Math 112 Calculus Learning Goals

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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) 

1. 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? 

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 

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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. 15
         * 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. 
  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  
        1. algebraically  01
        2. graphically    02
  2. Recognize when you can directly substitute to compute a limit and when you cannot.  10, 03, 04, 05, 06
  3. Be able to use algebra to simply and find limits that are undefined. 15, 19, 20, 21, 22, 28, 29, 07, 08, 09, 010, 011

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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.

Goals

   1. Use the Squeeze Theorem to find limits.  36, 37, 38, o5, o6, o7
   2. Evaluate limits that involve absolute value. 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 

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Section 2.4 (Jessica)

Kill

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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)

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 3.2 (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 3.9 (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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