# Difference between revisions of "Math 112 Calculus Learning Goals"

## Section 1.1

Homework:
Written: 25, 31, 40, 43, 44, 56, 64, 65
Online: o1 - o10

Goals:

1. Given a function described algebraically, find the
- domain 31, o4, o5
- range,
- value at a given number o2, o7
- value when we plug in another function, e.g. difference quotient 25, o2, o3.
2. Convert one representation of a function to another.
- verbal to/from graphical. o1
-algebraic to/from graphical. o6
-verbal to/from algebraic 55, 56, o9, o10
3. Piecewise functions: Given an algebraic description of a piecewise function, find its domain and sketch its graph. 43.
Be able to write the absolute value as a piecewise function, and use this to sketch its graph. 40, o6.
4. Use the definition of an even/odd function to decide whether a function described algebraically is even or odd. 65, o8
Use symmetry to decide whether a graph is an even or odd function. 64

## Section 1.2

Homework:
Written: 4, 5, 7, 9, 12 and Appdx D: 24, 37*, 46
Online: o1 - o7

Goals

1. Write down a linear function given a set of information. 5, 12, o3, o4.
2. Recognize what the graph of linear, polynomial, power, trig, exponential, and log functions graph should look like. 4,7,o1
3. Given roots of a polynomial and one other point, write down its equation. 9, o2.
4. Know trig goals below. Appdx D:24, 37, 46, o5, o6, o7.

## Trig to know, including problems to skim for practice.

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, 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

Homework:
Written: 1, 7, 13, 14, 22, 39, 46, 60, 65
Online: o1 - o10

Goals

1. Apply and recognize transformations of functions:
- Given a graph, find an algebraic function. 1, 7, o1, o5, o9
- Given a formula, find a graph. 13, 14, 22, o6, o7
2. Given functions f and g, find: f+g, f-g, fg, f/g, and their domains. o2, o8.
3. Understand and apply compositions of functions:
- Given functions f and g, find g o f and f o g. 28, 39, 60, o3, o10
- Given f o g, find functions f and g. 46, 65, o4

## Section 1.5

Homework:
Written: 11, 15, 18, 19, 21, 25, 29
Online: o1 - o5

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, o5.

## Section 1.6

Homework:
Written: 12,16,18,19,23,35,45,48,54,60,67,71,73
Online: o1 - o14

Goals

1. Tell if a function is one-to-one when it is described
-algebraically 12
-graphically o1
2. Know the definition of an inverse function, and use it to find the value of the inverse function at a point 16, 18, o2, o6
3. Given a function, find its inverse
-algebraically 19,23,54,o3,o4,o5
-graphically 45,73,o7
4. Know the laws of logarithms and be able to solve and simplify logarithmic and exponential expressions. 35,48,o6,o8,o9,o10,o11
5. Know the inverses of the trigonometric functions
- domain and range 71
- values at a point 60, o12, o13
- simplify a trig function composed with an inverse trig function 67, o14

## Section 2.1

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

Homework:
Written: 6, 7, 9, 16, 27, 32, 34a, 40
Online: o1 - o6

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, o2, 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, o2, o3, o4, o5, o6
3. Explain how one-sided and two-sided limits are connected. Use this to compute limits. 6, 7, o1, o2, o3
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, 40, o4, o5, o6

## Section 2.3a

Homework:
Written: 10,15,19,20,21,22,28,29
Online: o1 - o11

Goals

1. Apply limit laws to simplify limits
- algebraically o1
- graphically o2
2. Recognize when you can directly substitute to compute a limit and when you cannot. 10, o3, o4, o5, o6
3. Use algebra to simply and find limits.
- for polynomials over polynomials. 15, 19, 20, o4, o5, o6, o7, o8, o9
- expressions with square roots. 21, 22, 29, o10
- expressions with multiple fractions. 28, 29, o11

## Section 2.3b

Homework:
Written: 36-39, 42, 55, 56, 58
Online: o1 - o8

Goals

1. Use the Squeeze Theorem to find limits. 36, 37, 38, o4, o5, o6
2. Evaluate limits that involve absolute values and other piecewise functions. 39, 42, o1, o2, o3
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). o7, o8

