one-to-one
-algebraically 12
-graphically 01
2) Know the definition of an inverse function and be able to use the steps to find the inverse
-algebraically 16,19,23,54,02,03,04,05,06
-graphically 18,45,73,07
3)Know the laws of logarithms and be able to solve and simplify logarithmic and exponential expressions. 35,48,06,08,010,011,012
4)Know the inverses of the trigonometric functions
- domain and range 71
- values at a point 60, 013,016
- simplify a trig function composed with an inverse trig function 67, o18

----

== Section 2.1 (Tyler) ==

Homework

Written:4*, 5*? (require calculator)

Goals

1. Know definitions of tangent and secant line. Compute the slope of a secant line. 4
2. Know definition of average velocity and the idea of instantaneous velocity. Compute an average velocity. 5
3. Explain how average velocity is equivalent to slope of a secant line, same for instantaneous velocity and slope of tangent line.

----

== Section 2.2 (Tyler) ==

Homework

Written: 6, 7, 9, 16, 27, 32, 34a, 40
Online: kill O2, O5. Modify: O3 fix so different parts don't have same answer.
O6, O7, O8: Use the same input notation for infinity. O8 Delete irrelevant lecture about asymptotes.

Goals:

1. Idea of a limit: (Note: the "definition" of a limit in this section is sloppy and is not a real definition of a limit. See section 2.4)
* Find the limit of a function at a point from its graph, including when the limit does not exist. 6,7, O1, O3
* Recognize the difference between the limit of a function at a point and the value of the function at a point. 6, 7, O1
* Give examples of functions that have prescribed limits at certain points. 16
* Understand that calculators and tables can entice you to guess a wrong limit. Give examples of functions whose limit is not what you would guess from plugging in values to your calculator. (no problems)
2. Idea (and notation) for one-sided limit: Do the same things for one-sided limits as done before for regular limits. 6, 7, O1, O3, O4, O6, O7, O8
3. Explain how one-sided and two-sided limits are connected. Use this to compute limits. 6, 7, O1, O3, O4, O6, O7, O8
4. Infinite limits:
* Explain why infinity is not a number and how the definition of an infinite limit gets around this.
* Find all vertical asymptotes for a function. 9, 34a
* For rational functions, find when a limit is infinity and when it is negative infinity. 27, 32, O6, O7, O8, 40

----

== Section 2.3a (Rebecca) ==

Homework

Written: 10,15,19,20,21,22,28,29
Online: Get rid of 3

Goals

1. Be able to apply limit laws to simplify limits
- algebraically 01
- graphically 02
2. Recognize when you can directly substitute to compute a limit and when you cannot. 10, 03, 04, 05, 06
3. Use algebra to simply and find limits.
- for polynomials over polynomials. 15, 19, 20, o5, o6, o7, o8, o9, o10
- expressions with square roots. 21, 22, 29, o11
- expressions with multiple fractions. 28, 29, o12

----

== Section 2.3b (Drew) ==

Homework

Written: 36-39, 42, 55, 56, 58
Online: Add o2 from 2.2. Kill o3, o4. Modify o6 so that direct substitution doesn't work. Add two problems using substitution.

Goals

1. Use the Squeeze Theorem to find limits. 36, 37, 38, o5, o6, o7
2. Evaluate limits that involve absolute values and other piecewise functions. 39, 42, o1, o2, o2 from 2.2
3. Use limit laws to find lim f(x) given that lim g(f(x)) = L. 55, 56, 58
4. Use substitution to rewrite lim g(f(x)) as lim g(u). 2 new online problems.

----

== Section 2.4 (Jessica) ==
Homework:
Written: 2, 14, 15, 16, 19, 20
Online: kill o6 (done), modify problems to find the *largest* delta (done).

Goals:
1. Use a graph showing a function and lines y=L+epsilon, y=L-epsilon to find the largest value of delta so that if 0<|x-a|<delta, then |f(x)-L|<epsilon. 2
2. Know the epsilon-delta definition of a limit. Given a limit and a value for epsilon, be able to find the largest value of delta corresponding to that epsilon. 14, o1, o2, o3, o4, o5
3. Use the epsilon-delta definition of a limit to prove a statement about the limit of a linear function. 15, 16, 19, 20.

----

== Section 2.5a (McKay/Savannah) ==

Written:4, 6, 15, 18, 22, 23, 43ab, 58
Online - Kill O2-O4, O11-O16; Change O8, O9, O10 to ask, "Where is the following function continuous?"; Add a graph problem where the student must identify where the graph is discontinuous; Add a graph problem where the student must identify the types of discontinuities

1. Given the graph of a function, be able to tell:
a) where it is continuous. 4
b) where it is discontinuous, and the type of discontinuity. 6
2. Given a function described algebraically, be able to tell:
a) where it is continuous. 22, 23, O1, O8, O9, O10
b) where it is discontinuous and types of discontinuities. 15, 18, 43(a,b), O5, O6, O7
This includes functions from Theorem 7, piecewise functions, and combinations of functions.
3. Use limit laws and the definition of continuity to prove facts about continuity (e.g. Theorem 4). 58

----

== Section 2.5b (Mark) ==

Homework

Written: 28, 31, 32, 36, 37, 39, 41, 45, 47, 49, 65
Online: Kill 3, 5, 6, 7. Add 3 problems: Find intervals where f is positive, negative. One quadratic, easy factor; one rational with linear top and bottom; one rational with quadratic top, perfect square bottom.

Goals

1. Use continuity to evaluate limits 31,32
2. Continuity of piecewise functions
-determine if a piecewise function is continuous (or continuous from the right/left) 36,37,39
-determine parameters to make a piecewise function continuous 41,O1,O2,O4
3. Given a composition of functions, tell where it is continuous/discontinuous. 28
4. Use the Intermediate Value Theorem to prove the existence of solutions to equations. 45,47,49,65
5. Find intervals where a continuous function is positive/negative. (3 new online problems)

----

== Section 2.6 (Skyler) ==

Homework:
Written: 4, 5, 7, 13, 33, 43, 48
Online: Drop #15

Goals

1. Understand definition of limit at +/- inf
2. Find the limit of a rational function as x -> +/-inf and equations of horizontal asymptotes (when they exist)
3. Identify functions whose limit at +/- inf does not exist or whose limit is infinite
4. Compute limits at infinity by graphs and algebraic techniques

----

== Section 2.7 (Sam) ==
Homework:
Written: 5,9,19, 20, 43,44, 46
Online: eliminate o7,o9,o12

Goals:
1. Given an algebraic function, take the derivative (using the definition of the derivative)
Write down an equation for a tangent line through a point on a graph 3, 5, 05, o10, o11
2. Given a real life situation, interpret instantaneous change or average of change 43, 44, 46
Given a position function (or graph thereof), interpret the graph or find (average) velocity o1, o2, o3
3. Sketch the graph of a function such that certain criteria are met (values at certain points, derivatives, etc.) 19, 20
4. Given a limit expression, write down the point and function whose derivative it represents o6, o8
5. Write both formal definitions of a derivative
Explain how the derivative can be interpreted as slope of a line, velocity, instantaneous change

Changes:
Added: 43, 44
Eliminating: 2,25, o7, o9, o12

----

== Section 2.8 (Savannah) ==

Homework

Written: 1, 3, 5, 6, 7, 9, 11, 19, 20, 25, 35, 36, 37, 38, 44, Prove that if a function f is differentiable at a, then f is continuous at a.
Online: Kill O2, O3, Change O9 to f'(x)

Goals

1. Given a function f, find a formula for the derivative of f using f'(x) = lim (h->0) [f(x+h)-f(x)]/h, and be able to state the domain of f'(x). 19, 20, 25, O9, O10
2. Given a graph of a function f, be able to find the graph of f', f''. Given a graph of a function f', f'', be able to find th graph of f. 1, 3, 5, 6, 7, 9, 11, O1, O4, O5
3. Recognize when and why a function fails to be differentiable: 35, 36, 37, 38
a) corner
b) discontinuity (Theorem 4)
c) vertical tangent line
4. Be able to compute and apply higher derivatives: O6, O7, O8, 44
- (f')'=f''
- position [s(t)], velocity [v(t)=s'(t)], acceleration [a(t)=v'(t)=s''(t)]
5. Prove that if f is differentiable at a, then f is continuous at a. [Written homework]

----

== Section 3.1 (Drew) ==

Homework:
Written: 2, 13, 17, 20, 25, 28, 29, 33, 55, 61, 77
Online: Kill o6 last part, o14, o8

Goals:
1. Compute derivatives using:
a. The power rule. Especially, be able to rewrite functions so the power rule can be applied. 13, 17, 20, 25, 28, 29, o1, o3, o7, o8, o9, o10, o11, o12
b. The derivative of e^x. 2, 17, 28, o12
c. Sums, differences, and scalar multiples of functions you can differentiate with these rules. (almost all problems)
2. Use the definition of the derivative and limit laws to:
a. Prove simple rules, including the power rule for n=-2, -1, -1/2, 1/2, 1, 2, as well as sum and difference rules. 61
b. Compute certain limits. 77
3. Find a tangent line to the graph of a function at a given point. 33, 55, o2, o4, o5

----

== Section 3.2 (Rebecca) ==

Homework

Written: 2,11,13,23,24,32,33,47,49,55,57
Online: As listed.

Goals

1. Be able to apply the product rule to find the derivative of functions. 11,24,33,47,49,55,57, 02,04,05,06,07,09
2. Be able to apply the quotient rule to find the derivative of functions. 2,13,23,24,32,47,49, 01,03,08,010,011
3. Be able to prove the product and quotient rules and variations. 55,57*

----

== Section 3.3 (Skyler/James) ==

Homework

Written: 9, 10, 18, 20, 35, 42, 45, 49
Online: As listed. Change ordinal number suffixes on #3 (e.g. 83th makes no sense)

Goals

1. Know derivatives for the 6 basic trig functions. Be able to use these to compute derivatives of more complicated functions.
2. Know limits [2] and [3] in the book. Be able to use these to solve other limits involving trig functions
3. Know the "periodicity" of differentiation of trig functions--i.e. the fourth derivative of the sine function is again the sine function
4. Be able to use trig identities to find derivatives and limits
5. Use trig derivatives in various real world problems.

----

== Section 3.4 (Savannah) ==

Homework

Written: 7, 24, 25, 31, 37, 47, 63, 65, 71, 84, 89
Online: Kill O6, O7, O8; Reorder problems (see Dr. Purcell); Add #77 from the book to online (change "particle" to "point")

Goals

1. Fluency in chain rule: Given a composition of functions, use the chain rule and other rules to compute its derivative quickly. 7, 25, 37, 47, O1, O2, O3, O9, O10, O11, O12, O13, O14, O15, O16
2. Given values of two functions and their derivatives at certain points, use the formal statement of the chain rule to compute the derivative of compositions of these functions. 63, 65, 71, O4, O5
3. Given a word problem involving the composition of functions, compute rates of change. Interpret the meaning of the derivative as a particular rate of change. 84
4. Use the chain rule and definition of even/odd functions to prove properties of derivatives of even/odd functions. 89
5. Compute the derivative of a^x. 24, 31

----

== Section 3.5 (Rebecca) ==
Homework

Written: 3,11,12,15,18,21,23,34,41,57 (Add 43,67)
Online: Kill 03,014; Replace 08 with 35 from the book and make it online

Goals

1) For an implicitly defined equation find dy/dx, dx/dy, and d^2y/dx^2 using implicit differentiation.
2) Find the tangent line to an implicitly defined function.
3) Know the derivatives of the inverse trig functions.
----

== Section 3.6 (Sam) ==

----

== Section 3.8 (Jessica) ==

Homework:

Written: 14, 15
Online: kill o5 (pH scale)

Goals:

Use exponential functions to model each of the following:
1. Population growth. o1, o3, o4
2. Radioactive decay. o6, o7
Find half-life, or given half-life, find equation.
3. Cooling. 14, 15
4. Interest. o2, o8
- Derive the formula for compounding interest in n periods o2, o8 (see also section 1.3, number 60)
- Use expression of e as a limit to find formula for continuously compounded interest.
----

== Section 3.9 (Mark) ==

Homework

Written: 5, 22, 33, 35, 37, 42
Online: Kill O1, O4, O7, O8, O11, O12, O13; remove geometry formulas from online questions.

Goals

1. Solve related rates problems. (all homework problems)
a) Draw picture
b) Recognize variables from the problem, and rates of change as their derivatives.
c) Write equations relating variables (using geometry, trigonometry)
d) Differentiate, remembering to use the chain rule.
e) Plug in specific values to determine the requested answer.

----

== Section 3.10 (Tyler) ==

Homework
Written 1,2,3,5,23,28,43,44
Online Kill O4, O5, Renumber so that O3 is first and O1 is third. Possibly add one more. Change the hint in O1 to read "Try using linear approximation..."

Goals
1. Compute the linearization of a function at a point. 1,2,3,5, O2, O3
- Explain the relationship between the linearization and the tangent line. 5, O2
- Explain how linearization is useful. 23,28
2. Use the linearization at a point x=a to approximate the value of a function f(a+epsilon). 23,28, O1, O2

Suggestion: merge 3.10 and 3.11 for written as well as online.
----

== Section 3.11 (Savannah) ==

----

== Section 4.1 ==

----

== Section 4.2 ==

----

== Section 4.3 ==

Goals:

1. Know and apply the increasing/decreasing test o1, o5
2. Find local maxima and minima (First and Second derivative tests, endpoints) 21,23
3. Know the definition of concavity and how to determine concavity (The concavity test) o2, o5, 72
4. Know what an inflection point is and how to find it o2
5. Know how the above information affects the graph of a function
Draw a graph such that.... 25, 28
Given a graph, where is concavity positive, etc.? 1, 6, 7, 31 and a new online problem similar to problem 8 on page 295

Changes:

o3 and o4 are to be moved to section 4.5 and a problem is to be added online that is like problem 8.
----

== Section 4.4 ==

----

== Section 4.5 (Skyler) ==

Goals:

Use all of the following to draw graphs of functions:

* Algebraic Properties (Domain, intercepts, even/odd/periodic)

* Limit Properties (Asymptotes, singularities, discontinuities)

* Derivative Properties (Increasing/decreasing, extrema, concavity/inflection)

Written homework - as listed
Online homework - pulled from 4.3 (O3 and O4)

----

== Section 4.7 ==

----

== Section 4.8 (Mark) ==

Goals

1. Use Newton's Method to approximate roots/solutions of equations.
2. Derive the formula for Newton's Method from the tangent line equation.
3. Give an example of a function 'f' and a starting point 'x_1' that makes Newton's Method fail.

Homework

Written: 1, 2, 3, 4, 29.
Online: Kill O1, Add new online problem with graph.

----

== Section 4.9 (Tyler)==

Goals:
1. Use the Mean value Theorem to prove Theorem 1.
2. Memorize the table of anti-differentiation formulas on page 341 and
relate it to the table of derivatives on the inside back cover of the book. 12, 15, 20, 27, 33, 49, 51, O1-O12
3. Use the table and Theorem 1 to find *all* antiderivatives of various functions. 12, 15, 20, 27, 49, 51
4. Use Theorem 1 and known differentiation formulas to find a unique anti-derivative with given values at specified points. 33, 53, 67,
69, 70, 74,O2,O3, O4,O5,O10, O11, O13
5. Use anti-differentiation to solve physical and economic problems, especially problems of motion. 67, 69, 70, 74, O4, O5

Changes to problems:
Written: Add story problems: 67, 69, 70, 74
Online: too long, Delete O1, O6, O12.
Adjust numbers in 04 to give a clean answer.
O10--change so that F(0) not equal to 0 (else they will get right answer for wrong reason).
O8 and O9 problem that Arctan/Arcsin not accepted--just arctan/arcsin.
O13 should not say "the anti-derivative, but rather "an anti-derivative".

----

== Appendix E (Paul)==
Homework:

Written: 1, 3, 5, 10, 11, 15, 17, 20, 23, 41, 43
Online: o1, o2, o3, o4, o5, o6, o7, o8, o9, o10

Goals

1. Write expanded sums in sigma notation (1, 3, 5, 10)
and expand sums written in sigma notation (11, 15, 17, 20).
Numerically evaluate sums (23). Apply appropriate rules to
distribute constants or expand sums of sums (o2, o4, o5, o8, o9, o10).
2. Explicitly evaluate $\sigma_{i=1}^n f(i)$ in terms of 'n', where 'f(x)'
is a polynomial in x of degree at most 2. (Degree 0 o3, o7; degree 1
o1, o2, o10; degree 2 o4, o5.)
Numerically evaluate this sum when 'i' runs over a given range of integers (o6, o8, o9).
3. Evaluate telescoping sums (41) and limits of sums (43).

----

== Section 5.1 ==

----

== Section 5.2 ==

----

== Section 5.3 ==

----

== Section 5.4 (Paul)==
Homework:

Written: 4, 9, 23, 32, 43, 61, 65
Online: o1, o2, o3, o4, o5, o6, o7, o8, o9, o10

Goals

1. Evaluate definite and indefinite integrals for all standard functions
(polynomials 9, o1a, o3, o6a, o6c, rational functions 4, o1b, o1c, o4, o6b, o7, o8,
trigonometric functions and inverses 32, o5, o8, o9abc, o10, exponentials 23, 32 and
logarithms o8, absolute value 43, etc.)
2. Solve problems about net change by setting up and evaluating appropriate
integrals of rates of change (density 61, marginal cost 65,
displacement/distance o2, etc.).
----

== Section 5.5 ==

----

Math 112 Calculus Learning Goals

2009-08-19T20:09:44Z

Pmj5: /* Appendix E (Paul) */

----

== Section 1.1 (Jessica) ==

Homework:

Written: 25, 31, 40, 43, 55, 56, 64, 65
Online: 3, 4, 5, 6, 8, 9, 11. Modify 12 (Given f like 66 or 67 in book, find f(-x). Is f even or odd or neither?) Add problem like 51, 57 in book.

Goals

1. Given a function described algebraically, find the
- domain 31, 07, 08,
- range,
- value at a given number 04,
- value when we plug in another function, e.g. difference quotient 04, 05, 25.
2. Convert one representation of a function to another.
- verbal to/from graphical 03
-algebraic to/from graphical 09
-verbal to/from algebraic 55, 56, 2 new online
3. Piecewise functions: Given an algebraic description of a piecewise function, find its domain and sketch its graph. 011, 43.
Be able to write the absolute value as a piecewise function, and use this to sketch its graph. 40, 09
4. Use the definition of an even/odd function to decide whether a function described algebraically is even or odd. 012, 65
Use symmetry to decide whether a graph is an even or odd function. 64

----

== Section 1.2 (Sam) ==

Homework:

Written: 4,5,7,9,12 and Appdx D: 24, 37*, 46
Online: O1, O2, O4, O5 (these are the numbers under the current numbering),
and Appdx D: 29, 60, 69
The * on problem 37 means a calculator will be necessary.

Goals

1. Writing down a linear function given a set of information 5,12,O4,O5
2. Recognizing what a function's graph should look like 4,7,O1
3. Based on a graph, write down a function O2
4. Writing down a polynomial based on information 9, O2
5. Trig review (definitions, proofs, values of trig functions at particular points) Appdx D:24, 37, 46, 29, 60, 69 (the latter three going online)

# Trig to know, including problems to skim for practice. (Jessica)

1. Given angles in degrees and radians, draw the angle, convert from radians to degrees and vice versa. (See appendix D: 1-12)
2. Write down sin, cos, tan, sec, scs, cot of any angle 0, pi/6, pi/4, pi/3, pi/2, and all angles obtained from these angles by adding a multiple of pi/2. (appendix D:23-28)
3. Given a right triangle with side measurements, write sin, cos, tan, sec, csc, cot of any of the angles of the triangle in terms of the side lengths. (Appendix D: 35-38)
Similarly, given one trig ratio, find the others. (Appendix D: 29-34)
4. Graph trig functions. (app D: 77-82)
5. Use trig identities to simplify expressions (appendix D: 43-57 odd, 59-63), and to solve equations and inequalities involving trig functions (app D: 65-76).
* Most important identities: sin^2(x) + cos^2(x) = 1
* sin(-x) = -sin(x)
* cos(-x) = cos(x)
* sin(x+y) = sin(x)cos(y) + cos(x)sin(y)
* cos(x+y) = cos(x)cos(y) - sin(x)sin(y)
Using the above, you can derive other identities:
* 1+tan^2(x)=sec^2(x); 1+cot^2(x) = csc^2(x)
* sin(x-y)
* cos(x-y)
* sin(2x)
* cos(2x)

----

== Section 1.3 (Savannah) ==

Homework

Written: 1,7,13, 14, 22, 39, 46, 60, 65
Online: Modify 1 to be y as a function of x. Replace 6 with a different word problem. Add 3 from section 1.2 online assignment.

Goals

1. Apply and recognize transformations of functions:
- Translating, Stretching, Reflecting, Absolute Value 1
- Graph --> Algebraic 7, O1, O5, 1.2-O3
- Algebraic --> Graph 13, 14, 22, O7, O8
2. Given functions f and g, find:
- f+g
- f-g
- fg
- f/g
- domains for above functions. O2, O9
3. Understand and apply compositions of functions:
- Given functions f and g, find g o f and f o g. 28, 39, 60, O3, O6
- Given f o g, find functions f and g. 46, 65, O4

----

== Section 1.5 (James) ==

Homework:

Written: 11, 15, 18, 19, 21, 25, 29
Online: o2, o3, o4, o5, one more word problem

Goals

1. Graph exponential functions a^x for a<1, a>1, as well as transformations of exponential functions from section 1.3. Find domains. 11, 15, 21, o2
2. Given two points that an exponential function passes through, find the equation of the exponential function. 18, o3.
3. Use laws of exponents to simplify exponential expressions, to show various facts about exponential functions. 19, 29.
4. Convert a word problem involving population change into an algebraic expression involving exponential functions. Use this to find sizes of populations at given times, and to graph. 25, o4, o5.

----

== Section 1.6 (McKay) ==

Homework

Written: 12,16,18,19,23,35,45,48,54,60,67,71,73
Online: 01,02,03,04,05,06,07,08,010,011,012,013,016,018

Goals

1) Be able to tell if a function is one-to-one
-algebraically 12
-graphically 01
2) Know the definition of an inverse function and be able to use the steps to find the inverse
-algebraically 16,19,23,54,02,03,04,05,06
-graphically 18,45,73,07
3)Know the laws of logarithms and be able to solve and simplify logarithmic and exponential expressions. 35,48,06,08,010,011,012
4)Know the inverses of the trigonometric functions
- domain and range 71
- values at a point 60, 013,016
- simplify a trig function composed with an inverse trig function 67, o18

----

== Section 2.1 (Tyler) ==

Homework

Written:4*, 5*? (require calculator)

Goals

1. Know definitions of tangent and secant line. Compute the slope of a secant line. 4
2. Know definition of average velocity and the idea of instantaneous velocity. Compute an average velocity. 5
3. Explain how average velocity is equivalent to slope of a secant line, same for instantaneous velocity and slope of tangent line.

----

== Section 2.2 (Tyler) ==

Homework

Written: 6, 7, 9, 16, 27, 32, 34a, 40
Online: kill O2, O5. Modify: O3 fix so different parts don't have same answer.
O6, O7, O8: Use the same input notation for infinity. O8 Delete irrelevant lecture about asymptotes.

Goals:

1. Idea of a limit: (Note: the "definition" of a limit in this section is sloppy and is not a real definition of a limit. See section 2.4)
* Find the limit of a function at a point from its graph, including when the limit does not exist. 6,7, O1, O3
* Recognize the difference between the limit of a function at a point and the value of the function at a point. 6, 7, O1
* Give examples of functions that have prescribed limits at certain points. 16
* Understand that calculators and tables can entice you to guess a wrong limit. Give examples of functions whose limit is not what you would guess from plugging in values to your calculator. (no problems)
2. Idea (and notation) for one-sided limit: Do the same things for one-sided limits as done before for regular limits. 6, 7, O1, O3, O4, O6, O7, O8
3. Explain how one-sided and two-sided limits are connected. Use this to compute limits. 6, 7, O1, O3, O4, O6, O7, O8
4. Infinite limits:
* Explain why infinity is not a number and how the definition of an infinite limit gets around this.
* Find all vertical asymptotes for a function. 9, 34a
* For rational functions, find when a limit is infinity and when it is negative infinity. 27, 32, O6, O7, O8, 40

----

== Section 2.3a (Rebecca) ==

Homework

Written: 10,15,19,20,21,22,28,29
Online: Get rid of 3

Goals

1. Be able to apply limit laws to simplify limits
- algebraically 01
- graphically 02
2. Recognize when you can directly substitute to compute a limit and when you cannot. 10, 03, 04, 05, 06
3. Use algebra to simply and find limits.
- for polynomials over polynomials. 15, 19, 20, o5, o6, o7, o8, o9, o10
- expressions with square roots. 21, 22, 29, o11
- expressions with multiple fractions. 28, 29, o12

----

== Section 2.3b (Drew) ==

Homework

Written: 36-39, 42, 55, 56, 58
Online: Add o2 from 2.2. Kill o3, o4. Modify o6 so that direct substitution doesn't work. Add two problems using substitution.

Goals

1. Use the Squeeze Theorem to find limits. 36, 37, 38, o5, o6, o7
2. Evaluate limits that involve absolute values and other piecewise functions. 39, 42, o1, o2, o2 from 2.2
3. Use limit laws to find lim f(x) given that lim g(f(x)) = L. 55, 56, 58
4. Use substitution to rewrite lim g(f(x)) as lim g(u). 2 new online problems.

----

== Section 2.4 (Jessica) ==
Homework:
Written: 2, 14, 15, 16, 19, 20
Online: kill o6 (done), modify problems to find the *largest* delta (done).

Goals:
1. Use a graph showing a function and lines y=L+epsilon, y=L-epsilon to find the largest value of delta so that if 0<|x-a|<delta, then |f(x)-L|<epsilon. 2
2. Know the epsilon-delta definition of a limit. Given a limit and a value for epsilon, be able to find the largest value of delta corresponding to that epsilon. 14, o1, o2, o3, o4, o5
3. Use the epsilon-delta definition of a limit to prove a statement about the limit of a linear function. 15, 16, 19, 20.

----

== Section 2.5a (McKay/Savannah) ==

Written:4, 6, 15, 18, 22, 23, 43ab, 58
Online - Kill O2-O4, O11-O16; Change O8, O9, O10 to ask, "Where is the following function continuous?"; Add a graph problem where the student must identify where the graph is discontinuous; Add a graph problem where the student must identify the types of discontinuities

1. Given the graph of a function, be able to tell:
a) where it is continuous. 4
b) where it is discontinuous, and the type of discontinuity. 6
2. Given a function described algebraically, be able to tell:
a) where it is continuous. 22, 23, O1, O8, O9, O10
b) where it is discontinuous and types of discontinuities. 15, 18, 43(a,b), O5, O6, O7
This includes functions from Theorem 7, piecewise functions, and combinations of functions.
3. Use limit laws and the definition of continuity to prove facts about continuity (e.g. Theorem 4). 58

----

== Section 2.5b (Mark) ==

Homework

Written: 28, 31, 32, 36, 37, 39, 41, 45, 47, 49, 65
Online: Kill 3, 5, 6, 7. Add 3 problems: Find intervals where f is positive, negative. One quadratic, easy factor; one rational with linear top and bottom; one rational with quadratic top, perfect square bottom.

