Suggested Exercises for MATH 1350 Survey of Calculus

2017 - 2018 Spring Semester

Textbook Information

The goal of the course is for you to be able to solve the 373 problems on this list.

Notice that the exercises are presented in groups enclosed in parentheses, and each group is a link. Click on the group to go to a web page with study guides to help you organize your studying of each group of exercises.

Textbook Section | Suggested Exercises

2.1 Introduction to Limits

Section 2.1 Group 1: Limit Behavior of Functions Given by Graphs

Exercises: Section 2.1 # 15, 16, 21, 23
Remarks: In these exercises, the student is asked to find function values and limits for functions f and g that are given by graphs (not given by formulas). One could visualize the idea of these problems with an arrow diagram:
Graph ==> Description of Limit Behavior.
Relevant Reading: Section 2.1 pages 95 - 98
Useful Book Examples: Section 2.1 Example 4 (and corresponding Matched Problem)
Useful Lecture Notes: Class Meeting #1
Useful Class Drills: Class Drill #1

Section 2.1 Group 2: Drawing a Graph that has Prescribed Limit Behavior

Exercises: Section 2.1 # 47, 49
Remarks: In these exercises, the student is given a description of Limit Behavior, and is asked to draw a graph that has that behavior. One could visualize the idea of these problems with an arrow diagram:
Description of Limit Behavior ==> Graph.
Relevant Reading: Section 2.1 pages 95 - 98 contain the general exposition about Limits, but there is no discussion of problems such as these.
Useful Book Examples: There are no similar book examples.
Useful Lecture Notes: Class Meeting #1 lecture notes will be very important, because there are no similar book examples!

Section 2.1 Group 3: Basic Limits using Theorem 2 Properties of Limits

Exercises: Section 2.1 # 33, 35, 37
Remarks: In these exercises, the student is asked to find function values and limits for functions that are given by graphs formulas. The idea is to use Theorem 2 (Properties of Limits) and Theorem 3 (Limits of Polynomial and Rational Functions). The main point of the "solution" of problems like these is to show each application of a Theorem on its own line, with the particular Theorem cited by number (such as Theorem 2.6). (See book examples and class notes for models.) In all three of these basic exercises, the value of the limit at x=c ends up equaling the function value at x=c. But again, the limit is to be computed by using the Theorems and citing them in the written solution, not simply by just substituting in x=c.
Relevant Reading: Section 2.1 pages 95 - 101
Useful Book Examples: Section 2.1 Examples 5,6 (and corresponding Matched Problems)
Useful Lecture Notes: Class Meeting #2
Useful Reference Pages: Reference 2: Facts about Limits from Section 2.1 on page 2 of the Course Packet.

Section 2.1 Group 4: Limits of Piecewise-Defined Functions

Exercises: Section 2.1 # 51, 53, 57, 91
Remarks: In these exercises, the function is a piecewise defined function. That is, the domain of the function is broken up into two or more pieces, and the formula that describes the function is different on each piece. In problems #51 and #53, the piecewise definition is explicit: The pieces of the domain are presented clearly, and for each piece of the domain, the formula for the function is given clearly. Problem #57 is not presented as a piecewise-defined function: the formula is just a single expression. But notice that the expression involves the Absolute Value Function. In order to work with the Absolute Value fFnction, you will have to realize that it, too, is a piecewise-defined function. (The piecewise definition of the Absolute Value Function is given just before Example #3 on page 97.) Problem #91, about Telephone Rates, asks the student to come up with a piecewise definition of the charge function F, and then graph it and answer questions about its limit behavior.
Relevant Reading: Section 2.1 pages 95 - 101
Useful Book Examples: Section 2.1 Examples 3, 7, 8B (and corresponding Matched Problems)
Useful Lecture Notes: Not covered in lecture: be sure to study the book!

Section 2.1 Group 5: Limits of Rational Functions

Exercises: Section 2.1 # 61, 63, 77, 79
Remarks: In these exercises, the function is a rational function. A crucial concept is the distinction between the function value at x=c and the limit at x=c. In some problems, the function value does not exist at x=c but the limit does exist at x=c. But in other problems, niether the function value nor the limit exists at x=c. Perhaps the most important concept of the first month of the course is the idea that sometimes one cannot cancel expressions when computing function values (if the expressions result in 0/0), but can cancel those same expressions when computing limits. The book does a lousy job with this concept in Section 2.1. In Example 8, the book says that "...Algebraic simplification is often useful ...", but this is very vague and does not address the underlying issue. The "Conceptual Insight" at the top of page 103 attempts to explain the concept. But nothing is mentioned about the fact that when one is actually computing a function value (not computing a limit), one is not allowed to cancel. In lecture, I will be very explicit about the important underlying concept. And I will be very clear in lectures about how I expect these sorts of problems to be written. It is important that in your solutions, you explain the important underlying concept clearly. (Again, note that the book solutions do not explain the underlying concept at all. So I will be expecting your written solutions to be more thorough than the solutions presented in the book!)
Relevant Reading: Section 2.1 pages 95 - 104
Useful Book Examples: Section 2.1 Examples 8A, 9 (and corresponding Matched Problems)
Useful Lecture Notes: Class Meeting #2
Useful Reference Pages: Reference 2: Facts about Limits from Section 2.1 on page 2 of the Course Packet.