## Section 2.4 (Jessica)

Homework:
Written: 2, 14, 15, 16, 19, 20
Online: o1 - o5

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

Homework:
Written: 4, 6, 15, 18, 22, 23, 43ab, 58
Online: o1 - o9

Goals

1. Know the definition of continuity. Use it to determine why a given function is discontinuous, or limits and values of continuous functions. o1, o2, o3, o4
2. Given the graph of a function, be able to tell:
-where it is continuous. 4, o8
-where it is discontinuous, and the type of discontinuity. 6, o9
3. Given a function described algebraically, be able to tell:
-where it is continuous. 22, 23, o5, o6, o7
-where it is discontinuous and types of discontinuities. 15, 18, 43(a,b)
4. Use limit laws and the definition of continuity to prove facts about continuity (e.g. Theorem 4). 58

## Section 2.5b

Homework:
Written: 28, 31, 32, 36, 37, 39, 41, 45, 47, 49, 65
Online: o1 - o7

Goals

1. Use continuity to evaluate limits 31, 32, o7
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,o3
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. o4, o5, o6

## Section 2.6

Homework:
Written: 4, 5, 7, 13, 23, 24, 33, 43, 48
Online: o1 - o 14

Goals

1. Find the limit as x approaches +/-infinity, and equations of any horizontal asymptotes
-For a rational function, 13, 43, o1, o2, o6, o9, o11, o12, o13 -For a function described graphically, 4, o14, -For a function with exponentials in the numerator and denominator, 33, o7, o8 -For a function with square roots in the numerator and denominator. Be able to approach both infinity and negative infinity correctly in this case. 23, 24, o3, o4, o12 -Using the squeeze theorem. o10
2. Give examples of functions with prescribed horizontal and vertical asymptotes. 5, 7, 48

## Section 2.7

Homework:
Written: 5, 9, 19, 20, 43, 44, 46
Online: o1 - o9

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. 5, o4, o5, o8, o9
2. Given a word problem, interpret instantaneous or average rate of change. 43, 44, 46
Given a position function, find average and instantaneous velocity. o1, o2, o3
3. Sketch the graph of a function with prescribed derivatives. 19, 20
4. Given a limit expression, write down the point and function whose derivative it represents. o6, o7
5. Write both formal definitions of a derivative. Explain how the derivative can be interpreted as slope of a line, velocity, instantaneous change.

## Section 2.8

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: o1 - o8

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, o7, o8
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 the graph of f. 1, 3, 5, 6, 7, 9, 11, o1, o4, o5, o6
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: 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

Homework:
Written: 2, 13, 17, 20, 25, 28, 29, 33, 55, 61, 77
Online: o1 - o12

Goals

1. Compute derivatives using:
• The power rule. Especially, be able to rewrite functions so the power rule can be applied. 13, 17, 20, 25, 28, 29, all online problems.
• The derivative of e^x. 2, 17, 28, o12
• 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:
• 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
• 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

Homework:
Written: 2,11,13,23,24,32,33,47,49,55,57
Online: o1 - o11

Goals

1. Apply the product rule to find the derivative of functions. 11,24,33,47,49,55,57, o2,o4,o5,o6,o7
2. Apply the quotient rule to find the derivative of functions. 2,13,23,24,32,47,49 ,o3,o8,o10,o11
3. Be able to prove the product and quotient rules and variations. 55,57*

## Section 3.3

Homework:
Written: 9, 10, 18, 20, 35, 42, 45, 49
Online: o1 - o15

Goals

1. Know derivatives for the 6 basic trig functions. Combine these with other differentiation rules to take derivatives. 9, 10, 18, 35, 49, o1, o2, o4, o8, o9, o10, o11, o12, o13, o14, o15
2. Know limits  and  in the book. Be able to use these to solve other limits involving trig functions. 42, 45, o5, o6, o7
3. Know the "periodicity" of differentiation of trig functions--i.e. the fourth derivative of the sine function is again the sine function. o3
4. Use the definition of the derivative plus the limits  and  to prove that the derivative of sin(x) is cos(x), and to prove that the derivative of cos(x) is sin(x). 20