Goals

1. Use continuity to evaluate limits 31,32
2. Continuity of piecewise functions
-determine if a piecewise function is continuous (or continuous from the right/left) 36,37,39
-determine parameters to make a piecewise function continuous 41,O1,O2,O4
3. Given a composition of functions, tell where it is continuous/discontinuous. 28
4. Use the Intermediate Value Theorem to prove the existence of solutions to equations. 45,47,49,65
5. Find intervals where a continuous function is positive/negative. (3 new online problems)

----

== Section 2.6 (Skyler) ==

Homework:
Written: 4, 5, 7, 13, 33, 43, 48
Online: Drop #15

Goals

1. Understand definition of limit at +/- inf
2. Find the limit of a rational function as x -> +/-inf and equations of horizontal asymptotes (when they exist)
3. Identify functions whose limit at +/- inf does not exist or whose limit is infinite
4. Compute limits at infinity by graphs and algebraic techniques

----

== Section 2.7 (Sam) ==
Homework:
Written: 5,9,19, 20, 43,44, 46
Online: eliminate o7,o9,o12

Goals:
1. Given an algebraic function, take the derivative (using the definition of the derivative)
Write down an equation for a tangent line through a point on a graph 3, 5, 05, o10, o11
2. Given a real life situation, interpret instantaneous change or average of change 43, 44, 46
Given a position function (or graph thereof), interpret the graph or find (average) velocity o1, o2, o3
3. Sketch the graph of a function such that certain criteria are met (values at certain points, derivatives, etc.) 19, 20
4. Given a limit expression, write down the point and function whose derivative it represents o6, o8
5. Write both formal definitions of a derivative
Explain how the derivative can be interpreted as slope of a line, velocity, instantaneous change

Changes:
Added: 43, 44
Eliminating: 2,25, o7, o9, o12

----

== Section 2.8 (Savannah) ==

Homework

Written: 1, 3, 5, 6, 7, 9, 11, 19, 20, 25, 35, 36, 37, 38, 44, Prove that if a function f is differentiable at a, then f is continuous at a.
Online: Kill O2, O3, Change O9 to f'(x)

Goals

1. Given a function f, find a formula for the derivative of f using f'(x) = lim (h->0) [f(x+h)-f(x)]/h, and be able to state the domain of f'(x). 19, 20, 25, O9, O10
2. Given a graph of a function f, be able to find the graph of f', f''. Given a graph of a function f', f'', be able to find th graph of f. 1, 3, 5, 6, 7, 9, 11, O1, O4, O5
3. Recognize when and why a function fails to be differentiable: 35, 36, 37, 38
a) corner
b) discontinuity (Theorem 4)
c) vertical tangent line
4. Be able to compute and apply higher derivatives: O6, O7, O8, 44
- (f')'=f''
- position [s(t)], velocity [v(t)=s'(t)], acceleration [a(t)=v'(t)=s''(t)]
5. Prove that if f is differentiable at a, then f is continuous at a. [Written homework]

----

== Section 3.1 (Drew) ==

Homework:
Written: 2, 13, 17, 20, 25, 28, 29, 33, 55, 61, 77
Online: Kill o6 last part, o14, o8

Goals:
1. Compute derivatives using:
a. The power rule. Especially, be able to rewrite functions so the power rule can be applied. 13, 17, 20, 25, 28, 29, o1, o3, o7, o8, o9, o10, o11, o12
b. The derivative of e^x. 2, 17, 28, o12
c. Sums, differences, and scalar multiples of functions you can differentiate with these rules. (almost all problems)
2. Use the definition of the derivative and limit laws to:
a. Prove simple rules, including the power rule for n=-2, -1, -1/2, 1/2, 1, 2, as well as sum and difference rules. 61
b. Compute certain limits. 77
3. Find a tangent line to the graph of a function at a given point. 33, 55, o2, o4, o5

----

== Section 3.2 (Rebecca) ==

Homework

Written: 2,11,13,23,24,32,33,47,49,55,57
Online: As listed.

Goals

1. Be able to apply the product rule to find the derivative of functions. 11,24,33,47,49,55,57, 02,04,05,06,07,09
2. Be able to apply the quotient rule to find the derivative of functions. 2,13,23,24,32,47,49, 01,03,08,010,011
3. Be able to prove the product and quotient rules and variations. 55,57*

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== Section 3.3 (Skyler/James) ==

Homework

Written: 9, 10, 18, 20, 35, 42, 45, 49
Online: As listed. Change ordinal number suffixes on #3 (e.g. 83th makes no sense)

Goals

1. Know derivatives for the 6 basic trig functions. Be able to use these to compute derivatives of more complicated functions.
2. Know limits [2] and [3] in the book. Be able to use these to solve other limits involving trig functions
3. Know the "periodicity" of differentiation of trig functions--i.e. the fourth derivative of the sine function is again the sine function
4. Be able to use trig identities to find derivatives and limits
5. Use trig derivatives in various real world problems.

----

== Section 3.4 (Savannah) ==

Homework

Written: 7, 24, 25, 31, 37, 47, 63, 65, 71, 84, 89
Online: Kill O6, O7, O8; Reorder problems (see Dr. Purcell); Add #77 from the book to online (change "particle" to "point")

Goals

1. Fluency in chain rule: Given a composition of functions, use the chain rule and other rules to compute its derivative quickly. 7, 25, 37, 47, O1, O2, O3, O9, O10, O11, O12, O13, O14, O15, O16
2. Given values of two functions and their derivatives at certain points, use the formal statement of the chain rule to compute the derivative of compositions of these functions. 63, 65, 71, O4, O5
3. Given a word problem involving the composition of functions, compute rates of change. Interpret the meaning of the derivative as a particular rate of change. 84
4. Use the chain rule and definition of even/odd functions to prove properties of derivatives of even/odd functions. 89
5. Compute the derivative of a^x. 24, 31

----

== Section 3.5 (Rebecca) ==
Homework

Written: 3,11,12,15,18,21,23,34,41,57 (Add 43,67)
Online: Kill 03,014; Replace 08 with 35 from the book and make it online

Goals

1) For an implicitly defined equation find dy/dx, dx/dy, and d^2y/dx^2 using implicit differentiation.
2) Find the tangent line to an implicitly defined function.
3) Know the derivatives of the inverse trig functions.
----

== Section 3.6 (Sam) ==

----

== Section 3.8 (Jessica) ==

Homework:

Written: 14, 15
Online: kill o5 (pH scale)

Goals:

Use exponential functions to model each of the following:
1. Population growth. o1, o3, o4
2. Radioactive decay. o6, o7
Find half-life, or given half-life, find equation.
3. Cooling. 14, 15
4. Interest. o2, o8
- Derive the formula for compounding interest in n periods o2, o8 (see also section 1.3, number 60)
- Use expression of e as a limit to find formula for continuously compounded interest.
----

== Section 3.9 (Mark) ==

Homework

Written: 5, 22, 33, 35, 37, 42
Online: Kill O1, O4, O7, O8, O11, O12, O13; remove geometry formulas from online questions.

Goals

1. Solve related rates problems. (all homework problems)
a) Draw picture
b) Recognize variables from the problem, and rates of change as their derivatives.
c) Write equations relating variables (using geometry, trigonometry)
d) Differentiate, remembering to use the chain rule.
e) Plug in specific values to determine the requested answer.

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== Section 3.10 (Tyler) ==

Homework
Written 1,2,3,5,23,28,43,44
Online Kill O4, O5, Renumber so that O3 is first and O1 is third. Possibly add one more. Change the hint in O1 to read "Try using linear approximation..."

Goals
1. Compute the linearization of a function at a point. 1,2,3,5, O2, O3
- Explain the relationship between the linearization and the tangent line. 5, O2
- Explain how linearization is useful. 23,28
2. Use the linearization at a point x=a to approximate the value of a function f(a+epsilon). 23,28, O1, O2

Suggestion: merge 3.10 and 3.11 for written as well as online.
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== Section 3.11 (Savannah) ==

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== Section 4.1 ==

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== Section 4.2 ==

----

== Section 4.3 ==

Goals:

1. Know and apply the increasing/decreasing test o1, o5
2. Find local maxima and minima (First and Second derivative tests, endpoints) 21,23
3. Know the definition of concavity and how to determine concavity (The concavity test) o2, o5, 72
4. Know what an inflection point is and how to find it o2
5. Know how the above information affects the graph of a function
Draw a graph such that.... 25, 28
Given a graph, where is concavity positive, etc.? 1, 6, 7, 31 and a new online problem similar to problem 8 on page 295

Changes:

o3 and o4 are to be moved to section 4.5 and a problem is to be added online that is like problem 8.
----

== Section 4.4 ==

----

== Section 4.5 (Skyler) ==

Goals:

Use all of the following to draw graphs of functions:

* Algebraic Properties (Domain, intercepts, even/odd/periodic)

* Limit Properties (Asymptotes, singularities, discontinuities)

* Derivative Properties (Increasing/decreasing, extrema, concavity/inflection)

Written homework - as listed
Online homework - pulled from 4.3 (O3 and O4)

----

== Section 4.7 ==

----

== Section 4.8 (Mark) ==

Goals

1. Use Newton's Method to approximate roots/solutions of equations.
2. Derive the formula for Newton's Method from the tangent line equation.
3. Give an example of a function 'f' and a starting point 'x_1' that makes Newton's Method fail.

Homework

Written: 1, 2, 3, 4, 29.
Online: Kill O1, Add new online problem with graph.

----

== Section 4.9 (Tyler)==

Goals:
1. Use the Mean value Theorem to prove Theorem 1.
2. Memorize the table of anti-differentiation formulas on page 341 and
relate it to the table of derivatives on the inside back cover of the book. 12, 15, 20, 27, 33, 49, 51, O1-O12
3. Use the table and Theorem 1 to find *all* antiderivatives of various functions. 12, 15, 20, 27, 49, 51
4. Use Theorem 1 and known differentiation formulas to find a unique anti-derivative with given values at specified points. 33, 53, 67,
69, 70, 74,O2,O3, O4,O5,O10, O11, O13
5. Use anti-differentiation to solve physical and economic problems, especially problems of motion. 67, 69, 70, 74, O4, O5

Changes to problems:
Written: Add story problems: 67, 69, 70, 74
Online: too long, Delete O1, O6, O12.
Adjust numbers in 04 to give a clean answer.
O10--change so that F(0) not equal to 0 (else they will get right answer for wrong reason).
O8 and O9 problem that Arctan/Arcsin not accepted--just arctan/arcsin.
O13 should not say "the anti-derivative, but rather "an anti-derivative".

----

== Appendix E (Paul)==
Homework:

Written: 1, 3, 5, 10, 11, 15, 17, 20, 23, 41, 43
Online: o1, o2, o3, o4, o5, o6, o7, o8, o9, o10

Goals

1. Write expanded sums in sigma notation (1, 3, 5, 10)
and expand sums written in sigma notation (11, 15, 17, 20).
Numerically evaluate sums (23). Apply appropriate rules to
distribute constants or expand sums of sums (o2, o4, o5, o8, o9, o10).
2. Explicitly evaluate $\sigma_{i=1}^n f(i)$ in terms of 'n', where 'f(x)'
is a polynomial in x of degree at most 2. (Degree 0 o3, o7; degree 1
o1, o2, o10; degree 2 o4, o5.)
Numerically evaluate this sum when 'i' runs over a given range of integers (o6, o8, o9).
3. Evaluate telescoping sums (41) and limits of sums (43).

----

== Section 5.1 ==

----

== Section 5.2 ==

----

== Section 5.3 ==

----

== Section 5.4 ==

----

== Section 5.5 ==

----

Math 112 Calculus Learning Goals

2009-08-19T20:07:45Z

Pmj5: /* Appendix E (Paul) */

----

== Section 1.1 (Jessica) ==

Homework:

Written: 25, 31, 40, 43, 55, 56, 64, 65
Online: 3, 4, 5, 6, 8, 9, 11. Modify 12 (Given f like 66 or 67 in book, find f(-x). Is f even or odd or neither?) Add problem like 51, 57 in book.

Goals

1. Given a function described algebraically, find the
- domain 31, 07, 08,
- range,
- value at a given number 04,
- value when we plug in another function, e.g. difference quotient 04, 05, 25.
2. Convert one representation of a function to another.
- verbal to/from graphical 03
-algebraic to/from graphical 09
-verbal to/from algebraic 55, 56, 2 new online
3. Piecewise functions: Given an algebraic description of a piecewise function, find its domain and sketch its graph. 011, 43.
Be able to write the absolute value as a piecewise function, and use this to sketch its graph. 40, 09
4. Use the definition of an even/odd function to decide whether a function described algebraically is even or odd. 012, 65
Use symmetry to decide whether a graph is an even or odd function. 64

----

== Section 1.2 (Sam) ==

Homework:

Written: 4,5,7,9,12 and Appdx D: 24, 37*, 46
Online: O1, O2, O4, O5 (these are the numbers under the current numbering),
and Appdx D: 29, 60, 69
The * on problem 37 means a calculator will be necessary.

Goals

1. Writing down a linear function given a set of information 5,12,O4,O5
2. Recognizing what a function's graph should look like 4,7,O1
3. Based on a graph, write down a function O2
4. Writing down a polynomial based on information 9, O2
5. Trig review (definitions, proofs, values of trig functions at particular points) Appdx D:24, 37, 46, 29, 60, 69 (the latter three going online)

# Trig to know, including problems to skim for practice. (Jessica)

1. Given angles in degrees and radians, draw the angle, convert from radians to degrees and vice versa. (See appendix D: 1-12)
2. Write down sin, cos, tan, sec, scs, cot of any angle 0, pi/6, pi/4, pi/3, pi/2, and all angles obtained from these angles by adding a multiple of pi/2. (appendix D:23-28)
3. Given a right triangle with side measurements, write sin, cos, tan, sec, csc, cot of any of the angles of the triangle in terms of the side lengths. (Appendix D: 35-38)
Similarly, given one trig ratio, find the others. (Appendix D: 29-34)
4. Graph trig functions. (app D: 77-82)
5. Use trig identities to simplify expressions (appendix D: 43-57 odd, 59-63), and to solve equations and inequalities involving trig functions (app D: 65-76).
* Most important identities: sin^2(x) + cos^2(x) = 1
* sin(-x) = -sin(x)
* cos(-x) = cos(x)
* sin(x+y) = sin(x)cos(y) + cos(x)sin(y)
* cos(x+y) = cos(x)cos(y) - sin(x)sin(y)
Using the above, you can derive other identities:
* 1+tan^2(x)=sec^2(x); 1+cot^2(x) = csc^2(x)
* sin(x-y)
* cos(x-y)
* sin(2x)
* cos(2x)

----

== Section 1.3 (Savannah) ==

Homework

Written: 1,7,13, 14, 22, 39, 46, 60, 65
Online: Modify 1 to be y as a function of x. Replace 6 with a different word problem. Add 3 from section 1.2 online assignment.

Goals

1. Apply and recognize transformations of functions:
- Translating, Stretching, Reflecting, Absolute Value 1
- Graph --> Algebraic 7, O1, O5, 1.2-O3
- Algebraic --> Graph 13, 14, 22, O7, O8
2. Given functions f and g, find:
- f+g
- f-g
- fg
- f/g
- domains for above functions. O2, O9
3. Understand and apply compositions of functions:
- Given functions f and g, find g o f and f o g. 28, 39, 60, O3, O6
- Given f o g, find functions f and g. 46, 65, O4

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== Section 1.5 (James) ==

Homework:

Written: 11, 15, 18, 19, 21, 25, 29
Online: o2, o3, o4, o5, one more word problem

Goals

1. Graph exponential functions a^x for a<1, a>1, as well as transformations of exponential functions from section 1.3. Find domains. 11, 15, 21, o2
2. Given two points that an exponential function passes through, find the equation of the exponential function. 18, o3.
3. Use laws of exponents to simplify exponential expressions, to show various facts about exponential functions. 19, 29.
4. Convert a word problem involving population change into an algebraic expression involving exponential functions. Use this to find sizes of populations at given times, and to graph. 25, o4, o5.

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== Section 1.6 (McKay) ==

Homework

Written: 12,16,18,19,23,35,45,48,54,60,67,71,73
Online: 01,02,03,04,05,06,07,08,010,011,012,013,016,018

Goals

1) Be able to tell if a function is one-to-one
-algebraically 12
-graphically 01
2) Know the definition of an inverse function and be able to use the steps to find the inverse
-algebraically 16,19,23,54,02,03,04,05,06
-graphically 18,45,73,07
3)Know the laws of logarithms and be able to solve and simplify logarithmic and exponential expressions. 35,48,06,08,010,011,012
4)Know the inverses of the trigonometric functions
- domain and range 71
- values at a point 60, 013,016
- simplify a trig function composed with an inverse trig function 67, o18

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== Section 2.1 (Tyler) ==

Homework

Written:4*, 5*? (require calculator)

Goals

1. Know definitions of tangent and secant line. Compute the slope of a secant line. 4
2. Know definition of average velocity and the idea of instantaneous velocity. Compute an average velocity. 5
3. Explain how average velocity is equivalent to slope of a secant line, same for instantaneous velocity and slope of tangent line.

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== Section 2.2 (Tyler) ==

Homework

Written: 6, 7, 9, 16, 27, 32, 34a, 40
Online: kill O2, O5. Modify: O3 fix so different parts don't have same answer.
O6, O7, O8: Use the same input notation for infinity. O8 Delete irrelevant lecture about asymptotes.

Goals:

1. Idea of a limit: (Note: the "definition" of a limit in this section is sloppy and is not a real definition of a limit. See section 2.4)
* Find the limit of a function at a point from its graph, including when the limit does not exist. 6,7, O1, O3
* Recognize the difference between the limit of a function at a point and the value of the function at a point. 6, 7, O1
* Give examples of functions that have prescribed limits at certain points. 16
* Understand that calculators and tables can entice you to guess a wrong limit. Give examples of functions whose limit is not what you would guess from plugging in values to your calculator. (no problems)
2. Idea (and notation) for one-sided limit: Do the same things for one-sided limits as done before for regular limits. 6, 7, O1, O3, O4, O6, O7, O8
3. Explain how one-sided and two-sided limits are connected. Use this to compute limits. 6, 7, O1, O3, O4, O6, O7, O8
4. Infinite limits:
* Explain why infinity is not a number and how the definition of an infinite limit gets around this.
* Find all vertical asymptotes for a function. 9, 34a
* For rational functions, find when a limit is infinity and when it is negative infinity. 27, 32, O6, O7, O8, 40

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== Section 2.3a (Rebecca) ==

Homework

Written: 10,15,19,20,21,22,28,29
Online: Get rid of 3

Goals

1. Be able to apply limit laws to simplify limits
- algebraically 01
- graphically 02
2. Recognize when you can directly substitute to compute a limit and when you cannot. 10, 03, 04, 05, 06
3. Use algebra to simply and find limits.
- for polynomials over polynomials. 15, 19, 20, o5, o6, o7, o8, o9, o10
- expressions with square roots. 21, 22, 29, o11
- expressions with multiple fractions. 28, 29, o12

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== Section 2.3b (Drew) ==

Homework

Written: 36-39, 42, 55, 56, 58
Online: Add o2 from 2.2. Kill o3, o4. Modify o6 so that direct substitution doesn't work. Add two problems using substitution.

Goals

1. Use the Squeeze Theorem to find limits. 36, 37, 38, o5, o6, o7
2. Evaluate limits that involve absolute values and other piecewise functions. 39, 42, o1, o2, o2 from 2.2
3. Use limit laws to find lim f(x) given that lim g(f(x)) = L. 55, 56, 58
4. Use substitution to rewrite lim g(f(x)) as lim g(u). 2 new online problems.

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== Section 2.4 (Jessica) ==
Homework:
Written: 2, 14, 15, 16, 19, 20
Online: kill o6 (done), modify problems to find the *largest* delta (done).

Goals:
1. Use a graph showing a function and lines y=L+epsilon, y=L-epsilon to find the largest value of delta so that if 0<|x-a|<delta, then |f(x)-L|<epsilon. 2
2. Know the epsilon-delta definition of a limit. Given a limit and a value for epsilon, be able to find the largest value of delta corresponding to that epsilon. 14, o1, o2, o3, o4, o5
3. Use the epsilon-delta definition of a limit to prove a statement about the limit of a linear function. 15, 16, 19, 20.

----

== Section 2.5a (McKay/Savannah) ==

Written:4, 6, 15, 18, 22, 23, 43ab, 58
Online - Kill O2-O4, O11-O16; Change O8, O9, O10 to ask, "Where is the following function continuous?"; Add a graph problem where the student must identify where the graph is discontinuous; Add a graph problem where the student must identify the types of discontinuities

1. Given the graph of a function, be able to tell:
a) where it is continuous. 4
b) where it is discontinuous, and the type of discontinuity. 6
2. Given a function described algebraically, be able to tell:
a) where it is continuous. 22, 23, O1, O8, O9, O10
b) where it is discontinuous and types of discontinuities. 15, 18, 43(a,b), O5, O6, O7
This includes functions from Theorem 7, piecewise functions, and combinations of functions.
3. Use limit laws and the definition of continuity to prove facts about continuity (e.g. Theorem 4). 58

----

== Section 2.5b (Mark) ==

Homework

Written: 28, 31, 32, 36, 37, 39, 41, 45, 47, 49, 65
Online: Kill 3, 5, 6, 7. Add 3 problems: Find intervals where f is positive, negative. One quadratic, easy factor; one rational with linear top and bottom; one rational with quadratic top, perfect square bottom.

Goals

1. Use continuity to evaluate limits 31,32
2. Continuity of piecewise functions
-determine if a piecewise function is continuous (or continuous from the right/left) 36,37,39
-determine parameters to make a piecewise function continuous 41,O1,O2,O4
3. Given a composition of functions, tell where it is continuous/discontinuous. 28
4. Use the Intermediate Value Theorem to prove the existence of solutions to equations. 45,47,49,65
5. Find intervals where a continuous function is positive/negative. (3 new online problems)

----

== Section 2.6 (Skyler) ==

Homework:
Written: 4, 5, 7, 13, 33, 43, 48
Online: Drop #15

Goals

1. Understand definition of limit at +/- inf
2. Find the limit of a rational function as x -> +/-inf and equations of horizontal asymptotes (when they exist)
3. Identify functions whose limit at +/- inf does not exist or whose limit is infinite
4. Compute limits at infinity by graphs and algebraic techniques

----

== Section 2.7 (Sam) ==
Homework:
Written: 5,9,19, 20, 43,44, 46
Online: eliminate o7,o9,o12

Goals:
1. Given an algebraic function, take the derivative (using the definition of the derivative)
Write down an equation for a tangent line through a point on a graph 3, 5, 05, o10, o11
2. Given a real life situation, interpret instantaneous change or average of change 43, 44, 46
Given a position function (or graph thereof), interpret the graph or find (average) velocity o1, o2, o3
3. Sketch the graph of a function such that certain criteria are met (values at certain points, derivatives, etc.) 19, 20
4. Given a limit expression, write down the point and function whose derivative it represents o6, o8
5. Write both formal definitions of a derivative
Explain how the derivative can be interpreted as slope of a line, velocity, instantaneous change

Changes:
Added: 43, 44
Eliminating: 2,25, o7, o9, o12

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== Section 2.8 (Savannah) ==

Homework

Written: 1, 3, 5, 6, 7, 9, 11, 19, 20, 25, 35, 36, 37, 38, 44, Prove that if a function f is differentiable at a, then f is continuous at a.
Online: Kill O2, O3, Change O9 to f'(x)

Goals

1. Given a function f, find a formula for the derivative of f using f'(x) = lim (h->0) [f(x+h)-f(x)]/h, and be able to state the domain of f'(x). 19, 20, 25, O9, O10
2. Given a graph of a function f, be able to find the graph of f', f''. Given a graph of a function f', f'', be able to find th graph of f. 1, 3, 5, 6, 7, 9, 11, O1, O4, O5
3. Recognize when and why a function fails to be differentiable: 35, 36, 37, 38
a) corner
b) discontinuity (Theorem 4)
c) vertical tangent line
4. Be able to compute and apply higher derivatives: O6, O7, O8, 44
- (f')'=f''
- position [s(t)], velocity [v(t)=s'(t)], acceleration [a(t)=v'(t)=s''(t)]
5. Prove that if f is differentiable at a, then f is continuous at a. [Written homework]

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== Section 3.1 (Drew) ==

Homework:
Written: 2, 13, 17, 20, 25, 28, 29, 33, 55, 61, 77
Online: Kill o6 last part, o14, o8

Goals:
1. Compute derivatives using:
a. The power rule. Especially, be able to rewrite functions so the power rule can be applied. 13, 17, 20, 25, 28, 29, o1, o3, o7, o8, o9, o10, o11, o12
b. The derivative of e^x. 2, 17, 28, o12
c. Sums, differences, and scalar multiples of functions you can differentiate with these rules. (almost all problems)
2. Use the definition of the derivative and limit laws to:
a. Prove simple rules, including the power rule for n=-2, -1, -1/2, 1/2, 1, 2, as well as sum and difference rules. 61
b. Compute certain limits. 77
3. Find a tangent line to the graph of a function at a given point. 33, 55, o2, o4, o5

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== Section 3.2 (Rebecca) ==

Homework

Written: 2,11,13,23,24,32,33,47,49,55,57
Online: As listed.

Goals

1. Be able to apply the product rule to find the derivative of functions. 11,24,33,47,49,55,57, 02,04,05,06,07,09
2. Be able to apply the quotient rule to find the derivative of functions. 2,13,23,24,32,47,49, 01,03,08,010,011
3. Be able to prove the product and quotient rules and variations. 55,57*

----

== Section 3.3 (Skyler/James) ==

Homework

Written: 9, 10, 18, 20, 35, 42, 45, 49
Online: As listed. Change ordinal number suffixes on #3 (e.g. 83th makes no sense)

Goals

1. Know derivatives for the 6 basic trig functions. Be able to use these to compute derivatives of more complicated functions.
2. Know limits [2] and [3] in the book. Be able to use these to solve other limits involving trig functions
3. Know the "periodicity" of differentiation of trig functions--i.e. the fourth derivative of the sine function is again the sine function
4. Be able to use trig identities to find derivatives and limits
5. Use trig derivatives in various real world problems.

----

== Section 3.4 (Savannah) ==

Homework

Written: 7, 24, 25, 31, 37, 47, 63, 65, 71, 84, 89
Online: Kill O6, O7, O8; Reorder problems (see Dr. Purcell); Add #77 from the book to online (change "particle" to "point")

Goals

1. Fluency in chain rule: Given a composition of functions, use the chain rule and other rules to compute its derivative quickly. 7, 25, 37, 47, O1, O2, O3, O9, O10, O11, O12, O13, O14, O15, O16
2. Given values of two functions and their derivatives at certain points, use the formal statement of the chain rule to compute the derivative of compositions of these functions. 63, 65, 71, O4, O5
3. Given a word problem involving the composition of functions, compute rates of change. Interpret the meaning of the derivative as a particular rate of change. 84
4. Use the chain rule and definition of even/odd functions to prove properties of derivatives of even/odd functions. 89
5. Compute the derivative of a^x. 24, 31

----

== Section 3.5 (Rebecca) ==
Homework

Written: 3,11,12,15,18,21,23,34,41,57 (Add 43,67)
Online: Kill 03,014; Replace 08 with 35 from the book and make it online

Goals

1) For an implicitly defined equation find dy/dx, dx/dy, and d^2y/dx^2 using implicit differentiation.
2) Find the tangent line to an implicitly defined function.
3) Know the derivatives of the inverse trig functions.
----

== Section 3.6 (Sam) ==

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== Section 3.8 (Jessica) ==

Homework:

Written: 14, 15
Online: kill o5 (pH scale)

Goals:

Use exponential functions to model each of the following:
1. Population growth. o1, o3, o4
2. Radioactive decay. o6, o7
Find half-life, or given half-life, find equation.
3. Cooling. 14, 15
4. Interest. o2, o8
- Derive the formula for compounding interest in n periods o2, o8 (see also section 1.3, number 60)
- Use expression of e as a limit to find formula for continuously compounded interest.
----

== Section 3.9 (Mark) ==

Homework

Written: 5, 22, 33, 35, 37, 42
Online: Kill O1, O4, O7, O8, O11, O12, O13; remove geometry formulas from online questions.