Section 2.1 Group 6: Limits of Difference Quotients

Exercises: Section 2.1 # 81, 83
Remarks: In these exercises, the student is given a formula for a function f and must build a Difference Quotient for f and then take a limit of that Difference Quotient. The calculations are difficult, and they seem rather random: the book mentions that Difference Quotients are "important", but does not explain why they are important. It turns out that Difference Quotients are central to the upcoming definition of the Derivative in Section 2.4.
Relevant Reading: Section 2.1 pages 95 - 104
Useful Book Examples: Section 2.1 Example 10 (and corresponding Matched Problem)
Useful Lecture Notes: Class Meeting #2

Section 2.1 Group 7: Conceptual Questions

Exercises: Section 2.1 # 67,68,69,70,71,72
Remarks: These exercises require no computation, but to answer them will require that you read Section 2.1 very carefully. The clues that you need to answer these six questions are scattered throughout the reading.
Relevant Reading: Section 2.1 pages 95 - 105
Useful Book Examples: Section 2.1 All 10 Examples (and corresponding Matched Problems)
Useful Lecture Notes: Class Meetings #1 and #2

2.2 Infinite Limits and Limits at Infinity

Section 2.2 Group 1: Limits Involving Infinity for a Function Given by a Graph

Exercises: Section 2.2 # 9,10,11,12,13,14,15,16
Remarks: In these exercises, the student is asked to find limits involving infinity for a function f that is given by a graph (not given by formulas). One could visualize the idea of these problems with an arrow diagram:
Graph of Function ==> Description of Limit Behavior
Relevant Reading: Section 2.2 pages 109 - 117 contain the general exposition about limits involving infinity, but there is no discussion of problems such as these.
Useful Book Examples: There are no similar book examples.
Useful Lecture Notes: Class Meeting #3 lecture notes will be very important, because there are no similar book examples!
Useful Class Drills: Class Drill #2

Section 2.2 Group 2: Limits Involving Infinity for a Function Given by a Formula

Exercises: Section 2.2 # 17,19,21,33,35,37,39
Remarks: In these exercises, the student is asked to find limits at a particular x-value for a function f that is given by a formula. One could visualize the idea of these problems with an arrow diagram:
Formula for Function ==> Description of Limit Behavior
In these exercises, sometimes the limits turn out to be real numbers. In these cases, the limits are the same as the kinds of limits that are discussed in Section 2.1. But in most of the exercises, the limits turn out to involve infinity or to not exist. Also note that in the textbook examples, limits like these are explored by substituting x-values into the function f(x) and computing the resulting y-value using a calculator. The reader would get the impression that without a calculator, there is no way to determine this sort of limit. But in Class Meeting #3 and Class Drill #3, a method of determining infinite limits without a calculator is presented. This method gives a much better understanding of how the limits work.
Relevant Reading: Section 2.2 pages 109 - 112, the subsections called Infinite Limits and Locating Vertical Asymptotes.
Useful Book Examples: Section 2.2 Examples 1,2 (and corresponding Matched Problems)
Useful Lecture Notes: Class Meeting #3 lecture notes will be very important, because they present a method of determining infinite limits without a calculator.
Useful Class Drills: Class Drill #3

Section 2.2 Group 3: Limits at Infinity for Polynomial Functions

Exercises: Section 2.2 # 27,31,73,83
Relevant Reading: Section 2.2 pages 112 - top of 115, the subsection called Limits at Infinity. This material is covered well in the book and is not as difficult as some of the other material in the section, so it will not be covered in class. Be sure to read the book.
Useful Book Examples: Section 2.2 Examples 3,4 (and corresponding Matched Problems)

Section 2.2 Group 4: Limits at Infinity for Rational Functions

Exercises: Section 2.2 # 43,47,65,67,69
Relevant Reading: Section 2.2 pages 115 - 117, the subsection called Finding Horizontal Asymptotes.
Useful Book Examples: Section 2.2 Example 5 (and corresponding Matched Problem)
Useful Lecture Notes: Class Meeting #4

Section 2.2 Group 5: Applications Problems Involving Limits at Infinity

Exercises: Section 2.2 # 89,92
Remarks: In both of these problems, a function is a model of a real-world situation. You must not only find the limit of the function as x goes to infinity, but also explain what the result tells us about the real-world situation that is being modeled by the function.
Relevant Reading: Section 2.2 pages 115 - 117, the subsection called Finding Horizontal Asymptotes.
Useful Book Examples: Section 2.2 Example 5 (and corresponding Matched Problem)
Useful Lecture Notes: Class Meeting #5

Section 2.2 Group 6: Finding All Horizontal and Vertical Asymptotes

Exercises: Section 2.2 # 51,53,63,64
Remarks: .
Relevant Reading: Section 2.2 pages 110 - 112 (subsection called Locating Vertical Asymptotes) and pages 115 - 117 (subsection called Finding Horizontal Asymptotes)
Useful Book Examples: Section 2.2 Examples 1,2,5,6 (and corresponding Matched Problems)
Useful Lecture Notes: Class Meeting #5

