Difference between revisions of "Math 112 Calculus Learning Goals"
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3. Identify functions whose limit at +/- inf does not exist or whose limit is infinite | 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 | 4. Compute limits at infinity by graphs and algebraic techniques | ||
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| + | == Section 3.1 (Drew) == | ||
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== Section 3.2 (Rebecca) == | == Section 3.2 (Rebecca) == | ||
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4. Be able to use trig identities to find derivatives and limits | 4. Be able to use trig identities to find derivatives and limits | ||
5. Use trig derivatives in various real world problems. | 5. Use trig derivatives in various real world problems. | ||
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| + | == Section 3.4 (Savannah) == | ||
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| + | == Section 3.5 (Rebecca) == | ||
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| + | == Section 3.6 (Sam) == | ||
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| + | == Section 3.8 (Jessica) == | ||
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| + | == Section 3.9 (Mark) == | ||
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| + | == Section 3.10 (Tyler) == | ||
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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 == | ||
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| + | == Section 4.4 == | ||
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| + | == Section 4.5 == | ||
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| + | == Section 4.7 == | ||
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| + | == Appendix E == | ||
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| + | == Section 5.1 == | ||
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Revision as of 10:31, 11 August 2009
Contents
- 1 Section 1.1 (Jessica)
- 2 Section 1.2 (Sam)
- 3 Section 1.3 (Savannah)
- 4 Section 1.5 (James)
- 5 Section 1.6 (McKay)
- 6 Section 2.1 (Tyler)
- 7 Section 2.2 (Tyler)
- 8 Section 2.3a (Rebecca)
- 9 Section 2.3b (Drew)
- 10 Section 2.4 (Jessica)
- 11 Section 2.5a (McKay/Savannah)
- 12 Section 2.5b (Mark)
- 13 Section 2.6 (Skyler)
- 14 Section 2.7 (Sam)
- 15 Section 2.8 (Savannah)
- 16 Section 3.1 (Drew)
- 17 Section 3.2 (Rebecca)
- 18 Section 3.3 (Skyler/James)
- 19 Section 3.4 (Savannah)
- 20 Section 3.5 (Rebecca)
- 21 Section 3.6 (Sam)
- 22 Section 3.8 (Jessica)
- 23 Section 3.9 (Mark)
- 24 Section 3.10 (Tyler)
- 25 Section 3.11 (Savannah)
- 26 Section 4.1
- 27 Section 4.2
- 28 Section 4.3
- 29 Section 4.4
- 30 Section 4.5
- 31 Section 4.7
- 32 Section 4.8
- 33 Section 4.9
- 34 Appendix E
- 35 Section 5.1
- 36 Section 5.2
- 37 Section 5.3
- 38 Section 5.4
- 39 Section 5.5
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, 30, 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, 47,29, 30, 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: o1, o2, o3, o4
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, o1, 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.
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 16,19,23,54,73,02,03,04,05,06
-graphically 18,45,07
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 and use them to solve and simplify trigonometric expressions. 60,67,71,013,016,018
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
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
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
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
Section 2.4 (Jessica)
Kill
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)
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.