Goals

1. Solve related rates problems. (all homework problems)
a) Draw picture
b) Recognize variables from the problem, and rates of change as their derivatives.
c) Write equations relating variables (using geometry, trigonometry)
d) Differentiate, remembering to use the chain rule.
e) Plug in specific values to determine the requested answer.

----

== Section 3.10 (Tyler) ==

Homework
Written 1,2,3,5,23,28,43,44
Online Kill O4, O5, Renumber so that O3 is first and O1 is third. Possibly add one more. Change the hint in O1 to read "Try using linear approximation..."

Goals
1. Compute the linearization of a function at a point. 1,2,3,5, O2, O3
- Explain the relationship between the linearization and the tangent line. 5, O2
- Explain how linearization is useful. 23,28
2. Use the linearization at a point x=a to approximate the value of a function f(a+epsilon). 23,28, O1, O2

Suggestion: merge 3.10 and 3.11 for written as well as online.
----

== Section 3.11 (Savannah) ==

----

== Section 4.1 ==

----

== Section 4.2 ==

----

== Section 4.3 ==

Goals:

1. Know and apply the increasing/decreasing test o1, o5
2. Find local maxima and minima (First and Second derivative tests, endpoints) 21,23
3. Know the definition of concavity and how to determine concavity (The concavity test) o2, o5, 72
4. Know what an inflection point is and how to find it o2
5. Know how the above information affects the graph of a function
Draw a graph such that.... 25, 28
Given a graph, where is concavity positive, etc.? 1, 6, 7, 31 and a new online problem similar to problem 8 on page 295

Changes:

o3 and o4 are to be moved to section 4.5 and a problem is to be added online that is like problem 8.
----

== Section 4.4 ==

----

== Section 4.5 (Skyler) ==

Goals:

Use all of the following to draw graphs of functions:

* Algebraic Properties (Domain, intercepts, even/odd/periodic)

* Limit Properties (Asymptotes, singularities, discontinuities)

* Derivative Properties (Increasing/decreasing, extrema, concavity/inflection)

Written homework - as listed
Online homework - pulled from 4.3 (O3 and O4)

----

== Section 4.7 ==

----

== Section 4.8 (Mark) ==

Goals

1. Use Newton's Method to approximate roots/solutions of equations.
2. Derive the formula for Newton's Method from the tangent line equation.
3. Give an example of a function 'f' and a starting point 'x_1' that makes Newton's Method fail.

Homework

Written: 1, 2, 3, 4, 29.
Online: Kill O1, Add new online problem with graph.

----

== Section 4.9 (Tyler)==

Goals:
1. Use the Mean value Theorem to prove Theorem 1.
2. Memorize the table of anti-differentiation formulas on page 341 and
relate it to the table of derivatives on the inside back cover of the book. 12, 15, 20, 27, 33, 49, 51, O1-O12
3. Use the table and Theorem 1 to find *all* antiderivatives of various functions. 12, 15, 20, 27, 49, 51
4. Use Theorem 1 and known differentiation formulas to find a unique anti-derivative with given values at specified points. 33, 53, 67,
69, 70, 74,O2,O3, O4,O5,O10, O11, O13
5. Use anti-differentiation to solve physical and economic problems, especially problems of motion. 67, 69, 70, 74, O4, O5

Changes to problems:
Written: Add story problems: 67, 69, 70, 74
Online: too long, Delete O1, O6, O12.
Adjust numbers in 04 to give a clean answer.
O10--change so that F(0) not equal to 0 (else they will get right answer for wrong reason).
O8 and O9 problem that Arctan/Arcsin not accepted--just arctan/arcsin.
O13 should not say "the anti-derivative, but rather "an anti-derivative".

----

== Appendix E (Paul)==
Homework:

Written: 1, 3, 5, 10, 11, 15, 17, 20, 23, 41, 43
Online: o1, o2, o3, o4, o5, o6, o7, o8, o9, o10

Goals

1. Write expanded sums in sigma notation (1, 3, 5, 10)
and expand sums written in sigma notation (11, 15, 17, 20).
Numerically evaluate sums (23, o6, o8, o9). Apply appropriate rules to
distribute constants or expand sums of sums (o2, o4, o5, o8, o9, o10).
2. Explicitly evaluate $\sigma_{i=1}^n f(i)$ in terms of 'n', where 'f(x)'
is a polynomial in x of degree at most 2. (Degree 0 o3, o7; degree 1
o1, o2, o8, o10; degree 2 o4, o5, o6, o9.)
Numerically evaluate this sum when 'i' runs over a given range of integers.
3. Evaluate telescoping sums (41) and limits of sums (43).

----

== Section 5.1 ==

----

== Section 5.2 ==

----

== Section 5.3 ==

----

== Section 5.4 ==

----

== Section 5.5 ==

----

Math 112 Calculus Learning Goals

2009-08-19T19:57:15Z

Pmj5: /* Appendix E (Paul) */

----

== Section 1.1 (Jessica) ==

Homework:

Written: 25, 31, 40, 43, 55, 56, 64, 65
Online: 3, 4, 5, 6, 8, 9, 11. Modify 12 (Given f like 66 or 67 in book, find f(-x). Is f even or odd or neither?) Add problem like 51, 57 in book.

Goals

1. Given a function described algebraically, find the
- domain 31, 07, 08,
- range,
- value at a given number 04,
- value when we plug in another function, e.g. difference quotient 04, 05, 25.
2. Convert one representation of a function to another.
- verbal to/from graphical 03
-algebraic to/from graphical 09
-verbal to/from algebraic 55, 56, 2 new online
3. Piecewise functions: Given an algebraic description of a piecewise function, find its domain and sketch its graph. 011, 43.
Be able to write the absolute value as a piecewise function, and use this to sketch its graph. 40, 09
4. Use the definition of an even/odd function to decide whether a function described algebraically is even or odd. 012, 65
Use symmetry to decide whether a graph is an even or odd function. 64

----

== Section 1.2 (Sam) ==

Homework:

Written: 4,5,7,9,12 and Appdx D: 24, 37*, 46
Online: O1, O2, O4, O5 (these are the numbers under the current numbering),
and Appdx D: 29, 60, 69
The * on problem 37 means a calculator will be necessary.

Goals

1. Writing down a linear function given a set of information 5,12,O4,O5
2. Recognizing what a function's graph should look like 4,7,O1
3. Based on a graph, write down a function O2
4. Writing down a polynomial based on information 9, O2
5. Trig review (definitions, proofs, values of trig functions at particular points) Appdx D:24, 37, 46, 29, 60, 69 (the latter three going online)

# Trig to know, including problems to skim for practice. (Jessica)

1. Given angles in degrees and radians, draw the angle, convert from radians to degrees and vice versa. (See appendix D: 1-12)
2. Write down sin, cos, tan, sec, scs, cot of any angle 0, pi/6, pi/4, pi/3, pi/2, and all angles obtained from these angles by adding a multiple of pi/2. (appendix D:23-28)
3. Given a right triangle with side measurements, write sin, cos, tan, sec, csc, cot of any of the angles of the triangle in terms of the side lengths. (Appendix D: 35-38)
Similarly, given one trig ratio, find the others. (Appendix D: 29-34)
4. Graph trig functions. (app D: 77-82)
5. Use trig identities to simplify expressions (appendix D: 43-57 odd, 59-63), and to solve equations and inequalities involving trig functions (app D: 65-76).
* Most important identities: sin^2(x) + cos^2(x) = 1
* sin(-x) = -sin(x)
* cos(-x) = cos(x)
* sin(x+y) = sin(x)cos(y) + cos(x)sin(y)
* cos(x+y) = cos(x)cos(y) - sin(x)sin(y)
Using the above, you can derive other identities:
* 1+tan^2(x)=sec^2(x); 1+cot^2(x) = csc^2(x)
* sin(x-y)
* cos(x-y)
* sin(2x)
* cos(2x)

----

== Section 1.3 (Savannah) ==

Homework

Written: 1,7,13, 14, 22, 39, 46, 60, 65
Online: Modify 1 to be y as a function of x. Replace 6 with a different word problem. Add 3 from section 1.2 online assignment.

Goals

1. Apply and recognize transformations of functions:
- Translating, Stretching, Reflecting, Absolute Value 1
- Graph --> Algebraic 7, O1, O5, 1.2-O3
- Algebraic --> Graph 13, 14, 22, O7, O8
2. Given functions f and g, find:
- f+g
- f-g
- fg
- f/g
- domains for above functions. O2, O9
3. Understand and apply compositions of functions:
- Given functions f and g, find g o f and f o g. 28, 39, 60, O3, O6
- Given f o g, find functions f and g. 46, 65, O4

----

== Section 1.5 (James) ==

Homework:

Written: 11, 15, 18, 19, 21, 25, 29
Online: o2, o3, o4, o5, one more word problem

Goals

1. Graph exponential functions a^x for a<1, a>1, as well as transformations of exponential functions from section 1.3. Find domains. 11, 15, 21, o2
2. Given two points that an exponential function passes through, find the equation of the exponential function. 18, o3.
3. Use laws of exponents to simplify exponential expressions, to show various facts about exponential functions. 19, 29.
4. Convert a word problem involving population change into an algebraic expression involving exponential functions. Use this to find sizes of populations at given times, and to graph. 25, o4, o5.

----

== Section 1.6 (McKay) ==

Homework

Written: 12,16,18,19,23,35,45,48,54,60,67,71,73
Online: 01,02,03,04,05,06,07,08,010,011,012,013,016,018

Goals

1) Be able to tell if a function is one-to-one
-algebraically 12
-graphically 01
2) Know the definition of an inverse function and be able to use the steps to find the inverse
-algebraically 16,19,23,54,02,03,04,05,06
-graphically 18,45,73,07
3)Know the laws of logarithms and be able to solve and simplify logarithmic and exponential expressions. 35,48,06,08,010,011,012
4)Know the inverses of the trigonometric functions
- domain and range 71
- values at a point 60, 013,016
- simplify a trig function composed with an inverse trig function 67, o18

----

== Section 2.1 (Tyler) ==

Homework

Written:4*, 5*? (require calculator)

Goals

1. Know definitions of tangent and secant line. Compute the slope of a secant line. 4
2. Know definition of average velocity and the idea of instantaneous velocity. Compute an average velocity. 5
3. Explain how average velocity is equivalent to slope of a secant line, same for instantaneous velocity and slope of tangent line.

----

== Section 2.2 (Tyler) ==

Homework

Written: 6, 7, 9, 16, 27, 32, 34a, 40
Online: kill O2, O5. Modify: O3 fix so different parts don't have same answer.
O6, O7, O8: Use the same input notation for infinity. O8 Delete irrelevant lecture about asymptotes.

Goals:

1. Idea of a limit: (Note: the "definition" of a limit in this section is sloppy and is not a real definition of a limit. See section 2.4)
* Find the limit of a function at a point from its graph, including when the limit does not exist. 6,7, O1, O3
* Recognize the difference between the limit of a function at a point and the value of the function at a point. 6, 7, O1
* Give examples of functions that have prescribed limits at certain points. 16
* Understand that calculators and tables can entice you to guess a wrong limit. Give examples of functions whose limit is not what you would guess from plugging in values to your calculator. (no problems)
2. Idea (and notation) for one-sided limit: Do the same things for one-sided limits as done before for regular limits. 6, 7, O1, O3, O4, O6, O7, O8
3. Explain how one-sided and two-sided limits are connected. Use this to compute limits. 6, 7, O1, O3, O4, O6, O7, O8
4. Infinite limits:
* Explain why infinity is not a number and how the definition of an infinite limit gets around this.
* Find all vertical asymptotes for a function. 9, 34a
* For rational functions, find when a limit is infinity and when it is negative infinity. 27, 32, O6, O7, O8, 40

----

== Section 2.3a (Rebecca) ==

Homework

Written: 10,15,19,20,21,22,28,29
Online: Get rid of 3

Goals

1. Be able to apply limit laws to simplify limits
- algebraically 01
- graphically 02
2. Recognize when you can directly substitute to compute a limit and when you cannot. 10, 03, 04, 05, 06
3. Use algebra to simply and find limits.
- for polynomials over polynomials. 15, 19, 20, o5, o6, o7, o8, o9, o10
- expressions with square roots. 21, 22, 29, o11
- expressions with multiple fractions. 28, 29, o12

----

== Section 2.3b (Drew) ==

Homework

Written: 36-39, 42, 55, 56, 58
Online: Add o2 from 2.2. Kill o3, o4. Modify o6 so that direct substitution doesn't work. Add two problems using substitution.

Goals

1. Use the Squeeze Theorem to find limits. 36, 37, 38, o5, o6, o7
2. Evaluate limits that involve absolute values and other piecewise functions. 39, 42, o1, o2, o2 from 2.2
3. Use limit laws to find lim f(x) given that lim g(f(x)) = L. 55, 56, 58
4. Use substitution to rewrite lim g(f(x)) as lim g(u). 2 new online problems.

----

== Section 2.4 (Jessica) ==
Homework:
Written: 2, 14, 15, 16, 19, 20
Online: kill o6 (done), modify problems to find the *largest* delta (done).

Goals:
1. Use a graph showing a function and lines y=L+epsilon, y=L-epsilon to find the largest value of delta so that if 0<|x-a|<delta, then |f(x)-L|<epsilon. 2
2. Know the epsilon-delta definition of a limit. Given a limit and a value for epsilon, be able to find the largest value of delta corresponding to that epsilon. 14, o1, o2, o3, o4, o5
3. Use the epsilon-delta definition of a limit to prove a statement about the limit of a linear function. 15, 16, 19, 20.

----

== Section 2.5a (McKay/Savannah) ==

Written:4, 6, 15, 18, 22, 23, 43ab, 58
Online - Kill O2-O4, O11-O16; Change O8, O9, O10 to ask, "Where is the following function continuous?"; Add a graph problem where the student must identify where the graph is discontinuous; Add a graph problem where the student must identify the types of discontinuities

1. Given the graph of a function, be able to tell:
a) where it is continuous. 4
b) where it is discontinuous, and the type of discontinuity. 6
2. Given a function described algebraically, be able to tell:
a) where it is continuous. 22, 23, O1, O8, O9, O10
b) where it is discontinuous and types of discontinuities. 15, 18, 43(a,b), O5, O6, O7
This includes functions from Theorem 7, piecewise functions, and combinations of functions.
3. Use limit laws and the definition of continuity to prove facts about continuity (e.g. Theorem 4). 58

----

== Section 2.5b (Mark) ==

Homework

Written: 28, 31, 32, 36, 37, 39, 41, 45, 47, 49, 65
Online: Kill 3, 5, 6, 7. Add 3 problems: Find intervals where f is positive, negative. One quadratic, easy factor; one rational with linear top and bottom; one rational with quadratic top, perfect square bottom.

Goals

1. Use continuity to evaluate limits 31,32
2. Continuity of piecewise functions
-determine if a piecewise function is continuous (or continuous from the right/left) 36,37,39
-determine parameters to make a piecewise function continuous 41,O1,O2,O4
3. Given a composition of functions, tell where it is continuous/discontinuous. 28
4. Use the Intermediate Value Theorem to prove the existence of solutions to equations. 45,47,49,65
5. Find intervals where a continuous function is positive/negative. (3 new online problems)

----

== Section 2.6 (Skyler) ==

Homework:
Written: 4, 5, 7, 13, 33, 43, 48
Online: Drop #15

Goals

1. Understand definition of limit at +/- inf
2. Find the limit of a rational function as x -> +/-inf and equations of horizontal asymptotes (when they exist)
3. Identify functions whose limit at +/- inf does not exist or whose limit is infinite
4. Compute limits at infinity by graphs and algebraic techniques

----

== Section 2.7 (Sam) ==
Homework:
Written: 5,9,19, 20, 43,44, 46
Online: eliminate o7,o9,o12

Goals:
1. Given an algebraic function, take the derivative (using the definition of the derivative)
Write down an equation for a tangent line through a point on a graph 3, 5, 05, o10, o11
2. Given a real life situation, interpret instantaneous change or average of change 43, 44, 46
Given a position function (or graph thereof), interpret the graph or find (average) velocity o1, o2, o3
3. Sketch the graph of a function such that certain criteria are met (values at certain points, derivatives, etc.) 19, 20
4. Given a limit expression, write down the point and function whose derivative it represents o6, o8
5. Write both formal definitions of a derivative
Explain how the derivative can be interpreted as slope of a line, velocity, instantaneous change

Changes:
Added: 43, 44
Eliminating: 2,25, o7, o9, o12

----

== Section 2.8 (Savannah) ==

Homework

Written: 1, 3, 5, 6, 7, 9, 11, 19, 20, 25, 35, 36, 37, 38, 44, Prove that if a function f is differentiable at a, then f is continuous at a.
Online: Kill O2, O3, Change O9 to f'(x)

Goals

1. Given a function f, find a formula for the derivative of f using f'(x) = lim (h->0) [f(x+h)-f(x)]/h, and be able to state the domain of f'(x). 19, 20, 25, O9, O10
2. Given a graph of a function f, be able to find the graph of f', f''. Given a graph of a function f', f'', be able to find th graph of f. 1, 3, 5, 6, 7, 9, 11, O1, O4, O5
3. Recognize when and why a function fails to be differentiable: 35, 36, 37, 38
a) corner
b) discontinuity (Theorem 4)
c) vertical tangent line
4. Be able to compute and apply higher derivatives: O6, O7, O8, 44
- (f')'=f''
- position [s(t)], velocity [v(t)=s'(t)], acceleration [a(t)=v'(t)=s''(t)]
5. Prove that if f is differentiable at a, then f is continuous at a. [Written homework]

----

== Section 3.1 (Drew) ==

Homework:
Written: 2, 13, 17, 20, 25, 28, 29, 33, 55, 61, 77
Online: Kill o6 last part, o14, o8

Goals:
1. Compute derivatives using:
a. The power rule. Especially, be able to rewrite functions so the power rule can be applied. 13, 17, 20, 25, 28, 29, o1, o3, o7, o8, o9, o10, o11, o12
b. The derivative of e^x. 2, 17, 28, o12
c. Sums, differences, and scalar multiples of functions you can differentiate with these rules. (almost all problems)
2. Use the definition of the derivative and limit laws to:
a. Prove simple rules, including the power rule for n=-2, -1, -1/2, 1/2, 1, 2, as well as sum and difference rules. 61
b. Compute certain limits. 77
3. Find a tangent line to the graph of a function at a given point. 33, 55, o2, o4, o5

----

== Section 3.2 (Rebecca) ==

Homework

Written: 2,11,13,23,24,32,33,47,49,55,57
Online: As listed.

Goals

1. Be able to apply the product rule to find the derivative of functions. 11,24,33,47,49,55,57, 02,04,05,06,07,09
2. Be able to apply the quotient rule to find the derivative of functions. 2,13,23,24,32,47,49, 01,03,08,010,011
3. Be able to prove the product and quotient rules and variations. 55,57*

----

== Section 3.3 (Skyler/James) ==

Homework

Written: 9, 10, 18, 20, 35, 42, 45, 49
Online: As listed. Change ordinal number suffixes on #3 (e.g. 83th makes no sense)

Goals

1. Know derivatives for the 6 basic trig functions. Be able to use these to compute derivatives of more complicated functions.
2. Know limits [2] and [3] in the book. Be able to use these to solve other limits involving trig functions
3. Know the "periodicity" of differentiation of trig functions--i.e. the fourth derivative of the sine function is again the sine function
4. Be able to use trig identities to find derivatives and limits
5. Use trig derivatives in various real world problems.

----

== Section 3.4 (Savannah) ==

Homework

Written: 7, 24, 25, 31, 37, 47, 63, 65, 71, 84, 89
Online: Kill O6, O7, O8; Reorder problems (see Dr. Purcell); Add #77 from the book to online (change "particle" to "point")

Goals

1. Fluency in chain rule: Given a composition of functions, use the chain rule and other rules to compute its derivative quickly. 7, 25, 37, 47, O1, O2, O3, O9, O10, O11, O12, O13, O14, O15, O16
2. Given values of two functions and their derivatives at certain points, use the formal statement of the chain rule to compute the derivative of compositions of these functions. 63, 65, 71, O4, O5
3. Given a word problem involving the composition of functions, compute rates of change. Interpret the meaning of the derivative as a particular rate of change. 84
4. Use the chain rule and definition of even/odd functions to prove properties of derivatives of even/odd functions. 89
5. Compute the derivative of a^x. 24, 31

----

== Section 3.5 (Rebecca) ==
Homework

Written: 3,11,12,15,18,21,23,34,41,57 (Add 43,67)
Online: Kill 03,014; Replace 08 with 35 from the book and make it online

Goals

1) For an implicitly defined equation find dy/dx, dx/dy, and d^2y/dx^2 using implicit differentiation.
2) Find the tangent line to an implicitly defined function.
3) Know the derivatives of the inverse trig functions.
----

== Section 3.6 (Sam) ==

----

== Section 3.8 (Jessica) ==

Homework:

Written: 14, 15
Online: kill o5 (pH scale)

Goals:

Use exponential functions to model each of the following:
1. Population growth. o1, o3, o4
2. Radioactive decay. o6, o7
Find half-life, or given half-life, find equation.
3. Cooling. 14, 15
4. Interest. o2, o8
- Derive the formula for compounding interest in n periods o2, o8 (see also section 1.3, number 60)
- Use expression of e as a limit to find formula for continuously compounded interest.
----

== Section 3.9 (Mark) ==

Homework

Written: 5, 22, 33, 35, 37, 42
Online: Kill O1, O4, O7, O8, O11, O12, O13; remove geometry formulas from online questions.

Goals

1. Solve related rates problems. (all homework problems)
a) Draw picture
b) Recognize variables from the problem, and rates of change as their derivatives.
c) Write equations relating variables (using geometry, trigonometry)
d) Differentiate, remembering to use the chain rule.
e) Plug in specific values to determine the requested answer.

----

== Section 3.10 (Tyler) ==

Homework
Written 1,2,3,5,23,28,43,44
Online Kill O4, O5, Renumber so that O3 is first and O1 is third. Possibly add one more. Change the hint in O1 to read "Try using linear approximation..."

Goals
1. Compute the linearization of a function at a point. 1,2,3,5, O2, O3
- Explain the relationship between the linearization and the tangent line. 5, O2
- Explain how linearization is useful. 23,28
2. Use the linearization at a point x=a to approximate the value of a function f(a+epsilon). 23,28, O1, O2

Suggestion: merge 3.10 and 3.11 for written as well as online.
----

== Section 3.11 (Savannah) ==

----

== Section 4.1 ==

----

== Section 4.2 ==

----

== Section 4.3 ==

Goals:

1. Know and apply the increasing/decreasing test o1, o5
2. Find local maxima and minima (First and Second derivative tests, endpoints) 21,23
3. Know the definition of concavity and how to determine concavity (The concavity test) o2, o5, 72
4. Know what an inflection point is and how to find it o2
5. Know how the above information affects the graph of a function
Draw a graph such that.... 25, 28
Given a graph, where is concavity positive, etc.? 1, 6, 7, 31 and a new online problem similar to problem 8 on page 295

Changes:

o3 and o4 are to be moved to section 4.5 and a problem is to be added online that is like problem 8.
----

== Section 4.4 ==

----

== Section 4.5 (Skyler) ==

Goals:

Use all of the following to draw graphs of functions:

* Algebraic Properties (Domain, intercepts, even/odd/periodic)

* Limit Properties (Asymptotes, singularities, discontinuities)

* Derivative Properties (Increasing/decreasing, extrema, concavity/inflection)

Written homework - as listed
Online homework - pulled from 4.3 (O3 and O4)

----

== Section 4.7 ==

----

== Section 4.8 (Mark) ==

Goals

1. Use Newton's Method to approximate roots/solutions of equations.
2. Derive the formula for Newton's Method from the tangent line equation.
3. Give an example of a function 'f' and a starting point 'x_1' that makes Newton's Method fail.

Homework

Written: 1, 2, 3, 4, 29.
Online: Kill O1, Add new online problem with graph.

----

== Section 4.9 (Tyler)==

Goals:
1. Use the Mean value Theorem to prove Theorem 1.
2. Memorize the table of anti-differentiation formulas on page 341 and
relate it to the table of derivatives on the inside back cover of the book. 12, 15, 20, 27, 33, 49, 51, O1-O12
3. Use the table and Theorem 1 to find *all* antiderivatives of various functions. 12, 15, 20, 27, 49, 51
4. Use Theorem 1 and known differentiation formulas to find a unique anti-derivative with given values at specified points. 33, 53, 67,
69, 70, 74,O2,O3, O4,O5,O10, O11, O13
5. Use anti-differentiation to solve physical and economic problems, especially problems of motion. 67, 69, 70, 74, O4, O5

Changes to problems:
Written: Add story problems: 67, 69, 70, 74
Online: too long, Delete O1, O6, O12.
Adjust numbers in 04 to give a clean answer.
O10--change so that F(0) not equal to 0 (else they will get right answer for wrong reason).
O8 and O9 problem that Arctan/Arcsin not accepted--just arctan/arcsin.
O13 should not say "the anti-derivative, but rather "an anti-derivative".

----

== Appendix E (Paul)==
Homework:

Written: 1, 3, 5, 10, 11, 15, 17, 20, 23, 41, 43
Online: o1, o2, o3, o4, o5, o6, o7, o8, o9, o10

Goals

1. Write expanded sums in sigma notation (1, 3, 5, 10)
and expand sums written in sigma notation (11, 15, 17, 20).
Numerically evaluate sums (23). Apply appropriate rules to
distribute constants or expand sums of sums (o2, o4, o5, o8, o9, o10).
2. Explicitly evaluate $\sigma_{i=1}^n f(i)$ in terms of 'n', where 'f(x)'
is a polynomial in x of degree at most 2. (Degree 0 o3, o7; degree 1
o1, o2, o8, o10; degree 2 o4, o5, o6, o9.)
Numerically evaluate this sum when 'i' runs over a given range of integers.
3. Evaluate telescoping sums (41) and limits of sums (43).

----

== Section 5.1 ==

----

== Section 5.2 ==

----

== Section 5.3 ==

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== Section 5.4 ==

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== Section 5.5 ==

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Math 112 Calculus Learning Goals

2009-08-19T19:56:43Z

Pmj5: /* Appendix E (Paul) */

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== Section 1.1 (Jessica) ==

Homework:

Written: 25, 31, 40, 43, 55, 56, 64, 65
Online: 3, 4, 5, 6, 8, 9, 11. Modify 12 (Given f like 66 or 67 in book, find f(-x). Is f even or odd or neither?) Add problem like 51, 57 in book.

Goals

1. Given a function described algebraically, find the
- domain 31, 07, 08,
- range,
- value at a given number 04,
- value when we plug in another function, e.g. difference quotient 04, 05, 25.
2. Convert one representation of a function to another.
- verbal to/from graphical 03
-algebraic to/from graphical 09
-verbal to/from algebraic 55, 56, 2 new online
3. Piecewise functions: Given an algebraic description of a piecewise function, find its domain and sketch its graph. 011, 43.
Be able to write the absolute value as a piecewise function, and use this to sketch its graph. 40, 09
4. Use the definition of an even/odd function to decide whether a function described algebraically is even or odd. 012, 65
Use symmetry to decide whether a graph is an even or odd function. 64

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== Section 1.2 (Sam) ==

Homework:

Written: 4,5,7,9,12 and Appdx D: 24, 37*, 46
Online: O1, O2, O4, O5 (these are the numbers under the current numbering),
and Appdx D: 29, 60, 69
The * on problem 37 means a calculator will be necessary.