Section 2.2 Group 7: Conceptual Questions about Asymptotes

Exercises: Section 2.2 # 77,78,79,80,81,82
Remarks: These exercises require no computation, but to answer them will require that you read Section 2.2 very carefully. The clues that you need to answer these six questions are scattered throughout the reading.
Relevant Reading: Section 2.2 pages 109 - 117
Useful Book Examples: Section 2.1 All 6 Examples (and corresponding Matched Problems)
Useful Lecture Notes: Class Meetings #3,4,5

2.3 Continuity

Section 2.3 Group 1: Questions about Continuity of a Function Given by a Graph

Exercises: Section 2.3 # 19,20,21,22
Remarks: In these exercises, the student is asked questions related to Continuity for a function f that is given by a graph (not given by formulas). One could visualize the idea of these problems with an arrow diagram:
Graph of Function ==> Description of Continuity Behavior
Relevant Reading: Section 2.3 pages 121 - 124 subsection titled Continuity
Useful Book Examples: Section 2.3 Example 1 (and corresponding Matched Problem)
Useful Lecture Notes: Class Meeting #6
Useful Class Drills: Class Drill #4

Section 2.3 Group 2: Drawing a Graph that has Prescribed Limit Behavior

Exercises: Section 2.3 # 11,14

Remarks: In these exercises, the student is given a description of Limit Behavior, and is asked to draw a graph that has that behavior. One could visualize the idea of these problems with an arrow diagram:
Description of Limit Behavior ==> Graph
This kind of problem has been encountered before, in Section 2.1 Exercises # 47,49. The added twist in the current section is that the student is also asked to discuss the continuity of the function.

Relevant Reading: Recall that in textbook Section 2.1 there was no discussion of problems involving drawing a graph that has prescribed limit behavior. WE discussed those kinds of problems in Class Meeting #1. Read Section 2.3 pages 121 - 124 subsection titled Continuity for a discussion of Continuity.

Useful Book Examples: There are no similar book examples.

Useful Lecture Notes: Class Meeting #1 lecture notes will be very important for recalling how to draw the graph, because there are no similar book examples!
Class Meeting #6 for discussion of Continuity

Section 2.3 Group 3: Questions about Continuity of a Function Given by a Formula

Exercises: Section 2.3 # 35,37,69
Remarks: In these exercises, the student is asked questions related to Continuity for a function f that is given by a formula. One could visualize the idea of these problems with an arrow diagram:
Formula for Function ==> Description of Continuity Behavior
This material is covered well in the book and is not as difficult as some of the other material in the section, so it will not be covered in class. Be sure to read the book.
Relevant Reading: Section 2.3 pages 124 - 126 subsections titled Continuity and Continuity Properties
Useful Book Examples: Section 2.3 Examples 2,3 (and corresponding Matched Problems)

Section 2.3 Group 4: Conceptual Questions about Continuity

Exercises: Section 2.3 # 77,78,79,80,81
Remarks: These exercises require no computation, but to answer them will require that you read the first half of Section 2.3 very carefully. The clues that you need to answer these six questions are scattered throughout the reading.
Relevant Reading: Section 2.3 pages 121 - 126 subsections titled Continuity and Continuity Properties
Useful Book Examples: Section 2.3 Examples 1,2,3 (and corresponding Matched Problems)
Useful Lecture Notes: Class Meeting #6

Section 2.3 Group 5: Simple Questions about Sign Behavior of Functions

Exercises: Section 2.3 # 55,85
Remarks: Question #55 is about Graph ==> Description of Sign Behavior
Question #85 is about Description of Sign Behavior ==> Sketch Possible Graph
These questions are simply meant to help emphasize to the student that a continuous function only changes sign at x-values where there is an x-intercept. On the intervals in between the x-intercepts, the sign of the function does not change.

Section 2.3 Group 6: Analyzing Sign Behavior of Functions Given By a Formula

Exercises: Section 2.3 # 47,49,51,53
Remarks: The sign chart techniques introduced in this section will be very important later in the course.
Relevant Reading: Section 2.3 pages 126 - 128 subsection titled Solving Inequalities Using Continuity Properties
Useful Book Examples: Section 2.3 Example 4 (and corresponding Matched Problem)
Useful Lecture Notes: Class Meetings #6,7

2.4 The Derivative

Section 2.4 Group 1: Slopes of Secant and Tangent Lines

Exercises: Section 2.4 # 9,13,43,45
Relevant Reading: Section 2.4 pages 132 - 137 subsections titled Rate of Change and Slope of the Tangent Line
Useful Book Examples: Section 2.4 Examples 1,2,3 (and corresponding Matched Problems)
Useful Lecture Notes: Class Meetings #8,9
Useful Class Drills: Class Drill #5 Representations of Slopes

Section 2.4 Group 2: Derivatives of Constant, Linear, and Quadratic Functions

Exercises: Section 2.4 # 19,21,27,55,87
Relevant Reading: Section 2.4 pages 138 - 139, the first part of the subsection titled The Derivative
Useful Book Examples: Section 2.4 Examples 4,5 (and corresponding Matched Problems)
Useful Lecture Notes: Class Meeting #9

Section 2.4 Group 3: Derivatives of 1/x type and square root type functions

Exercises: Section 2.4 # 33,35
Relevant Reading: Section 2.4 pages 138 - 141 subsection titled The Derivative
Useful Book Examples: Section 2.4 Examples 6,7 (and corresponding Matched Problems)
Useful Lecture Notes: Class Meeting #10