## Section 3.4

Homework:
Written: 7, 24, 25, 31, 37, 47, 63, 65, 71, 84, 89
Online: o1 - o 14

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, o4, o5, o6, o7, o8, o9, o10, o11
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, o12, o13
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, o14
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

Homework:
Written: 3, 11, 12, 15, 18, 21, 23, 34, 41, 43, 57, 67
Online: o1 - o13

Goals

1. For an implicitly defined equation, use implicit differentiation to find
- dy/dx. 3, 11, 12, 15, 18, 21, o1, o2, o8, o10, o12
- dx/dy. 23
- d^2y/dx^2. 34, o7
2. Find the tangent line to an implicitly defined function. 41, 43, o3, o9, o11, o13
3. Use implicit differentiation to prove formulas for inverse functions, including inverse trig functions. 57, 67
4. Use formulas for derivatives of inverse trig functions, as well as other differentiation rules, to compute derivatives. o4, o5, o6

## Section 3.8

Homework:
Written: 14, 15
Online: o1 - o7

Goals

Use exponential functions to model each of the following:
1. Population growth. o1, o3, o4
2. Radioactive decay. Especially, know how to find half-life, or given half-life, find an equation. o5, o6
3. Cooling. 14, 15
4. Interest. o2, o7
- Derive the formula for compounding interest in n periods (see section 1.3, number 60)
- Use expression of e as a limit to find formula for continuously compounded interest. (In text, p 239)

## Section 3.9 (Mark)

Homework:
Written: 5, 22, 33, 35, 37, 42
Online: o1 - o6

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 4.3 (Sam)

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

Homework: 4, 7, 15, 31, 33, 39, 47, 49, 57, 70, 71, 81

Goals:

 1.  Know when to use L'Hospital's rule to solve a limit.  (All problems, particularly 4, 71, others)
2.  Use L'Hospital's rule to find limits of functions of the following indeterminate forms:
a.  infinity/infinity or 0/0   7, 15, 33, 70, 81, o1, o3, o6
b.  Indeterminate products  39, 08
c.  Indeterminate differences  47, 49, o4, o7
d.  Indeterminate powers  57, o9, o10


## 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 4.8 (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.
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 5.1/5.2a (Drew)

Homework

  Written:  5.1: 1a, 12,18,21,26
Written:  5.2: 5ab, 17, 29, 30, 70
Online: o1, o2, 5.1.13, o1 from 5.2b modified, o2 from 5.2b modified


Goals

  1. Use left and right sums to approximate area under a continuous function (1a, o1).
2. Use the definitions of area and integral to write an area or integral as a limit, or write a limit as an area or an integral (18, 21, 5.2 17, 29, 30, 70, 5.2b o1, o2).
3. Approximate integrals by left and right sums. (5.2 5ab, o2)
4. Interpret total change problems as area problems (12, onlinified 13).


Killed 5, 15, o3, 5.2.6. Modify o1 and o2 from 5.2b and move them here. Added 5.2 17, 29, 30, and 70. Added 5.1.12. Onlinify 5.1.13 and add. Dropped midpoint sums.

## Section 5.3 (Tyler a, Ryan b)

Goals:

A. Understand and explain the relationship between area and derivative described by the Fundamental Theorem.
A. Evaluate and describe properties of functions defined by integrals 3, 5.3bO11,63,64
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.


Problems:

 A. written: 9,15,56,57,63,64
B. written: 22,31,36,41,43,59,65,66,74


Changes to Online HW:

A.  O5.  Make the hint part of the problem or make it not show until after they try once.
A.  Move 5.3b O2
A.   Kill O1

B. O3 change arbitrary to right endpoint.
B. Fix formatting
B. Add one more limit problem (e.g., 24)


## 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.).