Goals

1. Writing down a linear function given a set of information 5,12,O4,O5
2. Recognizing what a function's graph should look like 4,7,O1
3. Based on a graph, write down a function O2
4. Writing down a polynomial based on information 9, O2
5. Trig review (definitions, proofs, values of trig functions at particular points) Appdx D:24, 37, 46, 29, 60, 69 (the latter three going online)

# Trig to know, including problems to skim for practice. (Jessica)

1. Given angles in degrees and radians, draw the angle, convert from radians to degrees and vice versa. (See appendix D: 1-12)
2. Write down sin, cos, tan, sec, scs, cot of any angle 0, pi/6, pi/4, pi/3, pi/2, and all angles obtained from these angles by adding a multiple of pi/2. (appendix D:23-28)
3. Given a right triangle with side measurements, write sin, cos, tan, sec, csc, cot of any of the angles of the triangle in terms of the side lengths. (Appendix D: 35-38)
Similarly, given one trig ratio, find the others. (Appendix D: 29-34)
4. Graph trig functions. (app D: 77-82)
5. Use trig identities to simplify expressions (appendix D: 43-57 odd, 59-63), and to solve equations and inequalities involving trig functions (app D: 65-76).
* Most important identities: sin^2(x) + cos^2(x) = 1
* sin(-x) = -sin(x)
* cos(-x) = cos(x)
* sin(x+y) = sin(x)cos(y) + cos(x)sin(y)
* cos(x+y) = cos(x)cos(y) - sin(x)sin(y)
Using the above, you can derive other identities:
* 1+tan^2(x)=sec^2(x); 1+cot^2(x) = csc^2(x)
* sin(x-y)
* cos(x-y)
* sin(2x)
* cos(2x)

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== Section 1.3 (Savannah) ==

Homework

Written: 1,7,13, 14, 22, 39, 46, 60, 65
Online: Modify 1 to be y as a function of x. Replace 6 with a different word problem. Add 3 from section 1.2 online assignment.

Goals

1. Apply and recognize transformations of functions:
- Translating, Stretching, Reflecting, Absolute Value 1
- Graph --> Algebraic 7, O1, O5, 1.2-O3
- Algebraic --> Graph 13, 14, 22, O7, O8
2. Given functions f and g, find:
- f+g
- f-g
- fg
- f/g
- domains for above functions. O2, O9
3. Understand and apply compositions of functions:
- Given functions f and g, find g o f and f o g. 28, 39, 60, O3, O6
- Given f o g, find functions f and g. 46, 65, O4

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== Section 1.5 (James) ==

Homework:

Written: 11, 15, 18, 19, 21, 25, 29
Online: o2, o3, o4, o5, one more word problem

Goals

1. Graph exponential functions a^x for a<1, a>1, as well as transformations of exponential functions from section 1.3. Find domains. 11, 15, 21, o2
2. Given two points that an exponential function passes through, find the equation of the exponential function. 18, o3.
3. Use laws of exponents to simplify exponential expressions, to show various facts about exponential functions. 19, 29.
4. Convert a word problem involving population change into an algebraic expression involving exponential functions. Use this to find sizes of populations at given times, and to graph. 25, o4, o5.

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== Section 1.6 (McKay) ==

Homework

Written: 12,16,18,19,23,35,45,48,54,60,67,71,73
Online: 01,02,03,04,05,06,07,08,010,011,012,013,016,018

Goals

1) Be able to tell if a function is one-to-one
-algebraically 12
-graphically 01
2) Know the definition of an inverse function and be able to use the steps to find the inverse
-algebraically 16,19,23,54,02,03,04,05,06
-graphically 18,45,73,07
3)Know the laws of logarithms and be able to solve and simplify logarithmic and exponential expressions. 35,48,06,08,010,011,012
4)Know the inverses of the trigonometric functions
- domain and range 71
- values at a point 60, 013,016
- simplify a trig function composed with an inverse trig function 67, o18

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== Section 2.1 (Tyler) ==

Homework

Written:4*, 5*? (require calculator)

Goals

1. Know definitions of tangent and secant line. Compute the slope of a secant line. 4
2. Know definition of average velocity and the idea of instantaneous velocity. Compute an average velocity. 5
3. Explain how average velocity is equivalent to slope of a secant line, same for instantaneous velocity and slope of tangent line.

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== Section 2.2 (Tyler) ==

Homework

Written: 6, 7, 9, 16, 27, 32, 34a, 40
Online: kill O2, O5. Modify: O3 fix so different parts don't have same answer.
O6, O7, O8: Use the same input notation for infinity. O8 Delete irrelevant lecture about asymptotes.

Goals:

1. Idea of a limit: (Note: the "definition" of a limit in this section is sloppy and is not a real definition of a limit. See section 2.4)
* Find the limit of a function at a point from its graph, including when the limit does not exist. 6,7, O1, O3
* Recognize the difference between the limit of a function at a point and the value of the function at a point. 6, 7, O1
* Give examples of functions that have prescribed limits at certain points. 16
* Understand that calculators and tables can entice you to guess a wrong limit. Give examples of functions whose limit is not what you would guess from plugging in values to your calculator. (no problems)
2. Idea (and notation) for one-sided limit: Do the same things for one-sided limits as done before for regular limits. 6, 7, O1, O3, O4, O6, O7, O8
3. Explain how one-sided and two-sided limits are connected. Use this to compute limits. 6, 7, O1, O3, O4, O6, O7, O8
4. Infinite limits:
* Explain why infinity is not a number and how the definition of an infinite limit gets around this.
* Find all vertical asymptotes for a function. 9, 34a
* For rational functions, find when a limit is infinity and when it is negative infinity. 27, 32, O6, O7, O8, 40

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== Section 2.3a (Rebecca) ==

Homework

Written: 10,15,19,20,21,22,28,29
Online: Get rid of 3

Goals

1. Be able to apply limit laws to simplify limits
- algebraically 01
- graphically 02
2. Recognize when you can directly substitute to compute a limit and when you cannot. 10, 03, 04, 05, 06
3. Use algebra to simply and find limits.
- for polynomials over polynomials. 15, 19, 20, o5, o6, o7, o8, o9, o10
- expressions with square roots. 21, 22, 29, o11
- expressions with multiple fractions. 28, 29, o12

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== Section 2.3b (Drew) ==

Homework

Written: 36-39, 42, 55, 56, 58
Online: Add o2 from 2.2. Kill o3, o4. Modify o6 so that direct substitution doesn't work. Add two problems using substitution.

Goals

1. Use the Squeeze Theorem to find limits. 36, 37, 38, o5, o6, o7
2. Evaluate limits that involve absolute values and other piecewise functions. 39, 42, o1, o2, o2 from 2.2
3. Use limit laws to find lim f(x) given that lim g(f(x)) = L. 55, 56, 58
4. Use substitution to rewrite lim g(f(x)) as lim g(u). 2 new online problems.

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== Section 2.4 (Jessica) ==
Homework:
Written: 2, 14, 15, 16, 19, 20
Online: kill o6 (done), modify problems to find the *largest* delta (done).

Goals:
1. Use a graph showing a function and lines y=L+epsilon, y=L-epsilon to find the largest value of delta so that if 0<|x-a|<delta, then |f(x)-L|<epsilon. 2
2. Know the epsilon-delta definition of a limit. Given a limit and a value for epsilon, be able to find the largest value of delta corresponding to that epsilon. 14, o1, o2, o3, o4, o5
3. Use the epsilon-delta definition of a limit to prove a statement about the limit of a linear function. 15, 16, 19, 20.

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== Section 2.5a (McKay/Savannah) ==

Written:4, 6, 15, 18, 22, 23, 43ab, 58
Online - Kill O2-O4, O11-O16; Change O8, O9, O10 to ask, "Where is the following function continuous?"; Add a graph problem where the student must identify where the graph is discontinuous; Add a graph problem where the student must identify the types of discontinuities

1. Given the graph of a function, be able to tell:
a) where it is continuous. 4
b) where it is discontinuous, and the type of discontinuity. 6
2. Given a function described algebraically, be able to tell:
a) where it is continuous. 22, 23, O1, O8, O9, O10
b) where it is discontinuous and types of discontinuities. 15, 18, 43(a,b), O5, O6, O7
This includes functions from Theorem 7, piecewise functions, and combinations of functions.
3. Use limit laws and the definition of continuity to prove facts about continuity (e.g. Theorem 4). 58

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== Section 2.5b (Mark) ==

Homework

Written: 28, 31, 32, 36, 37, 39, 41, 45, 47, 49, 65
Online: Kill 3, 5, 6, 7. Add 3 problems: Find intervals where f is positive, negative. One quadratic, easy factor; one rational with linear top and bottom; one rational with quadratic top, perfect square bottom.

Goals

1. Use continuity to evaluate limits 31,32
2. Continuity of piecewise functions
-determine if a piecewise function is continuous (or continuous from the right/left) 36,37,39
-determine parameters to make a piecewise function continuous 41,O1,O2,O4
3. Given a composition of functions, tell where it is continuous/discontinuous. 28
4. Use the Intermediate Value Theorem to prove the existence of solutions to equations. 45,47,49,65
5. Find intervals where a continuous function is positive/negative. (3 new online problems)

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== Section 2.6 (Skyler) ==

Homework:
Written: 4, 5, 7, 13, 33, 43, 48
Online: Drop #15

Goals

1. Understand definition of limit at +/- inf
2. Find the limit of a rational function as x -> +/-inf and equations of horizontal asymptotes (when they exist)
3. Identify functions whose limit at +/- inf does not exist or whose limit is infinite
4. Compute limits at infinity by graphs and algebraic techniques

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== Section 2.7 (Sam) ==
Homework:
Written: 5,9,19, 20, 43,44, 46
Online: eliminate o7,o9,o12

Goals:
1. Given an algebraic function, take the derivative (using the definition of the derivative)
Write down an equation for a tangent line through a point on a graph 3, 5, 05, o10, o11
2. Given a real life situation, interpret instantaneous change or average of change 43, 44, 46
Given a position function (or graph thereof), interpret the graph or find (average) velocity o1, o2, o3
3. Sketch the graph of a function such that certain criteria are met (values at certain points, derivatives, etc.) 19, 20
4. Given a limit expression, write down the point and function whose derivative it represents o6, o8
5. Write both formal definitions of a derivative
Explain how the derivative can be interpreted as slope of a line, velocity, instantaneous change

Changes:
Added: 43, 44
Eliminating: 2,25, o7, o9, o12

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== Section 2.8 (Savannah) ==

Homework

Written: 1, 3, 5, 6, 7, 9, 11, 19, 20, 25, 35, 36, 37, 38, 44, Prove that if a function f is differentiable at a, then f is continuous at a.
Online: Kill O2, O3, Change O9 to f'(x)

Goals

1. Given a function f, find a formula for the derivative of f using f'(x) = lim (h->0) [f(x+h)-f(x)]/h, and be able to state the domain of f'(x). 19, 20, 25, O9, O10
2. Given a graph of a function f, be able to find the graph of f', f''. Given a graph of a function f', f'', be able to find th graph of f. 1, 3, 5, 6, 7, 9, 11, O1, O4, O5
3. Recognize when and why a function fails to be differentiable: 35, 36, 37, 38
a) corner
b) discontinuity (Theorem 4)
c) vertical tangent line
4. Be able to compute and apply higher derivatives: O6, O7, O8, 44
- (f')'=f''
- position [s(t)], velocity [v(t)=s'(t)], acceleration [a(t)=v'(t)=s''(t)]
5. Prove that if f is differentiable at a, then f is continuous at a. [Written homework]

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== Section 3.1 (Drew) ==

Homework:
Written: 2, 13, 17, 20, 25, 28, 29, 33, 55, 61, 77
Online: Kill o6 last part, o14, o8

Goals:
1. Compute derivatives using:
a. The power rule. Especially, be able to rewrite functions so the power rule can be applied. 13, 17, 20, 25, 28, 29, o1, o3, o7, o8, o9, o10, o11, o12
b. The derivative of e^x. 2, 17, 28, o12
c. Sums, differences, and scalar multiples of functions you can differentiate with these rules. (almost all problems)
2. Use the definition of the derivative and limit laws to:
a. Prove simple rules, including the power rule for n=-2, -1, -1/2, 1/2, 1, 2, as well as sum and difference rules. 61
b. Compute certain limits. 77
3. Find a tangent line to the graph of a function at a given point. 33, 55, o2, o4, o5

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== Section 3.2 (Rebecca) ==

Homework

Written: 2,11,13,23,24,32,33,47,49,55,57
Online: As listed.

Goals

1. Be able to apply the product rule to find the derivative of functions. 11,24,33,47,49,55,57, 02,04,05,06,07,09
2. Be able to apply the quotient rule to find the derivative of functions. 2,13,23,24,32,47,49, 01,03,08,010,011
3. Be able to prove the product and quotient rules and variations. 55,57*

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== Section 3.3 (Skyler/James) ==

Homework

Written: 9, 10, 18, 20, 35, 42, 45, 49
Online: As listed. Change ordinal number suffixes on #3 (e.g. 83th makes no sense)

Goals

1. Know derivatives for the 6 basic trig functions. Be able to use these to compute derivatives of more complicated functions.
2. Know limits [2] and [3] in the book. Be able to use these to solve other limits involving trig functions
3. Know the "periodicity" of differentiation of trig functions--i.e. the fourth derivative of the sine function is again the sine function
4. Be able to use trig identities to find derivatives and limits
5. Use trig derivatives in various real world problems.

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== Section 3.4 (Savannah) ==

Homework

Written: 7, 24, 25, 31, 37, 47, 63, 65, 71, 84, 89
Online: Kill O6, O7, O8; Reorder problems (see Dr. Purcell); Add #77 from the book to online (change "particle" to "point")

Goals

1. Fluency in chain rule: Given a composition of functions, use the chain rule and other rules to compute its derivative quickly. 7, 25, 37, 47, O1, O2, O3, O9, O10, O11, O12, O13, O14, O15, O16
2. Given values of two functions and their derivatives at certain points, use the formal statement of the chain rule to compute the derivative of compositions of these functions. 63, 65, 71, O4, O5
3. Given a word problem involving the composition of functions, compute rates of change. Interpret the meaning of the derivative as a particular rate of change. 84
4. Use the chain rule and definition of even/odd functions to prove properties of derivatives of even/odd functions. 89
5. Compute the derivative of a^x. 24, 31

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== Section 3.5 (Rebecca) ==
Homework

Written: 3,11,12,15,18,21,23,34,41,57 (Add 43,67)
Online: Kill 03,014; Replace 08 with 35 from the book and make it online

Goals

1) For an implicitly defined equation find dy/dx, dx/dy, and d^2y/dx^2 using implicit differentiation.
2) Find the tangent line to an implicitly defined function.
3) Know the derivatives of the inverse trig functions.
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== Section 3.6 (Sam) ==

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== Section 3.8 (Jessica) ==

Homework:

Written: 14, 15
Online: kill o5 (pH scale)

Goals:

Use exponential functions to model each of the following:
1. Population growth. o1, o3, o4
2. Radioactive decay. o6, o7
Find half-life, or given half-life, find equation.
3. Cooling. 14, 15
4. Interest. o2, o8
- Derive the formula for compounding interest in n periods o2, o8 (see also section 1.3, number 60)
- Use expression of e as a limit to find formula for continuously compounded interest.
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== Section 3.9 (Mark) ==

Homework

Written: 5, 22, 33, 35, 37, 42
Online: Kill O1, O4, O7, O8, O11, O12, O13; remove geometry formulas from online questions.

Goals

1. Solve related rates problems. (all homework problems)
a) Draw picture
b) Recognize variables from the problem, and rates of change as their derivatives.
c) Write equations relating variables (using geometry, trigonometry)
d) Differentiate, remembering to use the chain rule.
e) Plug in specific values to determine the requested answer.

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== Section 3.10 (Tyler) ==

Homework
Written 1,2,3,5,23,28,43,44
Online Kill O4, O5, Renumber so that O3 is first and O1 is third. Possibly add one more. Change the hint in O1 to read "Try using linear approximation..."

Goals
1. Compute the linearization of a function at a point. 1,2,3,5, O2, O3
- Explain the relationship between the linearization and the tangent line. 5, O2
- Explain how linearization is useful. 23,28
2. Use the linearization at a point x=a to approximate the value of a function f(a+epsilon). 23,28, O1, O2

Suggestion: merge 3.10 and 3.11 for written as well as online.
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== Section 3.11 (Savannah) ==

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== Section 4.1 ==

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== Section 4.2 ==

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== Section 4.3 ==

Goals:

1. Know and apply the increasing/decreasing test o1, o5
2. Find local maxima and minima (First and Second derivative tests, endpoints) 21,23
3. Know the definition of concavity and how to determine concavity (The concavity test) o2, o5, 72
4. Know what an inflection point is and how to find it o2
5. Know how the above information affects the graph of a function
Draw a graph such that.... 25, 28
Given a graph, where is concavity positive, etc.? 1, 6, 7, 31 and a new online problem similar to problem 8 on page 295

Changes:

o3 and o4 are to be moved to section 4.5 and a problem is to be added online that is like problem 8.
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== Section 4.4 ==

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== Section 4.5 (Skyler) ==

Goals:

Use all of the following to draw graphs of functions:

* Algebraic Properties (Domain, intercepts, even/odd/periodic)

* Limit Properties (Asymptotes, singularities, discontinuities)

* Derivative Properties (Increasing/decreasing, extrema, concavity/inflection)

Written homework - as listed
Online homework - pulled from 4.3 (O3 and O4)

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== Section 4.7 ==

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== Section 4.8 (Mark) ==

Goals

1. Use Newton's Method to approximate roots/solutions of equations.
2. Derive the formula for Newton's Method from the tangent line equation.
3. Give an example of a function 'f' and a starting point 'x_1' that makes Newton's Method fail.

Homework

Written: 1, 2, 3, 4, 29.
Online: Kill O1, Add new online problem with graph.

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== Section 4.9 (Tyler)==

Goals:
1. Use the Mean value Theorem to prove Theorem 1.
2. Memorize the table of anti-differentiation formulas on page 341 and
relate it to the table of derivatives on the inside back cover of the book. 12, 15, 20, 27, 33, 49, 51, O1-O12
3. Use the table and Theorem 1 to find *all* antiderivatives of various functions. 12, 15, 20, 27, 49, 51
4. Use Theorem 1 and known differentiation formulas to find a unique anti-derivative with given values at specified points. 33, 53, 67,
69, 70, 74,O2,O3, O4,O5,O10, O11, O13
5. Use anti-differentiation to solve physical and economic problems, especially problems of motion. 67, 69, 70, 74, O4, O5

Changes to problems:
Written: Add story problems: 67, 69, 70, 74
Online: too long, Delete O1, O6, O12.
Adjust numbers in 04 to give a clean answer.
O10--change so that F(0) not equal to 0 (else they will get right answer for wrong reason).
O8 and O9 problem that Arctan/Arcsin not accepted--just arctan/arcsin.
O13 should not say "the anti-derivative, but rather "an anti-derivative".

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== Appendix E (Paul)==
Homework:

Written: 1, 3, 5, 10, 11, 15, 17, 20, 23, 41, 43
Online: o1, o2, o3, o4, o5, o6, o7, o8, o9, o10

Goals

1. Write expanded sums in sigma notation (1, 3, 5, 10) and expand sums written in sigma notation (11, 15, 17, 20).
Numerically evaluate sums (23). Apply appropriate rules to distribute constants or expand sums of sums (o2, o4, o5, o8, o9, o10).
2. Explicitly evaluate $\sigma_{i=1}^n f(i)$ in terms of 'n', where 'f(x)' is a polynomial in x of degree at most 2. (Degree 0 o3, o7; degree 1 o1, o2, o8, o10; degree 2 o4, o5, o6, o9.)
Numerically evaluate this sum when 'i' runs over a given range of integers.
3. Evaluate telescoping sums (41) and limits of sums (43).

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== Section 5.1 ==

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== Section 5.2 ==

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== Section 5.3 ==

----

== Section 5.4 ==

----

== Section 5.5 ==

----

Math 112 Calculus Learning Goals

2009-08-19T19:53:20Z

Pmj5: /* Appendix E (Paul) */

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== Section 1.1 (Jessica) ==

Homework:

Written: 25, 31, 40, 43, 55, 56, 64, 65
Online: 3, 4, 5, 6, 8, 9, 11. Modify 12 (Given f like 66 or 67 in book, find f(-x). Is f even or odd or neither?) Add problem like 51, 57 in book.

Goals

1. Given a function described algebraically, find the
- domain 31, 07, 08,
- range,
- value at a given number 04,
- value when we plug in another function, e.g. difference quotient 04, 05, 25.
2. Convert one representation of a function to another.
- verbal to/from graphical 03
-algebraic to/from graphical 09
-verbal to/from algebraic 55, 56, 2 new online
3. Piecewise functions: Given an algebraic description of a piecewise function, find its domain and sketch its graph. 011, 43.
Be able to write the absolute value as a piecewise function, and use this to sketch its graph. 40, 09
4. Use the definition of an even/odd function to decide whether a function described algebraically is even or odd. 012, 65
Use symmetry to decide whether a graph is an even or odd function. 64

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== Section 1.2 (Sam) ==

Homework:

Written: 4,5,7,9,12 and Appdx D: 24, 37*, 46
Online: O1, O2, O4, O5 (these are the numbers under the current numbering),
and Appdx D: 29, 60, 69
The * on problem 37 means a calculator will be necessary.

Goals

1. Writing down a linear function given a set of information 5,12,O4,O5
2. Recognizing what a function's graph should look like 4,7,O1
3. Based on a graph, write down a function O2
4. Writing down a polynomial based on information 9, O2
5. Trig review (definitions, proofs, values of trig functions at particular points) Appdx D:24, 37, 46, 29, 60, 69 (the latter three going online)

# Trig to know, including problems to skim for practice. (Jessica)

1. Given angles in degrees and radians, draw the angle, convert from radians to degrees and vice versa. (See appendix D: 1-12)
2. Write down sin, cos, tan, sec, scs, cot of any angle 0, pi/6, pi/4, pi/3, pi/2, and all angles obtained from these angles by adding a multiple of pi/2. (appendix D:23-28)
3. Given a right triangle with side measurements, write sin, cos, tan, sec, csc, cot of any of the angles of the triangle in terms of the side lengths. (Appendix D: 35-38)
Similarly, given one trig ratio, find the others. (Appendix D: 29-34)
4. Graph trig functions. (app D: 77-82)
5. Use trig identities to simplify expressions (appendix D: 43-57 odd, 59-63), and to solve equations and inequalities involving trig functions (app D: 65-76).
* Most important identities: sin^2(x) + cos^2(x) = 1
* sin(-x) = -sin(x)
* cos(-x) = cos(x)
* sin(x+y) = sin(x)cos(y) + cos(x)sin(y)
* cos(x+y) = cos(x)cos(y) - sin(x)sin(y)
Using the above, you can derive other identities:
* 1+tan^2(x)=sec^2(x); 1+cot^2(x) = csc^2(x)
* sin(x-y)
* cos(x-y)
* sin(2x)
* cos(2x)

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== Section 1.3 (Savannah) ==

Homework

Written: 1,7,13, 14, 22, 39, 46, 60, 65
Online: Modify 1 to be y as a function of x. Replace 6 with a different word problem. Add 3 from section 1.2 online assignment.

Goals

1. Apply and recognize transformations of functions:
- Translating, Stretching, Reflecting, Absolute Value 1
- Graph --> Algebraic 7, O1, O5, 1.2-O3
- Algebraic --> Graph 13, 14, 22, O7, O8
2. Given functions f and g, find:
- f+g
- f-g
- fg
- f/g
- domains for above functions. O2, O9
3. Understand and apply compositions of functions:
- Given functions f and g, find g o f and f o g. 28, 39, 60, O3, O6
- Given f o g, find functions f and g. 46, 65, O4

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== Section 1.5 (James) ==

Homework:

Written: 11, 15, 18, 19, 21, 25, 29
Online: o2, o3, o4, o5, one more word problem

Goals

1. Graph exponential functions a^x for a<1, a>1, as well as transformations of exponential functions from section 1.3. Find domains. 11, 15, 21, o2
2. Given two points that an exponential function passes through, find the equation of the exponential function. 18, o3.
3. Use laws of exponents to simplify exponential expressions, to show various facts about exponential functions. 19, 29.
4. Convert a word problem involving population change into an algebraic expression involving exponential functions. Use this to find sizes of populations at given times, and to graph. 25, o4, o5.

----

== Section 1.6 (McKay) ==

Homework

Written: 12,16,18,19,23,35,45,48,54,60,67,71,73
Online: 01,02,03,04,05,06,07,08,010,011,012,013,016,018

Goals

1) Be able to tell if a function is one-to-one
-algebraically 12
-graphically 01
2) Know the definition of an inverse function and be able to use the steps to find the inverse
-algebraically 16,19,23,54,02,03,04,05,06
-graphically 18,45,73,07
3)Know the laws of logarithms and be able to solve and simplify logarithmic and exponential expressions. 35,48,06,08,010,011,012
4)Know the inverses of the trigonometric functions
- domain and range 71
- values at a point 60, 013,016
- simplify a trig function composed with an inverse trig function 67, o18

----

== Section 2.1 (Tyler) ==

Homework

Written:4*, 5*? (require calculator)

Goals

1. Know definitions of tangent and secant line. Compute the slope of a secant line. 4
2. Know definition of average velocity and the idea of instantaneous velocity. Compute an average velocity. 5
3. Explain how average velocity is equivalent to slope of a secant line, same for instantaneous velocity and slope of tangent line.

----

== Section 2.2 (Tyler) ==

Homework

Written: 6, 7, 9, 16, 27, 32, 34a, 40
Online: kill O2, O5. Modify: O3 fix so different parts don't have same answer.
O6, O7, O8: Use the same input notation for infinity. O8 Delete irrelevant lecture about asymptotes.

Goals:

1. Idea of a limit: (Note: the "definition" of a limit in this section is sloppy and is not a real definition of a limit. See section 2.4)
* Find the limit of a function at a point from its graph, including when the limit does not exist. 6,7, O1, O3
* Recognize the difference between the limit of a function at a point and the value of the function at a point. 6, 7, O1
* Give examples of functions that have prescribed limits at certain points. 16
* Understand that calculators and tables can entice you to guess a wrong limit. Give examples of functions whose limit is not what you would guess from plugging in values to your calculator. (no problems)
2. Idea (and notation) for one-sided limit: Do the same things for one-sided limits as done before for regular limits. 6, 7, O1, O3, O4, O6, O7, O8
3. Explain how one-sided and two-sided limits are connected. Use this to compute limits. 6, 7, O1, O3, O4, O6, O7, O8
4. Infinite limits:
* Explain why infinity is not a number and how the definition of an infinite limit gets around this.
* Find all vertical asymptotes for a function. 9, 34a
* For rational functions, find when a limit is infinity and when it is negative infinity. 27, 32, O6, O7, O8, 40

----

== Section 2.3a (Rebecca) ==

Homework

Written: 10,15,19,20,21,22,28,29
Online: Get rid of 3

Goals

1. Be able to apply limit laws to simplify limits
- algebraically 01
- graphically 02
2. Recognize when you can directly substitute to compute a limit and when you cannot. 10, 03, 04, 05, 06
3. Use algebra to simply and find limits.
- for polynomials over polynomials. 15, 19, 20, o5, o6, o7, o8, o9, o10
- expressions with square roots. 21, 22, 29, o11
- expressions with multiple fractions. 28, 29, o12

----

== Section 2.3b (Drew) ==

Homework

Written: 36-39, 42, 55, 56, 58
Online: Add o2 from 2.2. Kill o3, o4. Modify o6 so that direct substitution doesn't work. Add two problems using substitution.