Section 2.4 Group 4: Questions about the Nonexistence of the Derivative

Exercises: Section 2.4 # 47,48,49,50,51,52
Relevant Reading: Section 2.4 pages 142 - 143 subsection titled Nonexistence of the Derivative
Useful Book Examples: none
Useful Lecture Notes: Class Meeting #10

Section 2.4 Group 5: Conceptual Questions about the Derivative

Exercises: Section 2.4 # 61,62,63,64,66
Remarks: These exercises require no computation, but to answer them will require that you read Section 2.4 very carefully. The clues that you need to answer these five questions are scattered throughout the reading.
Relevant Reading: Section 2.3 pages 132 - 143

2.5 Basic Differentiation Properties

Section 2.5 Group 2: Basic Problems Involving Constant Multiple Property and Sum Property

Exercises: Section 2.5 # 35,37,49
Remarks: In these problems, the function is presented as a sum of terms that are each of the form of a constant times a power function. With the function in this form, the derivative can be found easily, with no rewriting of the function necessary.
Relevant Reading: Section 2.5 pages 150 - 151 subsections titled Constant Multiple Property and Sum Property
Useful Book Examples: Section 2.5 Examples 5bc (and corresponding Matched Problems)
Useful Lecture Notes: Class Meeting #12

Section 2.5 Group 3: Harder Problems Involving Constant Multiple Property and Sum Property

Exercises: Section 2.5 # 45,51,53,55,81
Remarks: In these problems, the function is presented in a form involving fractions. In this form, the derivative rules cannot be used to find the derivative. One must first rewrite the function as a sum of terms that are each of the form of a constant times a power function. With the function rewritten this way, the derivative can be found easily. The initial step of rewriting the function in a more useful form is a very important step, one that will be necessary in many problems throughout the rest of the course. This initial step relies only on arithmetic and algebra skills from prerequisite courses. Even so, many students have trouble with this step and as a result, are unable to find the derivatives in many of the problems that they encounter. So I encourage you to work very hard on this group of problems.
Relevant Reading: Section 2.5 pages 150 - 151 subsections titled Constant Multiple Property and Sum Property
Useful Book Examples: Section 2.5 Examples 4,5,7 (and corresponding Matched Problems)
Useful Lecture Notes: Class Meeting #12
Useful Class Drills: Class Drill #7

Section 2.5 Group 4: Velocity and Tangent Line Problems

Exercises: Section 2.5 # 59, 63
Relevant Reading: Section 2.5 pages 152 - 153 subsection titled Applications
Useful Book Examples: Section 2.5 Examples 7,8 (and corresponding Matched Problems)
Useful Lecture Notes: Class Meeting #12
Useful Class Drills: Class Drill #8

Section 2.5 Group 5: Conceptual Problems about Derivatives

Exercises: Section 2.5 # 83,84,85,86
Remarks: These exercises require no computation, but to answer them will require that you read Section 2.5 very carefully. The clues that you need to answer these four questions are scattered throughout the reading.
Relevant Reading: Section 2.5 pages 147 - 153

Section 2.5 Group 6: Applications Problems Involving Basic Differentiation Properties

Exercises: Section 2.5 # 89,91,97
Relevant Reading: Section 2.5 pages 147 - 152 presents the necessary derivative techniques.
Useful Book Examples: Section 2.5 does not include any examples asking the student to "...write a brief verbal interpretation of these results." But back in Section 2.4 Excample 8 (at the bottom of page 141), the student is asked to "...Find and interpret S(25) and S'(25)." The worked solution to that example gives an idea of what is meant by an "interpretation". You will need to read the solution to that example carefully to find the "interpretation". At the bottom of page 141, the author finds S(25) and S'(25). The "interpretation" of these two quantities is presented in the first sentence at the top of page 142.
- The author found that S(25) = 7, which is a number. The interpretation is that 25 months from now, the total sales will be $7 million. The units in this sentence are millions of dollars.
- The author found that S'(25) = 0.1, which is another number. The interpretation is that 25 months from now, the sales will be increasing at athe rate of $0.1 million per month. The units in this sentence are millions of dollars per month.

Observe that S(25) and S'(25) are simply numerical values, while the "interpretation" is a sentence that explains what these numerical values tell us about the company's sales. Also observe that the "interpretation" sentence uses appropriate units.
- Notice that the function S(t) was defined to represent the company's total sales (in millions of dollars) t months from now. And notice that the interpretation of the the meaning of the value S(25) consisted of a sentence that used the units millions of dollars.
- A key concept in this course is that the value of the derivative represents a rate of change. To figure out the correct units for the rate of change, one simply divides the unit of the function (millions of dollars, in this example) by the unit of time (months, in this example). But in my opinion, the author could have interpreted more clearly. To say that "...total sales ... are increasing at a rate of $100,000 per month" is not as clear as it should be. What does it mean that ...total sales...are increasing? How do total sales increase? They increase by selling products. So it might be clearer to say that "...the company's products are selling at a rate of $100,000 per month.".
Useful Lecture Notes: The idea of "interpretation" will not be covered in lecture during the discussion of Sections 2.4 and 2.5. You will need to read the book carefully (and read my remarks above) to learn about it when studying Section 2.5. But we will be discussing the topic quite a bit when we get to Chapter 3.