Goals

1. Use the Squeeze Theorem to find limits. 36, 37, 38, o5, o6, o7
2. Evaluate limits that involve absolute values and other piecewise functions. 39, 42, o1, o2, o2 from 2.2
3. Use limit laws to find lim f(x) given that lim g(f(x)) = L. 55, 56, 58
4. Use substitution to rewrite lim g(f(x)) as lim g(u). 2 new online problems.

----

== Section 2.4 (Jessica) ==
Homework:
Written: 2, 14, 15, 16, 19, 20
Online: kill o6 (done), modify problems to find the *largest* delta (done).

Goals:
1. Use a graph showing a function and lines y=L+epsilon, y=L-epsilon to find the largest value of delta so that if 0<|x-a|<delta, then |f(x)-L|<epsilon. 2
2. Know the epsilon-delta definition of a limit. Given a limit and a value for epsilon, be able to find the largest value of delta corresponding to that epsilon. 14, o1, o2, o3, o4, o5
3. Use the epsilon-delta definition of a limit to prove a statement about the limit of a linear function. 15, 16, 19, 20.

----

== Section 2.5a (McKay/Savannah) ==

Written:4, 6, 15, 18, 22, 23, 43ab, 58
Online - Kill O2-O4, O11-O16; Change O8, O9, O10 to ask, "Where is the following function continuous?"; Add a graph problem where the student must identify where the graph is discontinuous; Add a graph problem where the student must identify the types of discontinuities

1. Given the graph of a function, be able to tell:
a) where it is continuous. 4
b) where it is discontinuous, and the type of discontinuity. 6
2. Given a function described algebraically, be able to tell:
a) where it is continuous. 22, 23, O1, O8, O9, O10
b) where it is discontinuous and types of discontinuities. 15, 18, 43(a,b), O5, O6, O7
This includes functions from Theorem 7, piecewise functions, and combinations of functions.
3. Use limit laws and the definition of continuity to prove facts about continuity (e.g. Theorem 4). 58

----

== Section 2.5b (Mark) ==

Homework

Written: 28, 31, 32, 36, 37, 39, 41, 45, 47, 49, 65
Online: Kill 3, 5, 6, 7. Add 3 problems: Find intervals where f is positive, negative. One quadratic, easy factor; one rational with linear top and bottom; one rational with quadratic top, perfect square bottom.

Goals

1. Use continuity to evaluate limits 31,32
2. Continuity of piecewise functions
-determine if a piecewise function is continuous (or continuous from the right/left) 36,37,39
-determine parameters to make a piecewise function continuous 41,O1,O2,O4
3. Given a composition of functions, tell where it is continuous/discontinuous. 28
4. Use the Intermediate Value Theorem to prove the existence of solutions to equations. 45,47,49,65
5. Find intervals where a continuous function is positive/negative. (3 new online problems)

----

== Section 2.6 (Skyler) ==

Homework:
Written: 4, 5, 7, 13, 33, 43, 48
Online: Drop #15

Goals

1. Understand definition of limit at +/- inf
2. Find the limit of a rational function as x -> +/-inf and equations of horizontal asymptotes (when they exist)
3. Identify functions whose limit at +/- inf does not exist or whose limit is infinite
4. Compute limits at infinity by graphs and algebraic techniques

----

== Section 2.7 (Sam) ==
Homework:
Written: 5,9,19, 20, 43,44, 46
Online: eliminate o7,o9,o12

Goals:
1. Given an algebraic function, take the derivative (using the definition of the derivative)
Write down an equation for a tangent line through a point on a graph 3, 5, 05, o10, o11
2. Given a real life situation, interpret instantaneous change or average of change 43, 44, 46
Given a position function (or graph thereof), interpret the graph or find (average) velocity o1, o2, o3
3. Sketch the graph of a function such that certain criteria are met (values at certain points, derivatives, etc.) 19, 20
4. Given a limit expression, write down the point and function whose derivative it represents o6, o8
5. Write both formal definitions of a derivative
Explain how the derivative can be interpreted as slope of a line, velocity, instantaneous change

Changes:
Added: 43, 44
Eliminating: 2,25, o7, o9, o12

----

== Section 2.8 (Savannah) ==

Homework

Written: 1, 3, 5, 6, 7, 9, 11, 19, 20, 25, 35, 36, 37, 38, 44, Prove that if a function f is differentiable at a, then f is continuous at a.
Online: Kill O2, O3, Change O9 to f'(x)

Goals

1. Given a function f, find a formula for the derivative of f using f'(x) = lim (h->0) [f(x+h)-f(x)]/h, and be able to state the domain of f'(x). 19, 20, 25, O9, O10
2. Given a graph of a function f, be able to find the graph of f', f''. Given a graph of a function f', f'', be able to find th graph of f. 1, 3, 5, 6, 7, 9, 11, O1, O4, O5
3. Recognize when and why a function fails to be differentiable: 35, 36, 37, 38
a) corner
b) discontinuity (Theorem 4)
c) vertical tangent line
4. Be able to compute and apply higher derivatives: O6, O7, O8, 44
- (f')'=f''
- position [s(t)], velocity [v(t)=s'(t)], acceleration [a(t)=v'(t)=s''(t)]
5. Prove that if f is differentiable at a, then f is continuous at a. [Written homework]

----

== Section 3.1 (Drew) ==

Homework:
Written: 2, 13, 17, 20, 25, 28, 29, 33, 55, 61, 77
Online: Kill o6 last part, o14, o8

Goals:
1. Compute derivatives using:
a. The power rule. Especially, be able to rewrite functions so the power rule can be applied. 13, 17, 20, 25, 28, 29, o1, o3, o7, o8, o9, o10, o11, o12
b. The derivative of e^x. 2, 17, 28, o12
c. Sums, differences, and scalar multiples of functions you can differentiate with these rules. (almost all problems)
2. Use the definition of the derivative and limit laws to:
a. Prove simple rules, including the power rule for n=-2, -1, -1/2, 1/2, 1, 2, as well as sum and difference rules. 61
b. Compute certain limits. 77
3. Find a tangent line to the graph of a function at a given point. 33, 55, o2, o4, o5

----

== Section 3.2 (Rebecca) ==

Homework

Written: 2,11,13,23,24,32,33,47,49,55,57
Online: As listed.

Goals

1. Be able to apply the product rule to find the derivative of functions. 11,24,33,47,49,55,57, 02,04,05,06,07,09
2. Be able to apply the quotient rule to find the derivative of functions. 2,13,23,24,32,47,49, 01,03,08,010,011
3. Be able to prove the product and quotient rules and variations. 55,57*

----

== Section 3.3 (Skyler/James) ==

Homework

Written: 9, 10, 18, 20, 35, 42, 45, 49
Online: As listed. Change ordinal number suffixes on #3 (e.g. 83th makes no sense)

Goals

1. Know derivatives for the 6 basic trig functions. Be able to use these to compute derivatives of more complicated functions.
2. Know limits [2] and [3] in the book. Be able to use these to solve other limits involving trig functions
3. Know the "periodicity" of differentiation of trig functions--i.e. the fourth derivative of the sine function is again the sine function
4. Be able to use trig identities to find derivatives and limits
5. Use trig derivatives in various real world problems.

----

== Section 3.4 (Savannah) ==

Homework

Written: 7, 24, 25, 31, 37, 47, 63, 65, 71, 84, 89
Online: Kill O6, O7, O8; Reorder problems (see Dr. Purcell); Add #77 from the book to online (change "particle" to "point")

Goals

1. Fluency in chain rule: Given a composition of functions, use the chain rule and other rules to compute its derivative quickly. 7, 25, 37, 47, O1, O2, O3, O9, O10, O11, O12, O13, O14, O15, O16
2. Given values of two functions and their derivatives at certain points, use the formal statement of the chain rule to compute the derivative of compositions of these functions. 63, 65, 71, O4, O5
3. Given a word problem involving the composition of functions, compute rates of change. Interpret the meaning of the derivative as a particular rate of change. 84
4. Use the chain rule and definition of even/odd functions to prove properties of derivatives of even/odd functions. 89
5. Compute the derivative of a^x. 24, 31

----

== Section 3.5 (Rebecca) ==
Homework

Written: 3,11,12,15,18,21,23,34,41,57 (Add 43,67)
Online: Kill 03,014; Replace 08 with 35 from the book and make it online

Goals

1) For an implicitly defined equation find dy/dx, dx/dy, and d^2y/dx^2 using implicit differentiation.
2) Find the tangent line to an implicitly defined function.
3) Know the derivatives of the inverse trig functions.
----

== Section 3.6 (Sam) ==

----

== Section 3.8 (Jessica) ==

Homework:

Written: 14, 15
Online: kill o5 (pH scale)

Goals:

Use exponential functions to model each of the following:
1. Population growth. o1, o3, o4
2. Radioactive decay. o6, o7
Find half-life, or given half-life, find equation.
3. Cooling. 14, 15
4. Interest. o2, o8
- Derive the formula for compounding interest in n periods o2, o8 (see also section 1.3, number 60)
- Use expression of e as a limit to find formula for continuously compounded interest.
----

== Section 3.9 (Mark) ==

Homework

Written: 5, 22, 33, 35, 37, 42
Online: Kill O1, O4, O7, O8, O11, O12, O13; remove geometry formulas from online questions.

Goals

1. Solve related rates problems. (all homework problems)
a) Draw picture
b) Recognize variables from the problem, and rates of change as their derivatives.
c) Write equations relating variables (using geometry, trigonometry)
d) Differentiate, remembering to use the chain rule.
e) Plug in specific values to determine the requested answer.

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== Section 3.10 (Tyler) ==

Homework
Written 1,2,3,5,23,28,43,44
Online Kill O4, O5, Renumber so that O3 is first and O1 is third. Possibly add one more. Change the hint in O1 to read "Try using linear approximation..."

Goals
1. Compute the linearization of a function at a point. 1,2,3,5, O2, O3
- Explain the relationship between the linearization and the tangent line. 5, O2
- Explain how linearization is useful. 23,28
2. Use the linearization at a point x=a to approximate the value of a function f(a+epsilon). 23,28, O1, O2

Suggestion: merge 3.10 and 3.11 for written as well as online.
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== Section 3.11 (Savannah) ==

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== Section 4.1 ==

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== Section 4.2 ==

----

== Section 4.3 ==

Goals:

1. Know and apply the increasing/decreasing test o1, o5
2. Find local maxima and minima (First and Second derivative tests, endpoints) 21,23
3. Know the definition of concavity and how to determine concavity (The concavity test) o2, o5, 72
4. Know what an inflection point is and how to find it o2
5. Know how the above information affects the graph of a function
Draw a graph such that.... 25, 28
Given a graph, where is concavity positive, etc.? 1, 6, 7, 31 and a new online problem similar to problem 8 on page 295

Changes:

o3 and o4 are to be moved to section 4.5 and a problem is to be added online that is like problem 8.
----

== Section 4.4 ==

----

== Section 4.5 (Skyler) ==

Goals:

Use all of the following to draw graphs of functions:

* Algebraic Properties (Domain, intercepts, even/odd/periodic)

* Limit Properties (Asymptotes, singularities, discontinuities)

* Derivative Properties (Increasing/decreasing, extrema, concavity/inflection)

Written homework - as listed
Online homework - pulled from 4.3 (O3 and O4)

----

== Section 4.7 ==

----

== Section 4.8 (Mark) ==

Goals

1. Use Newton's Method to approximate roots/solutions of equations.
2. Derive the formula for Newton's Method from the tangent line equation.
3. Give an example of a function 'f' and a starting point 'x_1' that makes Newton's Method fail.

Homework

Written: 1, 2, 3, 4, 29.
Online: Kill O1, Add new online problem with graph.

----

== Section 4.9 (Tyler)==

Goals:
1. Use the Mean value Theorem to prove Theorem 1.
2. Memorize the table of anti-differentiation formulas on page 341 and
relate it to the table of derivatives on the inside back cover of the book. 12, 15, 20, 27, 33, 49, 51, O1-O12
3. Use the table and Theorem 1 to find *all* antiderivatives of various functions. 12, 15, 20, 27, 49, 51
4. Use Theorem 1 and known differentiation formulas to find a unique anti-derivative with given values at specified points. 33, 53, 67,
69, 70, 74,O2,O3, O4,O5,O10, O11, O13
5. Use anti-differentiation to solve physical and economic problems, especially problems of motion. 67, 69, 70, 74, O4, O5

Changes to problems:
Written: Add story problems: 67, 69, 70, 74
Online: too long, Delete O1, O6, O12.
Adjust numbers in 04 to give a clean answer.
O10--change so that F(0) not equal to 0 (else they will get right answer for wrong reason).
O8 and O9 problem that Arctan/Arcsin not accepted--just arctan/arcsin.
O13 should not say "the anti-derivative, but rather "an anti-derivative".

----

== Appendix E (Paul)==
Homework:

Written: 1, 3, 5, 10, 11, 15, 17, 20, 23, 41, 43
Online: o1, o2, o3, o4, o5, o6, o7, o8, o9, o10

Goals

1. Write expanded sums in sigma notation (1, 3, 5, 10) and expand sums written in sigma notation (11, 15, 17, 20).
Numerically evaluate sums (23). Apply appropriate rules to distribute constants or expand sums of sums.
2. Explicitly evaluate $\sigma_{i=1}^n f(i)$ in terms of 'n', where 'f(x)' is a polynomial in x of degree at most 3.
Numerically evaluate this sum when 'i' runs over a given range of integers.
3. Evaluate telescoping sums (41) and limits of sums (43).

----

== Section 5.1 ==

----

== Section 5.2 ==

----

== Section 5.3 ==

----

== Section 5.4 ==

----

== Section 5.5 ==

----

Math 112 Calculus Learning Goals

2009-08-19T19:50:46Z

Pmj5: /* Appendix E (Paul) */

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== Section 1.1 (Jessica) ==

Homework:

Written: 25, 31, 40, 43, 55, 56, 64, 65
Online: 3, 4, 5, 6, 8, 9, 11. Modify 12 (Given f like 66 or 67 in book, find f(-x). Is f even or odd or neither?) Add problem like 51, 57 in book.

Goals

1. Given a function described algebraically, find the
- domain 31, 07, 08,
- range,
- value at a given number 04,
- value when we plug in another function, e.g. difference quotient 04, 05, 25.
2. Convert one representation of a function to another.
- verbal to/from graphical 03
-algebraic to/from graphical 09
-verbal to/from algebraic 55, 56, 2 new online
3. Piecewise functions: Given an algebraic description of a piecewise function, find its domain and sketch its graph. 011, 43.
Be able to write the absolute value as a piecewise function, and use this to sketch its graph. 40, 09
4. Use the definition of an even/odd function to decide whether a function described algebraically is even or odd. 012, 65
Use symmetry to decide whether a graph is an even or odd function. 64

----

== Section 1.2 (Sam) ==

Homework:

Written: 4,5,7,9,12 and Appdx D: 24, 37*, 46
Online: O1, O2, O4, O5 (these are the numbers under the current numbering),
and Appdx D: 29, 60, 69
The * on problem 37 means a calculator will be necessary.

Goals

1. Writing down a linear function given a set of information 5,12,O4,O5
2. Recognizing what a function's graph should look like 4,7,O1
3. Based on a graph, write down a function O2
4. Writing down a polynomial based on information 9, O2
5. Trig review (definitions, proofs, values of trig functions at particular points) Appdx D:24, 37, 46, 29, 60, 69 (the latter three going online)

# Trig to know, including problems to skim for practice. (Jessica)

1. Given angles in degrees and radians, draw the angle, convert from radians to degrees and vice versa. (See appendix D: 1-12)
2. Write down sin, cos, tan, sec, scs, cot of any angle 0, pi/6, pi/4, pi/3, pi/2, and all angles obtained from these angles by adding a multiple of pi/2. (appendix D:23-28)
3. Given a right triangle with side measurements, write sin, cos, tan, sec, csc, cot of any of the angles of the triangle in terms of the side lengths. (Appendix D: 35-38)
Similarly, given one trig ratio, find the others. (Appendix D: 29-34)
4. Graph trig functions. (app D: 77-82)
5. Use trig identities to simplify expressions (appendix D: 43-57 odd, 59-63), and to solve equations and inequalities involving trig functions (app D: 65-76).
* Most important identities: sin^2(x) + cos^2(x) = 1
* sin(-x) = -sin(x)
* cos(-x) = cos(x)
* sin(x+y) = sin(x)cos(y) + cos(x)sin(y)
* cos(x+y) = cos(x)cos(y) - sin(x)sin(y)
Using the above, you can derive other identities:
* 1+tan^2(x)=sec^2(x); 1+cot^2(x) = csc^2(x)
* sin(x-y)
* cos(x-y)
* sin(2x)
* cos(2x)

----

== Section 1.3 (Savannah) ==

Homework

Written: 1,7,13, 14, 22, 39, 46, 60, 65
Online: Modify 1 to be y as a function of x. Replace 6 with a different word problem. Add 3 from section 1.2 online assignment.

Goals

1. Apply and recognize transformations of functions:
- Translating, Stretching, Reflecting, Absolute Value 1
- Graph --> Algebraic 7, O1, O5, 1.2-O3
- Algebraic --> Graph 13, 14, 22, O7, O8
2. Given functions f and g, find:
- f+g
- f-g
- fg
- f/g
- domains for above functions. O2, O9
3. Understand and apply compositions of functions:
- Given functions f and g, find g o f and f o g. 28, 39, 60, O3, O6
- Given f o g, find functions f and g. 46, 65, O4

----

== Section 1.5 (James) ==

Homework:

Written: 11, 15, 18, 19, 21, 25, 29
Online: o2, o3, o4, o5, one more word problem

Goals

1. Graph exponential functions a^x for a<1, a>1, as well as transformations of exponential functions from section 1.3. Find domains. 11, 15, 21, o2
2. Given two points that an exponential function passes through, find the equation of the exponential function. 18, o3.
3. Use laws of exponents to simplify exponential expressions, to show various facts about exponential functions. 19, 29.
4. Convert a word problem involving population change into an algebraic expression involving exponential functions. Use this to find sizes of populations at given times, and to graph. 25, o4, o5.

----

== Section 1.6 (McKay) ==

Homework

Written: 12,16,18,19,23,35,45,48,54,60,67,71,73
Online: 01,02,03,04,05,06,07,08,010,011,012,013,016,018

Goals

1) Be able to tell if a function is one-to-one
-algebraically 12
-graphically 01
2) Know the definition of an inverse function and be able to use the steps to find the inverse
-algebraically 16,19,23,54,02,03,04,05,06
-graphically 18,45,73,07
3)Know the laws of logarithms and be able to solve and simplify logarithmic and exponential expressions. 35,48,06,08,010,011,012
4)Know the inverses of the trigonometric functions
- domain and range 71
- values at a point 60, 013,016
- simplify a trig function composed with an inverse trig function 67, o18

----

== Section 2.1 (Tyler) ==

Homework

Written:4*, 5*? (require calculator)

Goals

1. Know definitions of tangent and secant line. Compute the slope of a secant line. 4
2. Know definition of average velocity and the idea of instantaneous velocity. Compute an average velocity. 5
3. Explain how average velocity is equivalent to slope of a secant line, same for instantaneous velocity and slope of tangent line.

----

== Section 2.2 (Tyler) ==

Homework

Written: 6, 7, 9, 16, 27, 32, 34a, 40
Online: kill O2, O5. Modify: O3 fix so different parts don't have same answer.
O6, O7, O8: Use the same input notation for infinity. O8 Delete irrelevant lecture about asymptotes.

Goals:

1. Idea of a limit: (Note: the "definition" of a limit in this section is sloppy and is not a real definition of a limit. See section 2.4)
* Find the limit of a function at a point from its graph, including when the limit does not exist. 6,7, O1, O3
* Recognize the difference between the limit of a function at a point and the value of the function at a point. 6, 7, O1
* Give examples of functions that have prescribed limits at certain points. 16
* Understand that calculators and tables can entice you to guess a wrong limit. Give examples of functions whose limit is not what you would guess from plugging in values to your calculator. (no problems)
2. Idea (and notation) for one-sided limit: Do the same things for one-sided limits as done before for regular limits. 6, 7, O1, O3, O4, O6, O7, O8
3. Explain how one-sided and two-sided limits are connected. Use this to compute limits. 6, 7, O1, O3, O4, O6, O7, O8
4. Infinite limits:
* Explain why infinity is not a number and how the definition of an infinite limit gets around this.
* Find all vertical asymptotes for a function. 9, 34a
* For rational functions, find when a limit is infinity and when it is negative infinity. 27, 32, O6, O7, O8, 40

----

== Section 2.3a (Rebecca) ==

Homework

Written: 10,15,19,20,21,22,28,29
Online: Get rid of 3

Goals

1. Be able to apply limit laws to simplify limits
- algebraically 01
- graphically 02
2. Recognize when you can directly substitute to compute a limit and when you cannot. 10, 03, 04, 05, 06
3. Use algebra to simply and find limits.
- for polynomials over polynomials. 15, 19, 20, o5, o6, o7, o8, o9, o10
- expressions with square roots. 21, 22, 29, o11
- expressions with multiple fractions. 28, 29, o12

----

== Section 2.3b (Drew) ==

Homework

Written: 36-39, 42, 55, 56, 58
Online: Add o2 from 2.2. Kill o3, o4. Modify o6 so that direct substitution doesn't work. Add two problems using substitution.

Goals

1. Use the Squeeze Theorem to find limits. 36, 37, 38, o5, o6, o7
2. Evaluate limits that involve absolute values and other piecewise functions. 39, 42, o1, o2, o2 from 2.2
3. Use limit laws to find lim f(x) given that lim g(f(x)) = L. 55, 56, 58
4. Use substitution to rewrite lim g(f(x)) as lim g(u). 2 new online problems.

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== Section 2.4 (Jessica) ==
Homework:
Written: 2, 14, 15, 16, 19, 20
Online: kill o6 (done), modify problems to find the *largest* delta (done).

Goals:
1. Use a graph showing a function and lines y=L+epsilon, y=L-epsilon to find the largest value of delta so that if 0<|x-a|<delta, then |f(x)-L|<epsilon. 2
2. Know the epsilon-delta definition of a limit. Given a limit and a value for epsilon, be able to find the largest value of delta corresponding to that epsilon. 14, o1, o2, o3, o4, o5
3. Use the epsilon-delta definition of a limit to prove a statement about the limit of a linear function. 15, 16, 19, 20.

----

== Section 2.5a (McKay/Savannah) ==

Written:4, 6, 15, 18, 22, 23, 43ab, 58
Online - Kill O2-O4, O11-O16; Change O8, O9, O10 to ask, "Where is the following function continuous?"; Add a graph problem where the student must identify where the graph is discontinuous; Add a graph problem where the student must identify the types of discontinuities

1. Given the graph of a function, be able to tell:
a) where it is continuous. 4
b) where it is discontinuous, and the type of discontinuity. 6
2. Given a function described algebraically, be able to tell:
a) where it is continuous. 22, 23, O1, O8, O9, O10
b) where it is discontinuous and types of discontinuities. 15, 18, 43(a,b), O5, O6, O7
This includes functions from Theorem 7, piecewise functions, and combinations of functions.
3. Use limit laws and the definition of continuity to prove facts about continuity (e.g. Theorem 4). 58

----

== Section 2.5b (Mark) ==

Homework

Written: 28, 31, 32, 36, 37, 39, 41, 45, 47, 49, 65
Online: Kill 3, 5, 6, 7. Add 3 problems: Find intervals where f is positive, negative. One quadratic, easy factor; one rational with linear top and bottom; one rational with quadratic top, perfect square bottom.

Goals

1. Use continuity to evaluate limits 31,32
2. Continuity of piecewise functions
-determine if a piecewise function is continuous (or continuous from the right/left) 36,37,39
-determine parameters to make a piecewise function continuous 41,O1,O2,O4
3. Given a composition of functions, tell where it is continuous/discontinuous. 28
4. Use the Intermediate Value Theorem to prove the existence of solutions to equations. 45,47,49,65
5. Find intervals where a continuous function is positive/negative. (3 new online problems)

----

== Section 2.6 (Skyler) ==

Homework:
Written: 4, 5, 7, 13, 33, 43, 48
Online: Drop #15

Goals

1. Understand definition of limit at +/- inf
2. Find the limit of a rational function as x -> +/-inf and equations of horizontal asymptotes (when they exist)
3. Identify functions whose limit at +/- inf does not exist or whose limit is infinite
4. Compute limits at infinity by graphs and algebraic techniques

----

== Section 2.7 (Sam) ==
Homework:
Written: 5,9,19, 20, 43,44, 46
Online: eliminate o7,o9,o12

Goals:
1. Given an algebraic function, take the derivative (using the definition of the derivative)
Write down an equation for a tangent line through a point on a graph 3, 5, 05, o10, o11
2. Given a real life situation, interpret instantaneous change or average of change 43, 44, 46
Given a position function (or graph thereof), interpret the graph or find (average) velocity o1, o2, o3
3. Sketch the graph of a function such that certain criteria are met (values at certain points, derivatives, etc.) 19, 20
4. Given a limit expression, write down the point and function whose derivative it represents o6, o8
5. Write both formal definitions of a derivative
Explain how the derivative can be interpreted as slope of a line, velocity, instantaneous change

Changes:
Added: 43, 44
Eliminating: 2,25, o7, o9, o12

----

== Section 2.8 (Savannah) ==

Homework

Written: 1, 3, 5, 6, 7, 9, 11, 19, 20, 25, 35, 36, 37, 38, 44, Prove that if a function f is differentiable at a, then f is continuous at a.
Online: Kill O2, O3, Change O9 to f'(x)

Goals

1. Given a function f, find a formula for the derivative of f using f'(x) = lim (h->0) [f(x+h)-f(x)]/h, and be able to state the domain of f'(x). 19, 20, 25, O9, O10
2. Given a graph of a function f, be able to find the graph of f', f''. Given a graph of a function f', f'', be able to find th graph of f. 1, 3, 5, 6, 7, 9, 11, O1, O4, O5
3. Recognize when and why a function fails to be differentiable: 35, 36, 37, 38
a) corner
b) discontinuity (Theorem 4)
c) vertical tangent line
4. Be able to compute and apply higher derivatives: O6, O7, O8, 44
- (f')'=f''
- position [s(t)], velocity [v(t)=s'(t)], acceleration [a(t)=v'(t)=s''(t)]
5. Prove that if f is differentiable at a, then f is continuous at a. [Written homework]

----

== Section 3.1 (Drew) ==

Homework:
Written: 2, 13, 17, 20, 25, 28, 29, 33, 55, 61, 77
Online: Kill o6 last part, o14, o8

Goals:
1. Compute derivatives using:
a. The power rule. Especially, be able to rewrite functions so the power rule can be applied. 13, 17, 20, 25, 28, 29, o1, o3, o7, o8, o9, o10, o11, o12
b. The derivative of e^x. 2, 17, 28, o12
c. Sums, differences, and scalar multiples of functions you can differentiate with these rules. (almost all problems)
2. Use the definition of the derivative and limit laws to:
a. Prove simple rules, including the power rule for n=-2, -1, -1/2, 1/2, 1, 2, as well as sum and difference rules. 61
b. Compute certain limits. 77
3. Find a tangent line to the graph of a function at a given point. 33, 55, o2, o4, o5

----

== Section 3.2 (Rebecca) ==

Homework

Written: 2,11,13,23,24,32,33,47,49,55,57
Online: As listed.

Goals

1. Be able to apply the product rule to find the derivative of functions. 11,24,33,47,49,55,57, 02,04,05,06,07,09
2. Be able to apply the quotient rule to find the derivative of functions. 2,13,23,24,32,47,49, 01,03,08,010,011
3. Be able to prove the product and quotient rules and variations. 55,57*

----

== Section 3.3 (Skyler/James) ==

Homework

Written: 9, 10, 18, 20, 35, 42, 45, 49
Online: As listed. Change ordinal number suffixes on #3 (e.g. 83th makes no sense)

Goals

1. Know derivatives for the 6 basic trig functions. Be able to use these to compute derivatives of more complicated functions.
2. Know limits [2] and [3] in the book. Be able to use these to solve other limits involving trig functions
3. Know the "periodicity" of differentiation of trig functions--i.e. the fourth derivative of the sine function is again the sine function
4. Be able to use trig identities to find derivatives and limits
5. Use trig derivatives in various real world problems.

----

== Section 3.4 (Savannah) ==

Homework

Written: 7, 24, 25, 31, 37, 47, 63, 65, 71, 84, 89
Online: Kill O6, O7, O8; Reorder problems (see Dr. Purcell); Add #77 from the book to online (change "particle" to "point")

Goals

1. Fluency in chain rule: Given a composition of functions, use the chain rule and other rules to compute its derivative quickly. 7, 25, 37, 47, O1, O2, O3, O9, O10, O11, O12, O13, O14, O15, O16
2. Given values of two functions and their derivatives at certain points, use the formal statement of the chain rule to compute the derivative of compositions of these functions. 63, 65, 71, O4, O5
3. Given a word problem involving the composition of functions, compute rates of change. Interpret the meaning of the derivative as a particular rate of change. 84
4. Use the chain rule and definition of even/odd functions to prove properties of derivatives of even/odd functions. 89
5. Compute the derivative of a^x. 24, 31

----

== Section 3.5 (Rebecca) ==
Homework

Written: 3,11,12,15,18,21,23,34,41,57 (Add 43,67)
Online: Kill 03,014; Replace 08 with 35 from the book and make it online

Goals

1) For an implicitly defined equation find dy/dx, dx/dy, and d^2y/dx^2 using implicit differentiation.
2) Find the tangent line to an implicitly defined function.
3) Know the derivatives of the inverse trig functions.
----

== Section 3.6 (Sam) ==

----

== Section 3.8 (Jessica) ==

Homework:

Written: 14, 15
Online: kill o5 (pH scale)

Goals:

Use exponential functions to model each of the following:
1. Population growth. o1, o3, o4
2. Radioactive decay. o6, o7
Find half-life, or given half-life, find equation.
3. Cooling. 14, 15
4. Interest. o2, o8
- Derive the formula for compounding interest in n periods o2, o8 (see also section 1.3, number 60)
- Use expression of e as a limit to find formula for continuously compounded interest.
----

== Section 3.9 (Mark) ==

Homework

Written: 5, 22, 33, 35, 37, 42
Online: Kill O1, O4, O7, O8, O11, O12, O13; remove geometry formulas from online questions.

Goals

1. Solve related rates problems. (all homework problems)
a) Draw picture
b) Recognize variables from the problem, and rates of change as their derivatives.
c) Write equations relating variables (using geometry, trigonometry)
d) Differentiate, remembering to use the chain rule.
e) Plug in specific values to determine the requested answer.

----

== Section 3.10 (Tyler) ==

Homework
Written 1,2,3,5,23,28,43,44
Online Kill O4, O5, Renumber so that O3 is first and O1 is third. Possibly add one more. Change the hint in O1 to read "Try using linear approximation..."

Goals
1. Compute the linearization of a function at a point. 1,2,3,5, O2, O3
- Explain the relationship between the linearization and the tangent line. 5, O2
- Explain how linearization is useful. 23,28
2. Use the linearization at a point x=a to approximate the value of a function f(a+epsilon). 23,28, O1, O2

Suggestion: merge 3.10 and 3.11 for written as well as online.
----

== Section 3.11 (Savannah) ==

----

== Section 4.1 ==

----

== Section 4.2 ==

----

== Section 4.3 ==

Goals:

1. Know and apply the increasing/decreasing test o1, o5
2. Find local maxima and minima (First and Second derivative tests, endpoints) 21,23
3. Know the definition of concavity and how to determine concavity (The concavity test) o2, o5, 72
4. Know what an inflection point is and how to find it o2
5. Know how the above information affects the graph of a function
Draw a graph such that.... 25, 28
Given a graph, where is concavity positive, etc.? 1, 6, 7, 31 and a new online problem similar to problem 8 on page 295

Changes:

o3 and o4 are to be moved to section 4.5 and a problem is to be added online that is like problem 8.
----

== Section 4.4 ==

----

== Section 4.5 (Skyler) ==

Goals:

Use all of the following to draw graphs of functions:

* Algebraic Properties (Domain, intercepts, even/odd/periodic)

* Limit Properties (Asymptotes, singularities, discontinuities)

* Derivative Properties (Increasing/decreasing, extrema, concavity/inflection)

Written homework - as listed
Online homework - pulled from 4.3 (O3 and O4)

----

== Section 4.7 ==

----

== Section 4.8 (Mark) ==

Goals

1. Use Newton's Method to approximate roots/solutions of equations.
2. Derive the formula for Newton's Method from the tangent line equation.
3. Give an example of a function 'f' and a starting point 'x_1' that makes Newton's Method fail.

Homework

Written: 1, 2, 3, 4, 29.
Online: Kill O1, Add new online problem with graph.

----

== Section 4.9 (Tyler)==

Goals:
1. Use the Mean value Theorem to prove Theorem 1.
2. Memorize the table of anti-differentiation formulas on page 341 and
relate it to the table of derivatives on the inside back cover of the book. 12, 15, 20, 27, 33, 49, 51, O1-O12
3. Use the table and Theorem 1 to find *all* antiderivatives of various functions. 12, 15, 20, 27, 49, 51
4. Use Theorem 1 and known differentiation formulas to find a unique anti-derivative with given values at specified points. 33, 53, 67,
69, 70, 74,O2,O3, O4,O5,O10, O11, O13
5. Use anti-differentiation to solve physical and economic problems, especially problems of motion. 67, 69, 70, 74, O4, O5

Changes to problems:
Written: Add story problems: 67, 69, 70, 74
Online: too long, Delete O1, O6, O12.
Adjust numbers in 04 to give a clean answer.
O10--change so that F(0) not equal to 0 (else they will get right answer for wrong reason).
O8 and O9 problem that Arctan/Arcsin not accepted--just arctan/arcsin.
O13 should not say "the anti-derivative, but rather "an anti-derivative".

----

== Appendix E (Paul)==
Homework:

Written: 1, 3, 5, 10, 11, 15, 17, 20, 23, 41, 43
Online: o1, o2, o3, o4, o5, o6, o7, o8, o9, o10

Goals

1. Write expanded sums in sigma notation and expand sums written in sigma notation.
Apply appropriate rules to distribute constants or expand sums of sums.
2. Explicitly evaluate $\sigma_{i=1}^n f(i)$ in terms of 'n', where 'f(x)' is a polynomial in x of degree at most 3.
Numerically evaluate this sum when 'i' runs over a given range of integers.
3. Evaluate telescoping sums and limits of sums.

----

== Section 5.1 ==

----

== Section 5.2 ==

----

== Section 5.3 ==

----

== Section 5.4 ==

----

== Section 5.5 ==

----

Math 112 Calculus Learning Goals

2009-08-19T19:50:14Z

Pmj5: /* Appendix E (Paul) */

----

== Section 1.1 (Jessica) ==

Homework:

Written: 25, 31, 40, 43, 55, 56, 64, 65
Online: 3, 4, 5, 6, 8, 9, 11. Modify 12 (Given f like 66 or 67 in book, find f(-x). Is f even or odd or neither?) Add problem like 51, 57 in book.

Goals

1. Given a function described algebraically, find the
- domain 31, 07, 08,
- range,
- value at a given number 04,
- value when we plug in another function, e.g. difference quotient 04, 05, 25.
2. Convert one representation of a function to another.
- verbal to/from graphical 03
-algebraic to/from graphical 09
-verbal to/from algebraic 55, 56, 2 new online
3. Piecewise functions: Given an algebraic description of a piecewise function, find its domain and sketch its graph. 011, 43.
Be able to write the absolute value as a piecewise function, and use this to sketch its graph. 40, 09
4. Use the definition of an even/odd function to decide whether a function described algebraically is even or odd. 012, 65
Use symmetry to decide whether a graph is an even or odd function. 64

----

== Section 1.2 (Sam) ==

Homework:

Written: 4,5,7,9,12 and Appdx D: 24, 37*, 46
Online: O1, O2, O4, O5 (these are the numbers under the current numbering),
and Appdx D: 29, 60, 69
The * on problem 37 means a calculator will be necessary.

Goals

1. Writing down a linear function given a set of information 5,12,O4,O5
2. Recognizing what a function's graph should look like 4,7,O1
3. Based on a graph, write down a function O2
4. Writing down a polynomial based on information 9, O2
5. Trig review (definitions, proofs, values of trig functions at particular points) Appdx D:24, 37, 46, 29, 60, 69 (the latter three going online)

# Trig to know, including problems to skim for practice. (Jessica)

1. Given angles in degrees and radians, draw the angle, convert from radians to degrees and vice versa. (See appendix D: 1-12)
2. Write down sin, cos, tan, sec, scs, cot of any angle 0, pi/6, pi/4, pi/3, pi/2, and all angles obtained from these angles by adding a multiple of pi/2. (appendix D:23-28)
3. Given a right triangle with side measurements, write sin, cos, tan, sec, csc, cot of any of the angles of the triangle in terms of the side lengths. (Appendix D: 35-38)
Similarly, given one trig ratio, find the others. (Appendix D: 29-34)
4. Graph trig functions. (app D: 77-82)
5. Use trig identities to simplify expressions (appendix D: 43-57 odd, 59-63), and to solve equations and inequalities involving trig functions (app D: 65-76).
* Most important identities: sin^2(x) + cos^2(x) = 1
* sin(-x) = -sin(x)
* cos(-x) = cos(x)
* sin(x+y) = sin(x)cos(y) + cos(x)sin(y)
* cos(x+y) = cos(x)cos(y) - sin(x)sin(y)
Using the above, you can derive other identities:
* 1+tan^2(x)=sec^2(x); 1+cot^2(x) = csc^2(x)
* sin(x-y)
* cos(x-y)
* sin(2x)
* cos(2x)

----

== Section 1.3 (Savannah) ==

Homework

Written: 1,7,13, 14, 22, 39, 46, 60, 65
Online: Modify 1 to be y as a function of x. Replace 6 with a different word problem. Add 3 from section 1.2 online assignment.

Goals

1. Apply and recognize transformations of functions:
- Translating, Stretching, Reflecting, Absolute Value 1
- Graph --> Algebraic 7, O1, O5, 1.2-O3
- Algebraic --> Graph 13, 14, 22, O7, O8
2. Given functions f and g, find:
- f+g
- f-g
- fg
- f/g
- domains for above functions. O2, O9
3. Understand and apply compositions of functions:
- Given functions f and g, find g o f and f o g. 28, 39, 60, O3, O6
- Given f o g, find functions f and g. 46, 65, O4

----

== Section 1.5 (James) ==

Homework:

Written: 11, 15, 18, 19, 21, 25, 29
Online: o2, o3, o4, o5, one more word problem

Goals

1. Graph exponential functions a^x for a<1, a>1, as well as transformations of exponential functions from section 1.3. Find domains. 11, 15, 21, o2
2. Given two points that an exponential function passes through, find the equation of the exponential function. 18, o3.
3. Use laws of exponents to simplify exponential expressions, to show various facts about exponential functions. 19, 29.
4. Convert a word problem involving population change into an algebraic expression involving exponential functions. Use this to find sizes of populations at given times, and to graph. 25, o4, o5.

----

== Section 1.6 (McKay) ==

Homework

Written: 12,16,18,19,23,35,45,48,54,60,67,71,73
Online: 01,02,03,04,05,06,07,08,010,011,012,013,016,018

Goals

1) Be able to tell if a function is one-to-one
-algebraically 12
-graphically 01
2) Know the definition of an inverse function and be able to use the steps to find the inverse
-algebraically 16,19,23,54,02,03,04,05,06
-graphically 18,45,73,07
3)Know the laws of logarithms and be able to solve and simplify logarithmic and exponential expressions. 35,48,06,08,010,011,012
4)Know the inverses of the trigonometric functions
- domain and range 71
- values at a point 60, 013,016
- simplify a trig function composed with an inverse trig function 67, o18

----

== Section 2.1 (Tyler) ==

Homework

Written:4*, 5*? (require calculator)

Goals

1. Know definitions of tangent and secant line. Compute the slope of a secant line. 4
2. Know definition of average velocity and the idea of instantaneous velocity. Compute an average velocity. 5
3. Explain how average velocity is equivalent to slope of a secant line, same for instantaneous velocity and slope of tangent line.

----

== Section 2.2 (Tyler) ==

Homework

Written: 6, 7, 9, 16, 27, 32, 34a, 40
Online: kill O2, O5. Modify: O3 fix so different parts don't have same answer.
O6, O7, O8: Use the same input notation for infinity. O8 Delete irrelevant lecture about asymptotes.

Goals:

1. Idea of a limit: (Note: the "definition" of a limit in this section is sloppy and is not a real definition of a limit. See section 2.4)
* Find the limit of a function at a point from its graph, including when the limit does not exist. 6,7, O1, O3
* Recognize the difference between the limit of a function at a point and the value of the function at a point. 6, 7, O1
* Give examples of functions that have prescribed limits at certain points. 16
* Understand that calculators and tables can entice you to guess a wrong limit. Give examples of functions whose limit is not what you would guess from plugging in values to your calculator. (no problems)
2. Idea (and notation) for one-sided limit: Do the same things for one-sided limits as done before for regular limits. 6, 7, O1, O3, O4, O6, O7, O8
3. Explain how one-sided and two-sided limits are connected. Use this to compute limits. 6, 7, O1, O3, O4, O6, O7, O8
4. Infinite limits:
* Explain why infinity is not a number and how the definition of an infinite limit gets around this.
* Find all vertical asymptotes for a function. 9, 34a
* For rational functions, find when a limit is infinity and when it is negative infinity. 27, 32, O6, O7, O8, 40

----

== Section 2.3a (Rebecca) ==

Homework

Written: 10,15,19,20,21,22,28,29
Online: Get rid of 3

Goals

1. Be able to apply limit laws to simplify limits
- algebraically 01
- graphically 02
2. Recognize when you can directly substitute to compute a limit and when you cannot. 10, 03, 04, 05, 06
3. Use algebra to simply and find limits.
- for polynomials over polynomials. 15, 19, 20, o5, o6, o7, o8, o9, o10
- expressions with square roots. 21, 22, 29, o11
- expressions with multiple fractions. 28, 29, o12

----

== Section 2.3b (Drew) ==

Homework

Written: 36-39, 42, 55, 56, 58
Online: Add o2 from 2.2. Kill o3, o4. Modify o6 so that direct substitution doesn't work. Add two problems using substitution.

Goals

1. Use the Squeeze Theorem to find limits. 36, 37, 38, o5, o6, o7
2. Evaluate limits that involve absolute values and other piecewise functions. 39, 42, o1, o2, o2 from 2.2
3. Use limit laws to find lim f(x) given that lim g(f(x)) = L. 55, 56, 58
4. Use substitution to rewrite lim g(f(x)) as lim g(u). 2 new online problems.

----

== Section 2.4 (Jessica) ==
Homework:
Written: 2, 14, 15, 16, 19, 20
Online: kill o6 (done), modify problems to find the *largest* delta (done).

Goals:
1. Use a graph showing a function and lines y=L+epsilon, y=L-epsilon to find the largest value of delta so that if 0<|x-a|<delta, then |f(x)-L|<epsilon. 2
2. Know the epsilon-delta definition of a limit. Given a limit and a value for epsilon, be able to find the largest value of delta corresponding to that epsilon. 14, o1, o2, o3, o4, o5
3. Use the epsilon-delta definition of a limit to prove a statement about the limit of a linear function. 15, 16, 19, 20.

----

== Section 2.5a (McKay/Savannah) ==

Written:4, 6, 15, 18, 22, 23, 43ab, 58
Online - Kill O2-O4, O11-O16; Change O8, O9, O10 to ask, "Where is the following function continuous?"; Add a graph problem where the student must identify where the graph is discontinuous; Add a graph problem where the student must identify the types of discontinuities

1. Given the graph of a function, be able to tell:
a) where it is continuous. 4
b) where it is discontinuous, and the type of discontinuity. 6
2. Given a function described algebraically, be able to tell:
a) where it is continuous. 22, 23, O1, O8, O9, O10
b) where it is discontinuous and types of discontinuities. 15, 18, 43(a,b), O5, O6, O7
This includes functions from Theorem 7, piecewise functions, and combinations of functions.
3. Use limit laws and the definition of continuity to prove facts about continuity (e.g. Theorem 4). 58

----

== Section 2.5b (Mark) ==

Homework

Written: 28, 31, 32, 36, 37, 39, 41, 45, 47, 49, 65
Online: Kill 3, 5, 6, 7. Add 3 problems: Find intervals where f is positive, negative. One quadratic, easy factor; one rational with linear top and bottom; one rational with quadratic top, perfect square bottom.

Goals

1. Use continuity to evaluate limits 31,32
2. Continuity of piecewise functions
-determine if a piecewise function is continuous (or continuous from the right/left) 36,37,39
-determine parameters to make a piecewise function continuous 41,O1,O2,O4
3. Given a composition of functions, tell where it is continuous/discontinuous. 28
4. Use the Intermediate Value Theorem to prove the existence of solutions to equations. 45,47,49,65
5. Find intervals where a continuous function is positive/negative. (3 new online problems)

----

== Section 2.6 (Skyler) ==

Homework:
Written: 4, 5, 7, 13, 33, 43, 48
Online: Drop #15

Goals

1. Understand definition of limit at +/- inf
2. Find the limit of a rational function as x -> +/-inf and equations of horizontal asymptotes (when they exist)
3. Identify functions whose limit at +/- inf does not exist or whose limit is infinite
4. Compute limits at infinity by graphs and algebraic techniques

----

== Section 2.7 (Sam) ==
Homework:
Written: 5,9,19, 20, 43,44, 46
Online: eliminate o7,o9,o12

Goals:
1. Given an algebraic function, take the derivative (using the definition of the derivative)
Write down an equation for a tangent line through a point on a graph 3, 5, 05, o10, o11
2. Given a real life situation, interpret instantaneous change or average of change 43, 44, 46
Given a position function (or graph thereof), interpret the graph or find (average) velocity o1, o2, o3
3. Sketch the graph of a function such that certain criteria are met (values at certain points, derivatives, etc.) 19, 20
4. Given a limit expression, write down the point and function whose derivative it represents o6, o8
5. Write both formal definitions of a derivative
Explain how the derivative can be interpreted as slope of a line, velocity, instantaneous change

Changes:
Added: 43, 44
Eliminating: 2,25, o7, o9, o12

----

== Section 2.8 (Savannah) ==

Homework

Written: 1, 3, 5, 6, 7, 9, 11, 19, 20, 25, 35, 36, 37, 38, 44, Prove that if a function f is differentiable at a, then f is continuous at a.
Online: Kill O2, O3, Change O9 to f'(x)

Goals

1. Given a function f, find a formula for the derivative of f using f'(x) = lim (h->0) [f(x+h)-f(x)]/h, and be able to state the domain of f'(x). 19, 20, 25, O9, O10
2. Given a graph of a function f, be able to find the graph of f', f''. Given a graph of a function f', f'', be able to find th graph of f. 1, 3, 5, 6, 7, 9, 11, O1, O4, O5
3. Recognize when and why a function fails to be differentiable: 35, 36, 37, 38
a) corner
b) discontinuity (Theorem 4)
c) vertical tangent line
4. Be able to compute and apply higher derivatives: O6, O7, O8, 44
- (f')'=f''
- position [s(t)], velocity [v(t)=s'(t)], acceleration [a(t)=v'(t)=s''(t)]
5. Prove that if f is differentiable at a, then f is continuous at a. [Written homework]

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== Section 3.1 (Drew) ==

Homework:
Written: 2, 13, 17, 20, 25, 28, 29, 33, 55, 61, 77
Online: Kill o6 last part, o14, o8

Goals:
1. Compute derivatives using:
a. The power rule. Especially, be able to rewrite functions so the power rule can be applied. 13, 17, 20, 25, 28, 29, o1, o3, o7, o8, o9, o10, o11, o12
b. The derivative of e^x. 2, 17, 28, o12
c. Sums, differences, and scalar multiples of functions you can differentiate with these rules. (almost all problems)
2. Use the definition of the derivative and limit laws to:
a. Prove simple rules, including the power rule for n=-2, -1, -1/2, 1/2, 1, 2, as well as sum and difference rules. 61
b. Compute certain limits. 77
3. Find a tangent line to the graph of a function at a given point. 33, 55, o2, o4, o5

----

== Section 3.2 (Rebecca) ==

Homework

Written: 2,11,13,23,24,32,33,47,49,55,57
Online: As listed.

Goals

1. Be able to apply the product rule to find the derivative of functions. 11,24,33,47,49,55,57, 02,04,05,06,07,09
2. Be able to apply the quotient rule to find the derivative of functions. 2,13,23,24,32,47,49, 01,03,08,010,011
3. Be able to prove the product and quotient rules and variations. 55,57*

----

== Section 3.3 (Skyler/James) ==

Homework

Written: 9, 10, 18, 20, 35, 42, 45, 49
Online: As listed. Change ordinal number suffixes on #3 (e.g. 83th makes no sense)

Goals

1. Know derivatives for the 6 basic trig functions. Be able to use these to compute derivatives of more complicated functions.
2. Know limits [2] and [3] in the book. Be able to use these to solve other limits involving trig functions
3. Know the "periodicity" of differentiation of trig functions--i.e. the fourth derivative of the sine function is again the sine function
4. Be able to use trig identities to find derivatives and limits
5. Use trig derivatives in various real world problems.

----

== Section 3.4 (Savannah) ==

Homework

Written: 7, 24, 25, 31, 37, 47, 63, 65, 71, 84, 89
Online: Kill O6, O7, O8; Reorder problems (see Dr. Purcell); Add #77 from the book to online (change "particle" to "point")

Goals

1. Fluency in chain rule: Given a composition of functions, use the chain rule and other rules to compute its derivative quickly. 7, 25, 37, 47, O1, O2, O3, O9, O10, O11, O12, O13, O14, O15, O16
2. Given values of two functions and their derivatives at certain points, use the formal statement of the chain rule to compute the derivative of compositions of these functions. 63, 65, 71, O4, O5
3. Given a word problem involving the composition of functions, compute rates of change. Interpret the meaning of the derivative as a particular rate of change. 84
4. Use the chain rule and definition of even/odd functions to prove properties of derivatives of even/odd functions. 89
5. Compute the derivative of a^x. 24, 31

----

== Section 3.5 (Rebecca) ==
Homework

Written: 3,11,12,15,18,21,23,34,41,57 (Add 43,67)
Online: Kill 03,014; Replace 08 with 35 from the book and make it online

Goals

1) For an implicitly defined equation find dy/dx, dx/dy, and d^2y/dx^2 using implicit differentiation.
2) Find the tangent line to an implicitly defined function.
3) Know the derivatives of the inverse trig functions.
----

== Section 3.6 (Sam) ==

----

== Section 3.8 (Jessica) ==

Homework:

Written: 14, 15
Online: kill o5 (pH scale)

Goals:

Use exponential functions to model each of the following:
1. Population growth. o1, o3, o4
2. Radioactive decay. o6, o7
Find half-life, or given half-life, find equation.
3. Cooling. 14, 15
4. Interest. o2, o8
- Derive the formula for compounding interest in n periods o2, o8 (see also section 1.3, number 60)
- Use expression of e as a limit to find formula for continuously compounded interest.
----

== Section 3.9 (Mark) ==

Homework

Written: 5, 22, 33, 35, 37, 42
Online: Kill O1, O4, O7, O8, O11, O12, O13; remove geometry formulas from online questions.

Goals

1. Solve related rates problems. (all homework problems)
a) Draw picture
b) Recognize variables from the problem, and rates of change as their derivatives.
c) Write equations relating variables (using geometry, trigonometry)
d) Differentiate, remembering to use the chain rule.
e) Plug in specific values to determine the requested answer.

----

== Section 3.10 (Tyler) ==

Homework
Written 1,2,3,5,23,28,43,44
Online Kill O4, O5, Renumber so that O3 is first and O1 is third. Possibly add one more. Change the hint in O1 to read "Try using linear approximation..."

Goals
1. Compute the linearization of a function at a point. 1,2,3,5, O2, O3
- Explain the relationship between the linearization and the tangent line. 5, O2
- Explain how linearization is useful. 23,28
2. Use the linearization at a point x=a to approximate the value of a function f(a+epsilon). 23,28, O1, O2

Suggestion: merge 3.10 and 3.11 for written as well as online.
----

== Section 3.11 (Savannah) ==

----

== Section 4.1 ==

----

== Section 4.2 ==

----

== Section 4.3 ==

Goals:

1. Know and apply the increasing/decreasing test o1, o5
2. Find local maxima and minima (First and Second derivative tests, endpoints) 21,23
3. Know the definition of concavity and how to determine concavity (The concavity test) o2, o5, 72
4. Know what an inflection point is and how to find it o2
5. Know how the above information affects the graph of a function
Draw a graph such that.... 25, 28
Given a graph, where is concavity positive, etc.? 1, 6, 7, 31 and a new online problem similar to problem 8 on page 295

Changes:

o3 and o4 are to be moved to section 4.5 and a problem is to be added online that is like problem 8.
----

== Section 4.4 ==

----

== Section 4.5 (Skyler) ==

Goals:

Use all of the following to draw graphs of functions:

* Algebraic Properties (Domain, intercepts, even/odd/periodic)

* Limit Properties (Asymptotes, singularities, discontinuities)

* Derivative Properties (Increasing/decreasing, extrema, concavity/inflection)

Written homework - as listed
Online homework - pulled from 4.3 (O3 and O4)

----

== Section 4.7 ==

----

== Section 4.8 (Mark) ==

Goals

1. Use Newton's Method to approximate roots/solutions of equations.
2. Derive the formula for Newton's Method from the tangent line equation.
3. Give an example of a function 'f' and a starting point 'x_1' that makes Newton's Method fail.

Homework

Written: 1, 2, 3, 4, 29.
Online: Kill O1, Add new online problem with graph.

----

== Section 4.9 (Tyler)==

Goals:
1. Use the Mean value Theorem to prove Theorem 1.
2. Memorize the table of anti-differentiation formulas on page 341 and
relate it to the table of derivatives on the inside back cover of the book. 12, 15, 20, 27, 33, 49, 51, O1-O12
3. Use the table and Theorem 1 to find *all* antiderivatives of various functions. 12, 15, 20, 27, 49, 51
4. Use Theorem 1 and known differentiation formulas to find a unique anti-derivative with given values at specified points. 33, 53, 67,
69, 70, 74,O2,O3, O4,O5,O10, O11, O13
5. Use anti-differentiation to solve physical and economic problems, especially problems of motion. 67, 69, 70, 74, O4, O5

Changes to problems:
Written: Add story problems: 67, 69, 70, 74
Online: too long, Delete O1, O6, O12.
Adjust numbers in 04 to give a clean answer.
O10--change so that F(0) not equal to 0 (else they will get right answer for wrong reason).
O8 and O9 problem that Arctan/Arcsin not accepted--just arctan/arcsin.
O13 should not say "the anti-derivative, but rather "an anti-derivative".