2.7 Marginal Analysis in Business and Economics

Section 2.7 Group 1: Finding The Exact Cost of Producing the 100th Bicycle

Exercises: Section 2.7 # 1, 2, 3
Relevant Reading: Section 2.7 pages 163 - top of 164
Useful Book Examples: Section 2.7 Example 1D (and corresponding Matched Problem)
Useful Lecture Notes: Class Meeting #13

Section 2.7 Group 2: Finding Marginal Cost, Marginal Revenue, and Marginal Profit

Exercises: Section 2.7 # 9, 13, 17
Relevant Reading: Section 2.7 pages 163 - middle of 168
Useful Book Examples: Section 2.7 Examples 1, 3 (and corresponding Matched Problems)
Useful Lecture Notes: Class Meeting #13

Section 2.7 Group 2: Applications Problems

Exercises: Section 2.7 # 33, 43, 47, 51
Relevant Reading: Section 2.7 pages 163 - middle of 168
Useful Book Examples: Section 2.7 Examples 1, 2, 3 (and corresponding Matched Problems)

3.1 The Constant e and Continuous Compound Interest

Section 3.1 Group 1: The Constant e

Exercises: Section 3.1 # 19
Relevant Reading: Section 3.1 page 181
Useful Book Examples: There are no relevant book examples. Be sure to read your class notes.
Useful Lecture Notes: Class Meeting #16,17

Section 3.1 Group 2: Using the Formula for Continually Compounded Interest

Exercises: Section 3.1 # 11, 13, 25, 27, 29, 33, 35, 39
Relevant Reading: Section 3.1 pages 181 - 184
Useful Book Examples: Section 3.1 Examples 1, 2, 3, 4 (and corresponding Matched Problems)
Useful Lecture Notes: Class Meeting #17

Section 3.1 Group 3: Radioactive Decay

Exercises: Section 3.1 # 41, 43
Relevant Reading: Section 3.1 pages 181 - 184
Useful Book Examples: No relevant book examples
Useful Lecture Notes: Not covered in class. You should be able to figure these two problems out using skills that you acquired in solving the problems in Group 2.

3.2 Derivatives of Exponential and Logarithmic Functions

Section 3.2 Group 1: Derivatives of Exponential Functions

Exercises: Section 3.2 # 9, 23, 49, 53
Relevant Reading: Section 3.2 pages 187 - 192
Useful Book Examples: Section 3.2 Examples 1, 2A, 3A (and corresponding Matched Problems)
Useful Lecture Notes: Class Meeting #18
Useful Class Drills: Class Drill #9: Derivatives of Functions Involving Exponents

Section 3.2 Group 2: Tangent Line and Rate of Change Problems Involving Exponential Functions

Exercises: Section 3.2 # 29, 63, 65, 67, 71)
Relevant Reading: Section 3.2 pages 187 - 192
Useful Book Examples: Section 3.2 Example 5 (and corresponding Matched Problem)
Useful Lecture Notes: Class Meeting #18

Section 3.2 Group 3: Derivatives of Logarithmic Functions

Exercises: Section 3.2 # 11, 17, 39, 40, 47, 51
Relevant Reading: Section 3.2 pages 187 - 192
Useful Book Examples: Section 3.2 Examples 2, 3 (and corresponding Matched Problems)
Useful Lecture Notes: Class Meeting #19
Useful Class Drills: Class Drill #10: Derivatives of Functions Involving Logarithms

Section 3.2 Group 4: Tangent Line Problems Involving Logarithmic Functions

Exercises: Section 3.2 # 27, 31
Relevant Reading: Section 3.2 pages 187 - 192
Useful Book Examples: There are no relevant book examples.
Useful Lecture Notes: Class Meeting #19
Useful Class Drills: Class Drill #10: Derivatives of Functions Involving Logarithms

3.3 Derivatives of Products and Quotients

Section 3.3 Group 1: Derivatives of Products

Exercises: Section 3.3 # 17, 19, 21, 55
Relevant Reading: Section 3.3 pages 196 - top of 198, the subsection titled Derivatives of Products
Useful Book Examples: Section 3.3 Examples 1, 3 (and corresponding Matched Problems)
Useful Lecture Notes: Class Meeting #20

Section 3.3 Group 2: Derivatives of Quotients

Exercises: Section 3.3 # 25, 31, 33, 59, 73, 87
Relevant Reading: Section 3.3 middle of page 198 to middle of page 200, the subsection titled Derivatives of Quotients
Useful Book Examples: Section 3.3 Examples 4, 5 (and corresponding Matched Problems)
Useful Lecture Notes: Class Meeting #21
Useful Class Drills: Class Drill #11: Don't Forget the Easy Derivative Rules

Section 3.3 Group 3: Tangent Line Problems Involving Quotients

Exercises: Section 3.3 # 63, 65, 69
Relevant Reading: Section 3.3 middle of page 198 to middle of page 200, the subsection titled Derivatives of Quotients
Useful Book Examples: No Relevant Book Examples: Be sure to read your class notes.
Useful Lecture Notes: Class Meeting #22

Section 3.3 Group 4: Applications Problems Involving Quotients

Exercises: Section 3.3 # 93, 95, 97
Relevant Reading: Section 3.3 middle of page 198 to middle of page 201, the subsection titled Derivatives of Quotients
Useful Book Examples: Section 3.3 Example 6 (and corresponding Matched Problem)
Useful Lecture Notes: Class Meeting #22
Useful Class Drills: Class Drills #13b, 13c

3.4 The Chain Rule

Section 3.4 Group 1: Chain Rule Problems Where the Outer Function is a Power Function

Exercises: Section 3.4 # 21, 29, 31, 37, 41, 59, 67, 71
Relevant Reading: Section 3.4 pages 204 - top of page 208, the subsections titled Composite Functions and General Power Rule
Useful Book Examples: Section 3.4 Example 3 (and corresponding Matched Problem)
Useful Lecture Notes: Class Meeting #23
Remarks: Book solves such problems using what it calls the General Power Rule. I won?t present that rule, because it is not helpful and not necessary. We?ll just use the chain rule.

Section 3.4 Group 2: Chain Rule Problems Where the Outer Function is a NOT a Power Function

Exercises: Section 3.4 # 25, 35, 43, 51
Relevant Reading: Section 3.4 middle of page 208 - top of page 211, the subsections titled The Chain Rule
Useful Book Examples: Section 3.4 Examples 4, 5, 6 (and corresponding Matched Problems)
Useful Lecture Notes: Class Meeting #24:
Useful Class Drills: Class Drill #12: Don?t Forget the Easy Derivative Rules Part II

Section 3.4 Group 3: Rate of Change Problems Involving the Chain Rule

Exercises: Section 3.4 # 91, 95, 97
Relevant Reading: Section 3.4 page 204 - top of page 211
Useful Book Examples: There are no relevant examples in Section 3.4, but there are examples involving Rate of Change in Sections 2.4, 3.2, and 3.3.
Useful Lecture Notes: Class Meeting #25
Useful Class Drills: Class Drills #13a, 13d

4.1 First Derivative and Graphs

Section 4.1 Group 1: Graphical Problems involving Increasing & Decreasing Functions

Exercises: Section 4.1 # 9, 10, 11, 12, 13, 14, 19, 21, 23, 25, 61, 65, 75, 79, 83, 91
Relevant Reading: Section 4.1 pages 238 - 249
Useful Book Examples: Section 4.1 Examples 6,9 (and corresponding Matched Problems)
Useful Lecture Notes: Class Meeting #27
Useful Class Drills: Class Drill 15 Determining Shape of the Graph of f ' by studying shape of graph of f
Remarks: I find the book's Section 4.1 to be very badly organized. The reading does not present concepts in a progressive manner, starting with simple concepts and then building on them to develop more sophisticated concepts. Rather, the concepts are presented in a rather random order. For instance, what I find to be the most important starting concept, that of correlating the increasing/decreasing behavior of a function f to the positive/negative behavior of its derivative f ', is scattered throughout the section. Simple examples involving recognizing increasing/decreasing behavior in a given graph of f do not show up until Examples 6 and 9. Furthermore, the ordering of exercises in the book does not correspond to the ordering of the presentation of the concepts in the reading. As a result, Section 4.1 tends to be one of the most difficult sections to cover in lecture, and one of the most difficult sections for students to understand. My lectures in Class Meetings #27, 28, 29 attempt to remedy this by re-ordering the presentation of the material. As much as possible, the exercise groups for Section 4.1 presented here correspond to the presentation of concepts in my lectures.

4.2 Second Derivative and Graphs

Section 4.2 Group 2: Analytical Problems Involving Increasing and Decreasing Functions

Exercises: Section 4.1 # 53, 55, 57, 59
Relevant Reading: Section 4.1 pages 238 - 240
Useful Book Examples: Section 4.1 Example 1 (and corresponding Matched Problem)
Useful Lecture Notes: Class Meeting #27

Section 4.1 Group 3: Partition Numbers and Critical Numbers

Exercises: Section 4.1 # 27, 29, 31
Relevant Reading: Section 4.1 pages 238 - middle of page 242, the subsection titled Increasing and Decreasing Functions
Useful Book Examples: Section 4.1 Examples # 2, 3, 4, 5 (and corresponding Matched Problems)
Useful Lecture Notes: Class Meeting #28
Useful Class Drills: Class Drill #16: Studying Graph Behavior

Section 4.1 Group 3: Basic Problems Involving the First Derivative Test

Exercises: Section 4.1 # 17, 41
Relevant Reading: Section 4.1 pages 242 - 246, the subsection titled First-Derivative Test
Useful Book Examples: Section 4.1 Example #7 (and corresponding Matched Problem)
Useful Lecture Notes: Class Meeting #28
Useful Class Drills: Class Drill #17: Using the First Derivative Test

Section 4.1 Group 4: More Sophisticated Problems Involving the First Derivative Test

Exercises: Section 4.1 # 43, 85, 97
Relevant Reading: Section 4.1 pages 242 - 246, the subsection titled First-Derivative Test
Useful Book Examples: Section 4.1 Example #7 (and corresponding Matched Problem)
Useful Lecture Notes: Class Meeting # 29
Useful Class Drills: Class Drill #17: Using the First Derivative Test

Section 4.2 Group 1: Graphical Problems Involving Concavity and Derivatives

Exercises: Section 4.2 # 9, 13
Relevant Reading: Section 4.2
- 254 - 257 (the subsection titled Using concavity as a graphing tool)
- 257 - middle of 258 (the first part of the subsection titled Finding Inflection Points)
- middle of 260 - middle of 261 (the first part of the subsection titled Analyzing Graphs)