----

== Appendix E (Paul)==
Homework:

Written: 1, 3, 5, 10, 11, 15, 17, 20, 23, 41, 43
Online: o1, o2, o3, o4, o5, o6, o7, o8, o9, o10

Goals

1. Write expanded sums in sigma notation and expand sums written in sigma notation.
Apply appropriate rules to distribute constants or expand sums of sums.
2. Explicitly evaluate $\sigma_{i=1}^n f(i)$ in terms of 'n', where 'f(x)' is a polynomial in x of degree at most 3. Numerically evaluate this sum when 'i' runs over a given range of integers.
3. Evaluate telescoping sums and limits of sums.

----

== Section 5.1 ==

----

== Section 5.2 ==

----

== Section 5.3 ==

----

== Section 5.4 ==

----

== Section 5.5 ==

----

Math 112 Calculus Learning Goals

2009-08-19T04:00:08Z

Pmj5: /* Appendix E (Paul) */

----

== Section 1.1 (Jessica) ==

Homework:

Written: 25, 31, 40, 43, 55, 56, 64, 65
Online: 3, 4, 5, 6, 8, 9, 11. Modify 12 (Given f like 66 or 67 in book, find f(-x). Is f even or odd or neither?) Add problem like 51, 57 in book.

Goals

1. Given a function described algebraically, find the
- domain 31, 07, 08,
- range,
- value at a given number 04,
- value when we plug in another function, e.g. difference quotient 04, 05, 25.
2. Convert one representation of a function to another.
- verbal to/from graphical 03
-algebraic to/from graphical 09
-verbal to/from algebraic 55, 56, 2 new online
3. Piecewise functions: Given an algebraic description of a piecewise function, find its domain and sketch its graph. 011, 43.
Be able to write the absolute value as a piecewise function, and use this to sketch its graph. 40, 09
4. Use the definition of an even/odd function to decide whether a function described algebraically is even or odd. 012, 65
Use symmetry to decide whether a graph is an even or odd function. 64

----

== Section 1.2 (Sam) ==

Homework:

Written: 4,5,7,9,12 and Appdx D: 24, 37*, 46
Online: O1, O2, O4, O5 (these are the numbers under the current numbering),
and Appdx D: 29, 60, 69
The * on problem 37 means a calculator will be necessary.

Goals

1. Writing down a linear function given a set of information 5,12,O4,O5
2. Recognizing what a function's graph should look like 4,7,O1
3. Based on a graph, write down a function O2
4. Writing down a polynomial based on information 9, O2
5. Trig review (definitions, proofs, values of trig functions at particular points) Appdx D:24, 37, 46, 29, 60, 69 (the latter three going online)

# Trig to know, including problems to skim for practice. (Jessica)

1. Given angles in degrees and radians, draw the angle, convert from radians to degrees and vice versa. (See appendix D: 1-12)
2. Write down sin, cos, tan, sec, scs, cot of any angle 0, pi/6, pi/4, pi/3, pi/2, and all angles obtained from these angles by adding a multiple of pi/2. (appendix D:23-28)
3. Given a right triangle with side measurements, write sin, cos, tan, sec, csc, cot of any of the angles of the triangle in terms of the side lengths. (Appendix D: 35-38)
Similarly, given one trig ratio, find the others. (Appendix D: 29-34)
4. Graph trig functions. (app D: 77-82)
5. Use trig identities to simplify expressions (appendix D: 43-57 odd, 59-63), and to solve equations and inequalities involving trig functions (app D: 65-76).
* Most important identities: sin^2(x) + cos^2(x) = 1
* sin(-x) = -sin(x)
* cos(-x) = cos(x)
* sin(x+y) = sin(x)cos(y) + cos(x)sin(y)
* cos(x+y) = cos(x)cos(y) - sin(x)sin(y)
Using the above, you can derive other identities:
* 1+tan^2(x)=sec^2(x); 1+cot^2(x) = csc^2(x)
* sin(x-y)
* cos(x-y)
* sin(2x)
* cos(2x)

----

== Section 1.3 (Savannah) ==

Homework

Written: 1,7,13, 14, 22, 39, 46, 60, 65
Online: Modify 1 to be y as a function of x. Replace 6 with a different word problem. Add 3 from section 1.2 online assignment.

Goals

1. Apply and recognize transformations of functions:
- Translating, Stretching, Reflecting, Absolute Value 1
- Graph --> Algebraic 7, O1, O5, 1.2-O3
- Algebraic --> Graph 13, 14, 22, O7, O8
2. Given functions f and g, find:
- f+g
- f-g
- fg
- f/g
- domains for above functions. O2, O9
3. Understand and apply compositions of functions:
- Given functions f and g, find g o f and f o g. 28, 39, 60, O3, O6
- Given f o g, find functions f and g. 46, 65, O4

----

== Section 1.5 (James) ==

Homework:

Written: 11, 15, 18, 19, 21, 25, 29
Online: o2, o3, o4, o5, one more word problem

Goals

1. Graph exponential functions a^x for a<1, a>1, as well as transformations of exponential functions from section 1.3. Find domains. 11, 15, 21, o2
2. Given two points that an exponential function passes through, find the equation of the exponential function. 18, o3.
3. Use laws of exponents to simplify exponential expressions, to show various facts about exponential functions. 19, 29.
4. Convert a word problem involving population change into an algebraic expression involving exponential functions. Use this to find sizes of populations at given times, and to graph. 25, o4, o5.

----

== Section 1.6 (McKay) ==

Homework

Written: 12,16,18,19,23,35,45,48,54,60,67,71,73
Online: 01,02,03,04,05,06,07,08,010,011,012,013,016,018

Goals

1) Be able to tell if a function is one-to-one
-algebraically 12
-graphically 01
2) Know the definition of an inverse function and be able to use the steps to find the inverse
-algebraically 16,19,23,54,02,03,04,05,06
-graphically 18,45,73,07
3)Know the laws of logarithms and be able to solve and simplify logarithmic and exponential expressions. 35,48,06,08,010,011,012
4)Know the inverses of the trigonometric functions
- domain and range 71
- values at a point 60, 013,016
- simplify a trig function composed with an inverse trig function 67, o18

----

== Section 2.1 (Tyler) ==

Homework

Written:4*, 5*? (require calculator)

Goals

1. Know definitions of tangent and secant line. Compute the slope of a secant line. 4
2. Know definition of average velocity and the idea of instantaneous velocity. Compute an average velocity. 5
3. Explain how average velocity is equivalent to slope of a secant line, same for instantaneous velocity and slope of tangent line.

----

== Section 2.2 (Tyler) ==

Homework

Written: 6, 7, 9, 16, 27, 32, 34a, 40
Online: kill O2, O5. Modify: O3 fix so different parts don't have same answer.
O6, O7, O8: Use the same input notation for infinity. O8 Delete irrelevant lecture about asymptotes.

Goals:

1. Idea of a limit: (Note: the "definition" of a limit in this section is sloppy and is not a real definition of a limit. See section 2.4)
* Find the limit of a function at a point from its graph, including when the limit does not exist. 6,7, O1, O3
* Recognize the difference between the limit of a function at a point and the value of the function at a point. 6, 7, O1
* Give examples of functions that have prescribed limits at certain points. 16
* Understand that calculators and tables can entice you to guess a wrong limit. Give examples of functions whose limit is not what you would guess from plugging in values to your calculator. (no problems)
2. Idea (and notation) for one-sided limit: Do the same things for one-sided limits as done before for regular limits. 6, 7, O1, O3, O4, O6, O7, O8
3. Explain how one-sided and two-sided limits are connected. Use this to compute limits. 6, 7, O1, O3, O4, O6, O7, O8
4. Infinite limits:
* Explain why infinity is not a number and how the definition of an infinite limit gets around this.
* Find all vertical asymptotes for a function. 9, 34a
* For rational functions, find when a limit is infinity and when it is negative infinity. 27, 32, O6, O7, O8, 40

----

== Section 2.3a (Rebecca) ==

Homework

Written: 10,15,19,20,21,22,28,29
Online: Get rid of 3

Goals

1. Be able to apply limit laws to simplify limits
- algebraically 01
- graphically 02
2. Recognize when you can directly substitute to compute a limit and when you cannot. 10, 03, 04, 05, 06
3. Use algebra to simply and find limits.
- for polynomials over polynomials. 15, 19, 20, o5, o6, o7, o8, o9, o10
- expressions with square roots. 21, 22, 29, o11
- expressions with multiple fractions. 28, 29, o12

----

== Section 2.3b (Drew) ==

Homework

Written: 36-39, 42, 55, 56, 58
Online: Add o2 from 2.2. Kill o3, o4. Modify o6 so that direct substitution doesn't work. Add two problems using substitution.

Goals

1. Use the Squeeze Theorem to find limits. 36, 37, 38, o5, o6, o7
2. Evaluate limits that involve absolute values and other piecewise functions. 39, 42, o1, o2, o2 from 2.2
3. Use limit laws to find lim f(x) given that lim g(f(x)) = L. 55, 56, 58
4. Use substitution to rewrite lim g(f(x)) as lim g(u). 2 new online problems.

----

== Section 2.4 (Jessica) ==
Homework:
Written: 2, 14, 15, 16, 19, 20
Online: kill o6 (done), modify problems to find the *largest* delta (done).

Goals:
1. Use a graph showing a function and lines y=L+epsilon, y=L-epsilon to find the largest value of delta so that if 0<|x-a|<delta, then |f(x)-L|<epsilon. 2
2. Know the epsilon-delta definition of a limit. Given a limit and a value for epsilon, be able to find the largest value of delta corresponding to that epsilon. 14, o1, o2, o3, o4, o5
3. Use the epsilon-delta definition of a limit to prove a statement about the limit of a linear function. 15, 16, 19, 20.

----

== Section 2.5a (McKay/Savannah) ==

Written:4, 6, 15, 18, 22, 23, 43ab, 58
Online - Kill O2-O4, O11-O16; Change O8, O9, O10 to ask, "Where is the following function continuous?"; Add a graph problem where the student must identify where the graph is discontinuous; Add a graph problem where the student must identify the types of discontinuities

1. Given the graph of a function, be able to tell:
a) where it is continuous. 4
b) where it is discontinuous, and the type of discontinuity. 6
2. Given a function described algebraically, be able to tell:
a) where it is continuous. 22, 23, O1, O8, O9, O10
b) where it is discontinuous and types of discontinuities. 15, 18, 43(a,b), O5, O6, O7
This includes functions from Theorem 7, piecewise functions, and combinations of functions.
3. Use limit laws and the definition of continuity to prove facts about continuity (e.g. Theorem 4). 58

----

== Section 2.5b (Mark) ==

Homework

Written: 28, 31, 32, 36, 37, 39, 41, 45, 47, 49, 65
Online: Kill 3, 5, 6, 7. Add 3 problems: Find intervals where f is positive, negative. One quadratic, easy factor; one rational with linear top and bottom; one rational with quadratic top, perfect square bottom.

Goals

1. Use continuity to evaluate limits 31,32
2. Continuity of piecewise functions
-determine if a piecewise function is continuous (or continuous from the right/left) 36,37,39
-determine parameters to make a piecewise function continuous 41,O1,O2,O4
3. Given a composition of functions, tell where it is continuous/discontinuous. 28
4. Use the Intermediate Value Theorem to prove the existence of solutions to equations. 45,47,49,65
5. Find intervals where a continuous function is positive/negative. (3 new online problems)

----

== Section 2.6 (Skyler) ==

Homework:
Written: 4, 5, 7, 13, 33, 43, 48
Online: Drop #15

Goals

1. Understand definition of limit at +/- inf
2. Find the limit of a rational function as x -> +/-inf and equations of horizontal asymptotes (when they exist)
3. Identify functions whose limit at +/- inf does not exist or whose limit is infinite
4. Compute limits at infinity by graphs and algebraic techniques

----

== Section 2.7 (Sam) ==
Homework:
Written: 5,9,19, 20, 43,44, 46
Online: eliminate o7,o9,o12

Goals:
1. Given an algebraic function, take the derivative (using the definition of the derivative)
Write down an equation for a tangent line through a point on a graph 3, 5, 05, o10, o11
2. Given a real life situation, interpret instantaneous change or average of change 43, 44, 46
Given a position function (or graph thereof), interpret the graph or find (average) velocity o1, o2, o3
3. Sketch the graph of a function such that certain criteria are met (values at certain points, derivatives, etc.) 19, 20
4. Given a limit expression, write down the point and function whose derivative it represents o6, o8
5. Write both formal definitions of a derivative
Explain how the derivative can be interpreted as slope of a line, velocity, instantaneous change

Changes:
Added: 43, 44
Eliminating: 2,25, o7, o9, o12

----

== Section 2.8 (Savannah) ==

Homework

Written: 1, 3, 5, 6, 7, 9, 11, 19, 20, 25, 35, 36, 37, 38, 44, Prove that if a function f is differentiable at a, then f is continuous at a.
Online: Kill O2, O3, Change O9 to f'(x)

Goals

1. Given a function f, find a formula for the derivative of f using f'(x) = lim (h->0) [f(x+h)-f(x)]/h, and be able to state the domain of f'(x). 19, 20, 25, O9, O10
2. Given a graph of a function f, be able to find the graph of f', f''. Given a graph of a function f', f'', be able to find th graph of f. 1, 3, 5, 6, 7, 9, 11, O1, O4, O5
3. Recognize when and why a function fails to be differentiable: 35, 36, 37, 38
a) corner
b) discontinuity (Theorem 4)
c) vertical tangent line
4. Be able to compute and apply higher derivatives: O6, O7, O8, 44
- (f')'=f''
- position [s(t)], velocity [v(t)=s'(t)], acceleration [a(t)=v'(t)=s''(t)]
5. Prove that if f is differentiable at a, then f is continuous at a. [Written homework]

----

== Section 3.1 (Drew) ==

----

== Section 3.2 (Rebecca) ==

Homework

Written: 2,11,13,23,24,32,33,47,49,55,57
Online: As listed.

Goals

1. Be able to apply the product rule to find the derivative of functions. 11,24,33,47,49,55,57, 02,04,05,06,07,09
2. Be able to apply the quotient rule to find the derivative of functions. 2,13,23,24,32,47,49, 01,03,08,010,011
3. Be able to prove the product and quotient rules and variations. 55,57*

----

== Section 3.3 (Skyler/James) ==

Homework

Written: 9, 10, 18, 20, 35, 42, 45, 49
Online: As listed. Change ordinal number suffixes on #3 (e.g. 83th makes no sense)

Goals

1. Know derivatives for the 6 basic trig functions. Be able to use these to compute derivatives of more complicated functions.
2. Know limits [2] and [3] in the book. Be able to use these to solve other limits involving trig functions
3. Know the "periodicity" of differentiation of trig functions--i.e. the fourth derivative of the sine function is again the sine function
4. Be able to use trig identities to find derivatives and limits
5. Use trig derivatives in various real world problems.

----

== Section 3.4 (Savannah) ==

Homework

Written: 7, 24, 25, 31, 37, 47, 63, 65, 71, 84, 89
Online: Kill O6, O7, O8; Reorder problems (see Dr. Purcell); Add #77 from the book to online (change "particle" to "point")

Goals

1. Fluency in chain rule: Given a composition of functions, use the chain rule and other rules to compute its derivative quickly. 7, 25, 37, 47, O1, O2, O3, O9, O10, O11, O12, O13, O14, O15, O16
2. Given values of two functions and their derivatives at certain points, use the formal statement of the chain rule to compute the derivative of compositions of these functions. 63, 65, 71, O4, O5
3. Given a word problem involving the composition of functions, compute rates of change. Interpret the meaning of the derivative as a particular rate of change. 84
4. Use the chain rule and definition of even/odd functions to prove properties of derivatives of even/odd functions. 89
5. Compute the derivative of a^x. 24, 31

----

== Section 3.5 (Rebecca) ==
Homework

Written: 3,11,12,15,18,21,23,34,41,57 (Add 43,67)
Online: Kill 03,014; Replace 08 with 35 from the book and make it online

Goals

1) For an implicitly defined equation find dy/dx, dx/dy, and d^2y/dx^2 using implicit differentiation.
2) Find the tangent line to an implicitly defined function.
3) Know the derivatives of the inverse trig functions.
----

== Section 3.6 (Sam) ==

----

== Section 3.8 (Jessica) ==

Homework:

Written: 14, 15
Online: kill o5 (pH scale)

Goals:

Use exponential functions to model each of the following:
1. Population growth. o1, o3, o4
2. Radioactive decay. o6, o7
Find half-life, or given half-life, find equation.
3. Cooling. 14, 15
4. Interest. o2, o8
- Derive the formula for compounding interest in n periods o2, o8 (see also section 1.3, number 60)
- Use expression of e as a limit to find formula for continuously compounded interest.
----

== Section 3.9 (Mark) ==

Homework

Written: 5, 22, 33, 35, 37, 42
Online: Kill O1, O4, O7, O8, O11, O12, O13; remove geometry formulas from online questions.

Goals

1. Solve related rates problems. (all homework problems)
a) Draw picture
b) Recognize variables from the problem, and rates of change as their derivatives.
c) Write equations relating variables (using geometry, trigonometry)
d) Differentiate, remembering to use the chain rule.
e) Plug in specific values to determine the requested answer.

----

== Section 3.10 (Tyler) ==

Homework
Written 1,2,3,5,23,28,43,44
Online Kill O4, O5, Renumber so that O3 is first and O1 is third. Possibly add one more. Change the hint in O1 to read "Try using linear approximation..."

Goals
1. Compute the linearization of a function at a point. 1,2,3,5, O2, O3
- Explain the relationship between the linearization and the tangent line. 5, O2
- Explain how linearization is useful. 23,28
2. Use the linearization at a point x=a to approximate the value of a function f(a+epsilon). 23,28, O1, O2

Suggestion: merge 3.10 and 3.11 for written as well as online.
----

== Section 3.11 (Savannah) ==

----

== Section 4.1 ==

----

== Section 4.2 ==

----

== Section 4.3 ==

Goals:

1. Know and apply the increasing/decreasing test o1, o5
2. Find local maxima and minima (First and Second derivative tests, endpoints) 21,23
3. Know the definition of concavity and how to determine concavity (The concavity test) o2, o5, 72
4. Know what an inflection point is and how to find it o2
5. Know how the above information affects the graph of a function
Draw a graph such that.... 25, 28
Given a graph, where is concavity positive, etc.? 1, 6, 7, 31 and a new online problem similar to problem 8 on page 295

Changes:

o3 and o4 are to be moved to section 4.5 and a problem is to be added online that is like problem 8.
----

== Section 4.4 ==

----

== Section 4.5 (Skyler) ==

Goals:

Use all of the following to draw graphs of functions:

* Algebraic Properties (Domain, intercepts, even/odd/periodic)

* Limit Properties (Asymptotes, singularities, discontinuities)

* Derivative Properties (Increasing/decreasing, extrema, concavity/inflection)

Written homework - as listed
Online homework - pulled from 4.3 (O3 and O4)

----

== Section 4.7 ==

----

== Section 4.8 (Mark) ==

Goals

1. Use Newton's Method to approximate roots/solutions of equations.
2. Derive the formula for Newton's Method from the tangent line equation.
3. Give an example of a function 'f' and a starting point 'x_1' that makes Newton's Method fail.

Homework

Written: 1, 2, 3, 4, 29.
Online: Kill O1, Add new online problem with graph.

----

== Section 4.9 (Tyler)==

Goals:
1. Use the Mean value Theorem to prove Theorem 1.
2. Memorize the table of anti-differentiation formulas on page 341 and
relate it to the table of derivatives on the inside back cover of the book. 12, 15, 20, 27, 33, 49, 51, O1-O12
3. Use the table and Theorem 1 to find *all* antiderivatives of various functions. 12, 15, 20, 27, 49, 51
4. Use Theorem 1 and known differentiation formulas to find a unique anti-derivative with given values at specified points. 33, 53, 67,
69, 70, 74,O2,O3, O4,O5,O10, O11, O13
5. Use anti-differentiation to solve physical and economic problems, especially problems of motion. 67, 69, 70, 74, O4, O5

Changes to problems:
Written: Add story problems: 67, 69, 70, 74
Online: too long, Delete O1, O6, O12.
Adjust numbers in 04 to give a clean answer.
O10--change so that F(0) not equal to 0 (else they will get right answer for wrong reason).
O8 and O9 problem that Arctan/Arcsin not accepted--just arctan/arcsin.
O13 should not say "the anti-derivative, but rather "an anti-derivative".

----

== Appendix E (Paul)==
Homework:

Written: 1, 3, 5, 10, 11, 15, 17, 20, 23, 41, 43
Online: o1, o2, o3, o4, o5, o6, o7, o8, o9, o10

Goals

1. Write expanded sums in sigma notation and expand sums written in sigma notation. Apply appropriate rules to distribute constants or expand sums of sums.
2. Explicitly evaluate $\sigma_{i=1}^n f(i)$ in terms of 'n', where 'f(x)' is a polynomial in x of degree at most 3. Numerically evaluate this sum when 'i' runs over a given range of integers.
3. Evaluate telescoping sums and limits of sums.

----

== Section 5.1 ==

----

== Section 5.2 ==

----

== Section 5.3 ==

----

== Section 5.4 ==

----

== Section 5.5 ==

----

Math 112 Calculus Learning Goals

2009-08-19T03:40:23Z

Pmj5: /* Appendix E */

----

== Section 1.1 (Jessica) ==

Homework:

Written: 25, 31, 40, 43, 55, 56, 64, 65
Online: 3, 4, 5, 6, 8, 9, 11. Modify 12 (Given f like 66 or 67 in book, find f(-x). Is f even or odd or neither?) Add problem like 51, 57 in book.

Goals

1. Given a function described algebraically, find the
- domain 31, 07, 08,
- range,
- value at a given number 04,
- value when we plug in another function, e.g. difference quotient 04, 05, 25.
2. Convert one representation of a function to another.
- verbal to/from graphical 03
-algebraic to/from graphical 09
-verbal to/from algebraic 55, 56, 2 new online
3. Piecewise functions: Given an algebraic description of a piecewise function, find its domain and sketch its graph. 011, 43.
Be able to write the absolute value as a piecewise function, and use this to sketch its graph. 40, 09
4. Use the definition of an even/odd function to decide whether a function described algebraically is even or odd. 012, 65
Use symmetry to decide whether a graph is an even or odd function. 64

----

== Section 1.2 (Sam) ==

Homework:

Written: 4,5,7,9,12 and Appdx D: 24, 37*, 46
Online: O1, O2, O4, O5 (these are the numbers under the current numbering),
and Appdx D: 29, 60, 69
The * on problem 37 means a calculator will be necessary.

Goals

1. Writing down a linear function given a set of information 5,12,O4,O5
2. Recognizing what a function's graph should look like 4,7,O1
3. Based on a graph, write down a function O2
4. Writing down a polynomial based on information 9, O2
5. Trig review (definitions, proofs, values of trig functions at particular points) Appdx D:24, 37, 46, 29, 60, 69 (the latter three going online)

# Trig to know, including problems to skim for practice. (Jessica)

1. Given angles in degrees and radians, draw the angle, convert from radians to degrees and vice versa. (See appendix D: 1-12)
2. Write down sin, cos, tan, sec, scs, cot of any angle 0, pi/6, pi/4, pi/3, pi/2, and all angles obtained from these angles by adding a multiple of pi/2. (appendix D:23-28)
3. Given a right triangle with side measurements, write sin, cos, tan, sec, csc, cot of any of the angles of the triangle in terms of the side lengths. (Appendix D: 35-38)
Similarly, given one trig ratio, find the others. (Appendix D: 29-34)
4. Graph trig functions. (app D: 77-82)
5. Use trig identities to simplify expressions (appendix D: 43-57 odd, 59-63), and to solve equations and inequalities involving trig functions (app D: 65-76).
* Most important identities: sin^2(x) + cos^2(x) = 1
* sin(-x) = -sin(x)
* cos(-x) = cos(x)
* sin(x+y) = sin(x)cos(y) + cos(x)sin(y)
* cos(x+y) = cos(x)cos(y) - sin(x)sin(y)
Using the above, you can derive other identities:
* 1+tan^2(x)=sec^2(x); 1+cot^2(x) = csc^2(x)
* sin(x-y)
* cos(x-y)
* sin(2x)
* cos(2x)

----

== Section 1.3 (Savannah) ==

Homework

Written: 1,7,13, 14, 22, 39, 46, 60, 65
Online: Modify 1 to be y as a function of x. Replace 6 with a different word problem. Add 3 from section 1.2 online assignment.

Goals

1. Apply and recognize transformations of functions:
- Translating, Stretching, Reflecting, Absolute Value 1
- Graph --> Algebraic 7, O1, O5, 1.2-O3
- Algebraic --> Graph 13, 14, 22, O7, O8
2. Given functions f and g, find:
- f+g
- f-g
- fg
- f/g
- domains for above functions. O2, O9
3. Understand and apply compositions of functions:
- Given functions f and g, find g o f and f o g. 28, 39, 60, O3, O6
- Given f o g, find functions f and g. 46, 65, O4

----

== Section 1.5 (James) ==

Homework:

Written: 11, 15, 18, 19, 21, 25, 29
Online: o2, o3, o4, o5, one more word problem

Goals

1. Graph exponential functions a^x for a<1, a>1, as well as transformations of exponential functions from section 1.3. Find domains. 11, 15, 21, o2
2. Given two points that an exponential function passes through, find the equation of the exponential function. 18, o3.
3. Use laws of exponents to simplify exponential expressions, to show various facts about exponential functions. 19, 29.
4. Convert a word problem involving population change into an algebraic expression involving exponential functions. Use this to find sizes of populations at given times, and to graph. 25, o4, o5.

----

== Section 1.6 (McKay) ==

Homework

Written: 12,16,18,19,23,35,45,48,54,60,67,71,73
Online: 01,02,03,04,05,06,07,08,010,011,012,013,016,018

Goals

1) Be able to tell if a function is one-to-one
-algebraically 12
-graphically 01
2) Know the definition of an inverse function and be able to use the steps to find the inverse
-algebraically 16,19,23,54,02,03,04,05,06
-graphically 18,45,73,07
3)Know the laws of logarithms and be able to solve and simplify logarithmic and exponential expressions. 35,48,06,08,010,011,012
4)Know the inverses of the trigonometric functions
- domain and range 71
- values at a point 60, 013,016
- simplify a trig function composed with an inverse trig function 67, o18

----

== Section 2.1 (Tyler) ==

Homework

Written:4*, 5*? (require calculator)

Goals

1. Know definitions of tangent and secant line. Compute the slope of a secant line. 4
2. Know definition of average velocity and the idea of instantaneous velocity. Compute an average velocity. 5
3. Explain how average velocity is equivalent to slope of a secant line, same for instantaneous velocity and slope of tangent line.

----

== Section 2.2 (Tyler) ==

Homework

Written: 6, 7, 9, 16, 27, 32, 34a, 40
Online: kill O2, O5. Modify: O3 fix so different parts don't have same answer.
O6, O7, O8: Use the same input notation for infinity. O8 Delete irrelevant lecture about asymptotes.