Useful Book Examples: Section 4.2 Example 1 (and corresponding Matched Problem)
Useful Lecture Notes: Class Meetings # 30, 31
Useful Class Drills:
- Class Drill 18 Identifying Three kinds of Graph Behavior
- Class Drill 19 Using a graph of f ' to get information about f

Section 4.2 Group 2: Analytical Problems Involving Concavity and the Second Derivative

Exercises: Section 4.2 # 17, 19, 33, 35, 37, 89, 89
Relevant Reading: Section 4.2 pages 254 - 261
Useful Book Examples: Section 4.2 Examples 2,3,4 (and corresponding Matched Problems)
Useful Lecture Notes: Class Meetings # 31, 32

Section 4.2 Group 3: Curve Sketching

Exercises: Section 4.2 # 41, 45, 53, 73
Relevant Reading: Section 4.2 pages 261 - 264
Useful Book Examples: Section 4.2 Example 5 (and corresponding Matched Problem)
Useful Lecture Notes: Class Meetings # 31, 32
Useful Class Drills:
- Class Drill 19 Using a graph of f ' to get information about f
- Class Drill 20 Sketching a Graph of a Function from Information about its Derivatives
- Class Drill 21 Using the Graphing Strategy to Graph a Polynomial

4.5 Absolute Maxima & Minima

Section 4.5 Group 1: Absolute Extrema and the Closed Interval Method

Exercises: Section 4.5 # 26, 67
Relevant Reading: Section 4.5 pages 293 - 296
Useful Book Examples: Section 4.5 Example 1
Useful Lecture Notes: Class Meeting # 33
Useful Class Drills: None

Section 4.5 Group 2: Graphical Problems Involving Absolute Extrema

Exercises: Section 4.5 # 9, 17
Relevant Reading: Section 4.5 pages 293 - 294
Useful Book Examples: None
Useful Lecture Notes: Class Meeting # 34
Useful Class Drills: Class Drill 22 Local and Absolute Extrema

Section 4.5 Group 3: Analytical Problems Involving Absolute Extrema

Exercises: Section 4.5 # 31, 33, 38, 39, 43, 51, 53, 73, 75
Relevant Reading: Section 4.5 pages 296 - 299
Useful Book Examples: Section 4.5 Examples # 2, 3
Useful Lecture Notes: Class Meeting # 34
Useful Class Drills: None

4.6 Optimization

Section 4.6 Group 1: Abstract Optimization Problems

Exercises: Section 4.6 # 9, 13, 15, 17
Remark: The math involved in these abstract problems is similar to the math involved in the fence problems (in Group 2). This is not mentioned in the book, but I make it explicit in my lectures in Class Meetings #35 and 36.
Relevant Reading: Section 4.6 pages 301 - middle of 304
Useful Book Examples: None
Useful Lecture Notes: Class Meetings # 35, 36
Useful Class Drills: None

Section 4.6 Group 2: Optimization Problems Involving a Fence

Exercises: Section 4.6 # 33, 34, 35, 36
Relevant Reading: Section 4.6 pages 301 - middle of 304 (subsection titled Area and Perimeter)
Useful Book Examples: Section 4.6 Examples # 1, 2 (and corresponding Matched Problems)
Useful Lecture Notes: Class Meetings # 35, 36
Useful Class Drills: None

Section 4.6 Group 3: Maximizing Revenue and Profit

Exercises: Section 4.6 # 19, 25, 27
Relevant Reading: Section 4.6 pages middle of 304 - middle of 308 (subsection titled Maximizing Revenue and Profit
Useful Book Examples: Section 4.6 Examples # 3, 4, 5, 6, 7 (and corresponding Matched Problems)
Useful Lecture Notes: Class Meetings # 36, 37
Useful Class Drills: None

5.1 Antiderivatives, Indefinite Integrals

Section 5.1 Group 1: Is One Given Function and Antiderivative of Another Given Function?

Exercises: Section 5.1 # 25, 27, 29, 31, 33, 34, 35, 36, 37, 38, 39, 41
Relevant Reading: Section 5.1 pages 320 - middle of 321 (subsection titled Antiderivatives
Useful Book Examples: Section 5.1 Example # 1 (and corresponding Matched Problem)
Useful Lecture Notes: Class Meeting # 39
Useful Class Drills: Class Drill #24 Is One Given Function an Antiderivative of Another Given Function?

Section 5.1 Group 2: Basic Indefinite Integral Problems

Exercises: Section 5.1 # 9, 17, 19, 21, 23
Relevant Reading: Section 5.1 middle of 321 - middle of 326 (subsection titled Indefinite Integrals: Formulas and Properties
Useful Book Examples: Section 5.1 Example # 2 (and corresponding Matched Problem)
Useful Lecture Notes: Class Meetings # 39, 40
Remark: In these problems, the student only needs to use Indefinite Integral Formulas 1,2,3 from page 322 and Property 4 from page 323. There are no tricks required in these problems, no rewriting of the integrand to put it into a better form. All of the integrands are in convenient form for applying Property 4 and Integral Formulas 1,2,3.