Goals:

1. Idea of a limit: (Note: the "definition" of a limit in this section is sloppy and is not a real definition of a limit. See section 2.4)
* Find the limit of a function at a point from its graph, including when the limit does not exist. 6,7, O1, O3
* Recognize the difference between the limit of a function at a point and the value of the function at a point. 6, 7, O1
* Give examples of functions that have prescribed limits at certain points. 16
* Understand that calculators and tables can entice you to guess a wrong limit. Give examples of functions whose limit is not what you would guess from plugging in values to your calculator. (no problems)
2. Idea (and notation) for one-sided limit: Do the same things for one-sided limits as done before for regular limits. 6, 7, O1, O3, O4, O6, O7, O8
3. Explain how one-sided and two-sided limits are connected. Use this to compute limits. 6, 7, O1, O3, O4, O6, O7, O8
4. Infinite limits:
* Explain why infinity is not a number and how the definition of an infinite limit gets around this.
* Find all vertical asymptotes for a function. 9, 34a
* For rational functions, find when a limit is infinity and when it is negative infinity. 27, 32, O6, O7, O8, 40

----

== Section 2.3a (Rebecca) ==

Homework

Written: 10,15,19,20,21,22,28,29
Online: Get rid of 3

Goals

1. Be able to apply limit laws to simplify limits
- algebraically 01
- graphically 02
2. Recognize when you can directly substitute to compute a limit and when you cannot. 10, 03, 04, 05, 06
3. Use algebra to simply and find limits.
- for polynomials over polynomials. 15, 19, 20, o5, o6, o7, o8, o9, o10
- expressions with square roots. 21, 22, 29, o11
- expressions with multiple fractions. 28, 29, o12

----

== Section 2.3b (Drew) ==

Homework

Written: 36-39, 42, 55, 56, 58
Online: Add o2 from 2.2. Kill o3, o4. Modify o6 so that direct substitution doesn't work. Add two problems using substitution.

Goals

1. Use the Squeeze Theorem to find limits. 36, 37, 38, o5, o6, o7
2. Evaluate limits that involve absolute values and other piecewise functions. 39, 42, o1, o2, o2 from 2.2
3. Use limit laws to find lim f(x) given that lim g(f(x)) = L. 55, 56, 58
4. Use substitution to rewrite lim g(f(x)) as lim g(u). 2 new online problems.

----

== Section 2.4 (Jessica) ==
Homework:
Written: 2, 14, 15, 16, 19, 20
Online: kill o6 (done), modify problems to find the *largest* delta (done).

Goals:
1. Use a graph showing a function and lines y=L+epsilon, y=L-epsilon to find the largest value of delta so that if 0<|x-a|<delta, then |f(x)-L|<epsilon. 2
2. Know the epsilon-delta definition of a limit. Given a limit and a value for epsilon, be able to find the largest value of delta corresponding to that epsilon. 14, o1, o2, o3, o4, o5
3. Use the epsilon-delta definition of a limit to prove a statement about the limit of a linear function. 15, 16, 19, 20.

----

== Section 2.5a (McKay/Savannah) ==

Written:4, 6, 15, 18, 22, 23, 43ab, 58
Online - Kill O2-O4, O11-O16; Change O8, O9, O10 to ask, "Where is the following function continuous?"; Add a graph problem where the student must identify where the graph is discontinuous; Add a graph problem where the student must identify the types of discontinuities

1. Given the graph of a function, be able to tell:
a) where it is continuous. 4
b) where it is discontinuous, and the type of discontinuity. 6
2. Given a function described algebraically, be able to tell:
a) where it is continuous. 22, 23, O1, O8, O9, O10
b) where it is discontinuous and types of discontinuities. 15, 18, 43(a,b), O5, O6, O7
This includes functions from Theorem 7, piecewise functions, and combinations of functions.
3. Use limit laws and the definition of continuity to prove facts about continuity (e.g. Theorem 4). 58

----

== Section 2.5b (Mark) ==

Homework

Written: 28, 31, 32, 36, 37, 39, 41, 45, 47, 49, 65
Online: Kill 3, 5, 6, 7. Add 3 problems: Find intervals where f is positive, negative. One quadratic, easy factor; one rational with linear top and bottom; one rational with quadratic top, perfect square bottom.

Goals

1. Use continuity to evaluate limits 31,32
2. Continuity of piecewise functions
-determine if a piecewise function is continuous (or continuous from the right/left) 36,37,39
-determine parameters to make a piecewise function continuous 41,O1,O2,O4
3. Given a composition of functions, tell where it is continuous/discontinuous. 28
4. Use the Intermediate Value Theorem to prove the existence of solutions to equations. 45,47,49,65
5. Find intervals where a continuous function is positive/negative. (3 new online problems)

----

== Section 2.6 (Skyler) ==

Homework:
Written: 4, 5, 7, 13, 33, 43, 48
Online: Drop #15

Goals

1. Understand definition of limit at +/- inf
2. Find the limit of a rational function as x -> +/-inf and equations of horizontal asymptotes (when they exist)
3. Identify functions whose limit at +/- inf does not exist or whose limit is infinite
4. Compute limits at infinity by graphs and algebraic techniques

----

== Section 2.7 (Sam) ==
Homework:
Written: 5,9,19, 20, 43,44, 46
Online: eliminate o7,o9,o12

Goals:
1. Given an algebraic function, take the derivative (using the definition of the derivative)
Write down an equation for a tangent line through a point on a graph 3, 5, 05, o10, o11
2. Given a real life situation, interpret instantaneous change or average of change 43, 44, 46
Given a position function (or graph thereof), interpret the graph or find (average) velocity o1, o2, o3
3. Sketch the graph of a function such that certain criteria are met (values at certain points, derivatives, etc.) 19, 20
4. Given a limit expression, write down the point and function whose derivative it represents o6, o8
5. Write both formal definitions of a derivative
Explain how the derivative can be interpreted as slope of a line, velocity, instantaneous change

Changes:
Added: 43, 44
Eliminating: 2,25, o7, o9, o12

----

== Section 2.8 (Savannah) ==

Homework

Written: 1, 3, 5, 6, 7, 9, 11, 19, 20, 25, 35, 36, 37, 38, 44, Prove that if a function f is differentiable at a, then f is continuous at a.
Online: Kill O2, O3, Change O9 to f'(x)

Goals

1. Given a function f, find a formula for the derivative of f using f'(x) = lim (h->0) [f(x+h)-f(x)]/h, and be able to state the domain of f'(x). 19, 20, 25, O9, O10
2. Given a graph of a function f, be able to find the graph of f', f''. Given a graph of a function f', f'', be able to find th graph of f. 1, 3, 5, 6, 7, 9, 11, O1, O4, O5
3. Recognize when and why a function fails to be differentiable: 35, 36, 37, 38
a) corner
b) discontinuity (Theorem 4)
c) vertical tangent line
4. Be able to compute and apply higher derivatives: O6, O7, O8, 44
- (f')'=f''
- position [s(t)], velocity [v(t)=s'(t)], acceleration [a(t)=v'(t)=s''(t)]
5. Prove that if f is differentiable at a, then f is continuous at a. [Written homework]

----

== Section 3.1 (Drew) ==

----

== Section 3.2 (Rebecca) ==

Homework

Written: 2,11,13,23,24,32,33,47,49,55,57
Online: As listed.

Goals

1. Be able to apply the product rule to find the derivative of functions. 11,24,33,47,49,55,57, 02,04,05,06,07,09
2. Be able to apply the quotient rule to find the derivative of functions. 2,13,23,24,32,47,49, 01,03,08,010,011
3. Be able to prove the product and quotient rules and variations. 55,57*

----

== Section 3.3 (Skyler/James) ==

Homework

Written: 9, 10, 18, 20, 35, 42, 45, 49
Online: As listed. Change ordinal number suffixes on #3 (e.g. 83th makes no sense)

Goals

1. Know derivatives for the 6 basic trig functions. Be able to use these to compute derivatives of more complicated functions.
2. Know limits [2] and [3] in the book. Be able to use these to solve other limits involving trig functions
3. Know the "periodicity" of differentiation of trig functions--i.e. the fourth derivative of the sine function is again the sine function
4. Be able to use trig identities to find derivatives and limits
5. Use trig derivatives in various real world problems.

----

== Section 3.4 (Savannah) ==

Homework

Written: 7, 24, 25, 31, 37, 47, 63, 65, 71, 84, 89
Online: Kill O6, O7, O8; Reorder problems (see Dr. Purcell); Add #77 from the book to online (change "particle" to "point")

Goals

1. Fluency in chain rule: Given a composition of functions, use the chain rule and other rules to compute its derivative quickly. 7, 25, 37, 47, O1, O2, O3, O9, O10, O11, O12, O13, O14, O15, O16
2. Given values of two functions and their derivatives at certain points, use the formal statement of the chain rule to compute the derivative of compositions of these functions. 63, 65, 71, O4, O5
3. Given a word problem involving the composition of functions, compute rates of change. Interpret the meaning of the derivative as a particular rate of change. 84
4. Use the chain rule and definition of even/odd functions to prove properties of derivatives of even/odd functions. 89
5. Compute the derivative of a^x. 24, 31

----

== Section 3.5 (Rebecca) ==
Homework

Written: 3,11,12,15,18,21,23,34,41,57 (Add 43,67)
Online: Kill 03,014; Replace 08 with 35 from the book and make it online

Goals

1) For an implicitly defined equation find dy/dx, dx/dy, and d^2y/dx^2 using implicit differentiation.
2) Find the tangent line to an implicitly defined function.
3) Know the derivatives of the inverse trig functions.
----

== Section 3.6 (Sam) ==

----

== Section 3.8 (Jessica) ==

Homework:

Written: 14, 15
Online: kill o5 (pH scale)

Goals:

Use exponential functions to model each of the following:
1. Population growth. o1, o3, o4
2. Radioactive decay. o6, o7
Find half-life, or given half-life, find equation.
3. Cooling. 14, 15
4. Interest. o2, o8
- Derive the formula for compounding interest in n periods o2, o8 (see also section 1.3, number 60)
- Use expression of e as a limit to find formula for continuously compounded interest.
----

== Section 3.9 (Mark) ==

Homework

Written: 5, 22, 33, 35, 37, 42
Online: Kill O1, O4, O7, O8, O11, O12, O13; remove geometry formulas from online questions.

Goals

1. Solve related rates problems. (all homework problems)
a) Draw picture
b) Recognize variables from the problem, and rates of change as their derivatives.
c) Write equations relating variables (using geometry, trigonometry)
d) Differentiate, remembering to use the chain rule.
e) Plug in specific values to determine the requested answer.

----

== Section 3.10 (Tyler) ==

Homework
Written 1,2,3,5,23,28,43,44
Online Kill O4, O5, Renumber so that O3 is first and O1 is third. Possibly add one more. Change the hint in O1 to read "Try using linear approximation..."

Goals
1. Compute the linearization of a function at a point. 1,2,3,5, O2, O3
- Explain the relationship between the linearization and the tangent line. 5, O2
- Explain how linearization is useful. 23,28
2. Use the linearization at a point x=a to approximate the value of a function f(a+epsilon). 23,28, O1, O2

Suggestion: merge 3.10 and 3.11 for written as well as online.
----

== Section 3.11 (Savannah) ==

----

== Section 4.1 ==

----

== Section 4.2 ==

----

== Section 4.3 ==

Goals:

1. Know and apply the increasing/decreasing test o1, o5
2. Find local maxima and minima (First and Second derivative tests, endpoints) 21,23
3. Know the definition of concavity and how to determine concavity (The concavity test) o2, o5, 72
4. Know what an inflection point is and how to find it o2
5. Know how the above information affects the graph of a function
Draw a graph such that.... 25, 28
Given a graph, where is concavity positive, etc.? 1, 6, 7, 31 and a new online problem similar to problem 8 on page 295

Changes:

o3 and o4 are to be moved to section 4.5 and a problem is to be added online that is like problem 8.
----

== Section 4.4 ==

----

== Section 4.5 (Skyler) ==

Goals:

Use all of the following to draw graphs of functions:

* Algebraic Properties (Domain, intercepts, even/odd/periodic)

* Limit Properties (Asymptotes, singularities, discontinuities)

* Derivative Properties (Increasing/decreasing, extrema, concavity/inflection)

Written homework - as listed
Online homework - pulled from 4.3 (O3 and O4)

----

== Section 4.7 ==

----

== Section 4.8 (Mark) ==

Goals

1. Use Newton's Method to approximate roots/solutions of equations.
2. Derive the formula for Newton's Method from the tangent line equation.
3. Give an example of a function 'f' and a starting point 'x_1' that makes Newton's Method fail.

Homework

Written: 1, 2, 3, 4, 29.
Online: Kill O1, Add new online problem with graph.

----

== Section 4.9 (Tyler)==

Goals:
1. Use the Mean value Theorem to prove Theorem 1.
2. Memorize the table of anti-differentiation formulas on page 341 and
relate it to the table of derivatives on the inside back cover of the book. 12, 15, 20, 27, 33, 49, 51, O1-O12
3. Use the table and Theorem 1 to find *all* antiderivatives of various functions. 12, 15, 20, 27, 49, 51
4. Use Theorem 1 and known differentiation formulas to find a unique anti-derivative with given values at specified points. 33, 53, 67,
69, 70, 74,O2,O3, O4,O5,O10, O11, O13
5. Use anti-differentiation to solve physical and economic problems, especially problems of motion. 67, 69, 70, 74, O4, O5

Changes to problems:
Written: Add story problems: 67, 69, 70, 74
Online: too long, Delete O1, O6, O12.
Adjust numbers in 04 to give a clean answer.
O10--change so that F(0) not equal to 0 (else they will get right answer for wrong reason).
O8 and O9 problem that Arctan/Arcsin not accepted--just arctan/arcsin.
O13 should not say "the anti-derivative, but rather "an anti-derivative".

----

== Appendix E (Paul)==

----

== Section 5.1 ==

----

== Section 5.2 ==

----

== Section 5.3 ==

----

== Section 5.4 ==

----

== Section 5.5 ==

----

Math 586: Introduction to Algebraic Number Theory.

2009-01-21T16:13:56Z

Pmj5: Added punctuation.

== Catalog Information ==

=== Title ===
Introduction to Algebraic Number Theory.

=== Credit Hours ===
3

=== Prerequisite ===
[[Math 372]] or equivalent; instructor's consent.

=== Description ===
Algebraic integers; different and discriminant; decomposition of primes; class group; Dirichlet unit theorem; Dedekind zeta function; cyclotomic fields; valuations; completions.

== Desired Learning Outcomes ==

=== Prerequisites ===
[[Math 372]] is a prerequisite for this course. In particular, students should be familiar with the concepts of groups and rings, and they should understand constructions of quotient groups and quotient rings. By this point in their mathematical career, students should be skilled at proving theorems by themselves.

=== Minimal learning outcomes ===
Students should achieve an advanced mastery of the topics listed below. This means that they should know all relevant definitions, correct statements and proofs of the major theorems (including their hypotheses and limitations), and examples and non-examples of the various concepts. The students should be able to demonstrate their mastery by solving difficult problems related to these concepts, and by proving theorems about the below concepts, even if the theorems go beyond the material in the text.
<div style="-moz-column-count:2; column-count:2;">
# Number Fields
#*Algebraic Numbers
#*Algebraic Integers
#*Cyclotomic Fields
#*Trace and Norm
#*Discriminants
#*Integral Bases
#*Computing Integral Bases
#Prime decomposition in rings of integers
#*Ideal theory of Dedekind domains
#*Splitting, ramification, inertia of primes
#*Computing prime decompositions
#*Decomposition and inertia groups
#*Frobenius maps
#*Functorial properties of the Frobenius
#Ideal Class Group
#*Finiteness of the class group
#*Minkowski bounds
#*Distribution of ideals in ideal classes
#*Class group computations in quadratic fields
#Dirichlet's unit theorem
#*Computation of fundamental units in quadratic fields
#Cebotarev Density Theorem (Statement)
#Dedekind zeta function
#* Class number formula
#Valuations
#*Completions of number rings

</div>

=== Additional topics ===
As time permits, additional topics that might be considered include Galois representations, class field theory, module theory over Dedekind domains, etc.

=== Courses for which this course is prerequisite ===

[[Category:Courses|586]]
This course is not a prerequisite for any other courses.

Math 371: Abstract Algebra 1.

2009-01-14T16:59:38Z

Pmj5: Punctuation edit

== Catalog Information ==

=== Title ===
Abstract Algebra.

=== (Credit Hours:Lecture Hours:Lab Hours) ===
(3:3:0)

=== Offered ===
F, W, Sp

=== Prerequisite ===
[[Math 190]], [[Math 343|343]].

=== Description ===
Groups, rings, fields, vector spaces, linear transformations, matrices, field extensions, etc.

== Desired Learning Outcomes ==
This course is aimed at undergraduate mathematics and mathematics education majors. It is a first course in abstract algebra. In addition to being an important branch of mathematics in its own right, abstract algebra is now an essential tool in number theory, geometry, topology, and, to a lesser extent, analysis. Thus it is a core requirement for all mathematics majors. Outside of mathematics, algebra also has applications in cryptography, coding theory, quantum chemistry, and physics.

=== Prerequisites ===
The chief prerequisite for this course is [[Math 190]]. In [[Math 190]], students should learn basic logic, basic set theory, the division algorithm, Euclidean algorithm, and unique factorization theorem for integers, equivalence relations, functions, and mathematical induction. As these topics are of high importance in Math 371, it might be prudent for the instructor to review them at the beginning of the semester.

=== Minimal learning outcomes ===
Students should achieve mastery of the topics listed below. This means that they should know all relevant definitions, correct statements of the major theorems (including their hypotheses and limitations), and examples and non-examples of the various concepts. The students should be able to demonstrate their mastery by solving non-trivial problems related to these concepts, and by proving simple (but non-trivial) theorems about the below concepts, related to, but not identical to, statements proven by the text or instructor.

<div style="-moz-column-count:2; column-count:2;">
# Group Theory
#* Basic Definitions
#* Examples of groups
#* Subgroups
#* Lagrange’s Theorem
#* Homomorphisms
#* Normal Subgroups
#* Quotient Groups
#* Isomorphism Theorems
#* Cauchy’s Theorem
#* Direct Products
#* The Symmetric Group
#* Even and odd Permutations
#* Cycle Decompositions
# Ring Theory
#* Basic Definitions
#* Examples of rings (both commutative and noncommutative)
#* Ideals
#* Ring homomorphisms
#* Quotient rings
#* Prime and maximal ideals
#* Polynomial rings
#* Factorization in polynomial rings
#* Field of fractions of a domain

</div>

=== Additional topics ===
Beyond the minimal learning outcomes, instructors are free to cover additional topics. These may include (but are certainly not limited to): Sylow’s theorems, the fundamental theorem of finite abelian groups, the simplicity of ''A<sub>n</sub>'', Burnside’s Theorem, Polya counting, isometries of '''R'''<sup>3</sup> and regular solids, straight-edge and compass constructions, the 2- and 4-squares theorems, the RSA algorithm, wallpaper groups, coding theory, or latin squares. Instructors are free to use new approaches to the teaching of the material, as long as the minimal learning outcomes are achieved.
=== Courses for which this course is prerequisite ===
This course is required for [[Math 350]], [[Math 372]], [[Math 387]], [[Math 561]], [[Math 586]]. As a result it is essential that all required learning objectives be covered.

[[Category:Courses|371]]

Talk:Math 485: Mathematical Cryptography

2008-08-28T20:14:51Z

Pmj5:

Maple and Mathematica do better than Matlab at doing arithmetic with large integers. I have had students encounter problems with rounding using Matlab, making implementing RSA difficult; perhaps the learning outcomes should not treat all three software packages equally. -pmj5

If the course focuses on the mathematics behind cryptography, would "Mathematical Cryptography" be a better course title than "Introduction to Cryptography"? -pmj5

The Math 371 prerequisite is a rather high barrier to entry. It concedes that we do not expect to have many non-math majors in this course. If, instead, the prerequisite was simply Math 343, this course could be a useful tool for recruiting students into the mathematics major or graduate program, by encouraging engineering/physics/etc. students to enroll. The material is almost certainly more likely to be interesting to the generic student than many of our other courses seem at first to non-mathematicians, because of its real-world applications, and we should take advantage of this to get more students enrolled in a mathematics course. We certainly shouldn't water down the material in an attempt to cater to students from other departments, but I believe that the course can be taught at a level that will both allow math majors to see connections between cryptography and other topics they have studied, and at the same time provide an opportunity for engineering/science students to be introduced to some beautiful mathematics. -pmj5

Number Theory

2008-08-20T22:02:07Z

Pmj5:

== Courses ==
* [[Math 387]]: Number Theory
* [[Math 485]]: Introduction to Cryptography
* [[Math 586]]: Introduction to Algebraic Number Theory
* [[Math 587]]: Introduction to Analytic Number Theory
* [[Math 686R]]: Topics in Algebraic Number Theory
* [[Math 687R]]: Topics in Analytic Number Theory

[[Category:Areas]]

Math 485: Mathematical Cryptography

2008-08-19T22:30:49Z

Pmj5:

== Catalog Information ==

=== Title ===
Introduction to Cryptography.

=== (Credit Hours:Lecture Hours:Lab Hours) ===
(3:3:0)

=== Prerequisite ===
[[Math 371]].

=== Description ===
A mathematical introduction to some of the high points of modern cryptography.

== Desired Learning Outcomes ==

This is a course in the mathematics and algorithms of modern cryptography. It complements, rather than being equivalent to, the current CS course on Computer Security (CS 465).

=== Prerequisites ===

The requirement for [[Math 371]] ensures both an appropriate level of mathematical maturity and a basic knowledge of linear algebra.

=== Minimal learning outcomes ===

The student should gain a understanding of the following topics. In particular this includes knowing the definitions, being familiar with standard examples, and being able to solve mathematical and algorithmic problems by directly using the material taught in the course. This includes appropriate use of Maple, Mathematica or Matlab.

<div style="-moz-column-count:2; column-count:2;">
# Classical systems, including:
#* Substitution theory
#* Block ciphers
#* Enigma
# Elementary number theory as follows:
#* Euclid's algorithm
#* Modular arithmetic and the algorithm for modular exponentiation
#* Chinese Remainder Theorem
#* Fermat and Euler Theorems
#* Primitive roots
#* Legendre and Jacobi symbols
#* Elementary continued fractions
#* Simple discussion of finite fields.
# The DES and AES encryption standards.
# RSA and its strengths and weaknesses; attacks on RSA.
#* Wiener's continued fraction attack on low decryption exponent.
# Primality testing algorithms.
# Factorization techniques.
#* The Quadratic Sieve.
# Discrete logarithms. Diffie-Hellman key exchange. ElGamal.
# Lattices and Lattice Algorithms. The LLL algorithm. The NTRU system. Lattice attacks on RSA.

</div>

=== Additional topics ===
If time allows, additional topics may include, but are not limited to: Elliptic curve cryptography, birthday attacks and probability, quantum cryptography (key distribution, Shor's algorithm), hash functions, digital signatures.

=== Courses for which this course is prerequisite ===
None.

[[Category:Courses|485]]

Talk:Math 485: Mathematical Cryptography

2008-08-19T22:29:10Z

Pmj5:

Talk:Math 485: Mathematical Cryptography

2008-08-19T22:28:50Z

Pmj5: New page: Maple and Mathematica do better than Matlab at doing arithmetic with large integers. I have had students encounter problems with rounding using Matlab, making implementing RSA difficult; ...

Maple and Mathematica do better than Matlab at doing arithmetic with large integers. I have had students encounter problems with rounding using Matlab, making implementing RSA difficult; perhaps the learning outcomes should not treat all three software packages equally.

If the course focuses on the mathematics behind cryptography, would "Mathematical Cryptography" be a better course title than "Introduction to Cryptography"?

The Math 371 prerequisite is a rather high barrier to entry. It concedes that we do not expect to have many non-math majors in this course. If, instead, the prerequisite was simply Math 343, this course could be a useful tool for recruiting students into the mathematics major or graduate program, by encouraging engineering/physics/etc. students to enroll. The material is almost certainly more likely to be interesting to the generic student than many of our other courses seem at first to non-mathematicians, because of its real-world applications, and we should take advantage of this to get more students enrolled in a mathematics course.

Math 485: Mathematical Cryptography

2008-08-19T22:20:10Z

Pmj5: Added additional topics

== Catalog Information ==

=== Title ===
Introduction to Cryptography.

=== (Credit Hours:Lecture Hours:Lab Hours) ===
(3:3:0)

=== Prerequisite ===
[[Math 371]].

=== Description ===
A mathematical introduction to some of the high points of modern cryptography.

== Desired Learning Outcomes ==

This is a course in the mathematics and algorithms of modern cryptography. It complements, rather than being equivalent to, the current CS course on Computer Security (CS 465).

=== Prerequisites ===

The requirement for [[Math 371]] ensures both an appropriate level of mathematical maturity and a basic knowledge of linear algebra.

=== Minimal learning outcomes ===

The student should gain a understanding of the following topics. In particular this includes knowing the definitions, being familiar with standard examples, and being able to solve mathematical and algorithmic problems by directly using the material taught in the course. This includes appropriate use of Maple, Mathematica or Matlab.

<div style="-moz-column-count:2; column-count:2;">
# Classical systems, including:
#* Substitution theory
#* Block ciphers
#* Enigma
# Elementary number theory as follows:
#* Euclid's algorithm
#* Modular arithmetic and the algorithm for modular exponentiation
#* Chinese Remainder Theorem
#* Fermat and Euler Theorems
#* Primitive roots
#* Elementary continued fractions
#* Simple discussion of finite fields.
# The DES and AES encryption standards.
# RSA and its strengths and weaknesses; attacks on RSA.
#* Wiener's continued fraction attack on low decryption exponent.
# Primality testing algorithms.
# Factorization techniques.
#* The Quadratic Sieve.
# Discrete logarithms. Diffie-Hellman key exchange. ElGamal.
# Lattices and Lattice Algorithms. The LLL algorithm. The NTRU system. Lattice attacks on RSA.

</div>

=== Additional topics ===
If time allows, additional topics may include, but are not limited to: Elliptic curve cryptography, birthday attacks and probability, quantum cryptography (key distribution, Shor's algorithm), hash functions, digital signatures.

=== Courses for which this course is prerequisite ===
None.

[[Category:Courses|485]]

Math 485: Mathematical Cryptography

2008-08-19T22:00:37Z

Pmj5:

== Catalog Information ==

=== Title ===
Introduction to Cryptography.

=== (Credit Hours:Lecture Hours:Lab Hours) ===
(3:3:0)

=== Prerequisite ===
[[Math 371]].

=== Description ===
A mathematical introduction to some of the high points of modern cryptography.

== Desired Learning Outcomes ==

This is a course in the mathematics and algorithms of modern cryptography. It complements, rather than being equivalent to, the current CS course on Computer Security (CS 465).

=== Prerequisites ===

The requirement for [[Math 371]] ensures both an appropriate level of mathematical maturity and a basic knowledge of linear algebra.

=== Minimal learning outcomes ===

The student should gain a understanding of the following topics. In particular this includes knowing the definitions, being familiar with standard examples, and being able to solve mathematical and algorithmic problems by directly using the material taught in the course. This includes appropriate use of Maple, Mathematica or Matlab.

<div style="-moz-column-count:2; column-count:2;">
# Classical systems, including:
#* Substitution theory
#* Block ciphers
#* Enigma
# Elementary number theory as follows:
#* Euclid's algorithm
#* Modular arithmetic and the algorithm for modular exponentiation
#* Chinese Remainder Theorem
#* Fermat and Euler Theorems
#* Primitive roots
#* Elementary continued fractions
#* Simple discussion of finite fields.
# The DES and AES encryption standards.
# RSA and its strengths and weaknesses; attacks on RSA.
#* Wiener's continued fraction attack on low decryption exponent.
# Primality testing algorithms.
# Factorization techniques.
#* The Quadratic Sieve.
# Discrete logarithms. Diffie-Hellman key exchange. ElGamal.
# Lattices and Lattice Algorithms. The LLL algorithm. The NTRU system. Lattice attacks on RSA.

</div>

=== Additional topics ===
If time allows: Elliptic curve cryptography.

=== Courses for which this course is prerequisite ===
None.

[[Category:Courses|485]]