Section 5.1 Group 3: Harder Indefinite Integral Problems

Exercises: Section 5.1 # 43, 45, 47, 49, 51, 53
Relevant Reading: Section 5.1 middle of 321 - middle of 326 (subsection titled Indefinite Integrals: Formulas and Properties
Useful Book Examples: Section 5.1 Example # 3 (and corresponding Matched Problem)
Useful Lecture Notes: Class Meetings # 40,41
Useful Class Drills: Class Drill #25 Good and Bad Indefinite Integral Solutions
Remark: In these problems, the integrand must be rewritten to put it into a better form before one can use the Integral Rules 1,2,3 and Properties 4,5.

Section 5.1 Group 4: Particular Antiderivative Satisfying an Extra Given Condition

Exercises: Section 5.1 # 55, 61
Relevant Reading: Section 5.1 middle of 321 - middle of 326 (subsection titled Indefinite Integrals: Formulas and Properties
Useful Book Examples: None
Useful Lecture Notes: Class Meetings # 41

5.2 Integration by Substitution

Section 5.2 Group 1: Is One Given Function and Antiderivative of Another Given Function?

Exercises: Section 5.2 # 51, 52, 53, 55, 56, 57
Remark: These problems are in the same style as the exercises in Section 5.1 Group 1. That is, the questions use the terminology of antiderivatives, but to answer the questions, one need only find derivatives. No indefinite integrals are needed. Notice that there is no discussion of this sort of problem in the current Section 5.2. The student need only review the textbook reading and class notes for the exercises in Section 5.1 Group 1. That material is listed again here:
Relevant Reading: Section 5.1 pages 320 - middle of 321 (subsection titled Antiderivatives
Useful Book Examples: Section 5.1 Example # 1 (and corresponding Matched Problem)
Useful Lecture Notes: Class Meeting # 39
Useful Class Drills: Class Drills #24, 26 Is One Given Function an Antiderivative of Another Given Function?

Section 5.2 Group 2: Substitution Problems With No Leftover Constant

Exercises: Section 5.2 # 11, 15, 17, 19, 67
Remark: In these problems, there are no leftover constants after the substitution step.
Relevant Reading: Section 5.2 pages 331 - middle of 336
Useful Book Examples: Section 5.1 Examples # 1, 3, 4 (and corresponding Matched Problems)
Useful Lecture Notes: Class Meeting # 42
Useful Reference: Reference 7: The Substitution Method

Section 5.2 Group 3: Substitution Problems With Leftover Constant

Exercises: Section 5.2 # 23, 27, 29, 31, 33, 41, 65
Remark: In these problems, there is a leftover constant after the substitution step.
Relevant Reading: Section 5.2 pages 331 - middle of 336
Useful Book Examples: Section 5.1 Example # 5 (and corresponding Matched Problem)
Useful Lecture Notes: Class Meetings # 43, 44
Useful Reference: Reference 7: The Substitution Method
Useful Class Drills: Class Drill #27: The Substitution Method

Section 5.2 Group 4: Applied Problems

Exercises: Section 5.2 # 81, 85
Remark: In these problems, there is a leftover constant after the substitution step.
Relevant Reading: Section 5.2 pages 331 - middle of 336
Useful Book Examples: Section 5.1 Example # 5 (and corresponding Matched Problem)
Useful Lecture Notes: Class Meetings # 43, 44
Useful Reference: Reference 7: The Substitution Method
Useful Class Drills: Class Drill #27: The Substitution Method

5.4 The Definite Integral

Section 5.4 Group 1: Left and Right Sums for a Function Given by a Graph

Exercises: Section 5.4 # 11, 13, 15, 17, 19
Relevant Reading: Section 5.4 pages 353 - Middle of 354
Useful Book Examples: None
Useful Lecture Notes: Class Meeting # 45
Useful Class Drills:
- Class Drill #28: Definite Integrals for a Simple Graph
- Class Drill #29: Estimating the Area Under a Graph Using Riemann Sums

Section 5.4 Group 2: Riemann Sum for a Function Given by a Formula or Table

Exercises: Section 5.4 # 23, 73
Relevant Reading: Section 5.2 pages 353 - 257
Useful Book Examples: Section 5.4 Examples 1 and 2 (and corresponding Matched Problem)
Remark: Note that the book discusses Left Sums and Right Sums mostly. And that's what we'll discuss in class. It is unfortunate (and silly) that the book does not include any exercises involving finding a Left Sum or Right Sum for a function given by a formula. Exercises #23,24,25,26 all deal with more complicated Riemann sums. Of these four exercises, Exercise #23 is the simplest, because it involve a midpoint sum. Example #2 on page 357 is similar.
Useful Lecture Notes: Class Meeting # 46
Useful Class Drills: None

Section 5.4 Group 3: Using Properties of Definite Integrals

Exercises: Section 5.4 # 33, 41, 45, 49, 51, 53
Relevant Reading: Section 5.4 pages bottom of 358 - 359 (the subsection entitled Properties of the Definite Integral)
Useful Book Examples: Section 5.4 Example 4 (and corresponding Matched Problem)
Remark: The book covers this material well, so I won't be discussing it in class. Be sure to read the book.
Useful Lecture Notes: not discussed in class
Useful Class Drills: non

Section 5.4 Group 4: Conceptual Problems about Definite Integrals

Exercises: Section 5.4 # 55, 56
Relevant Reading: Section 5.4 pages 353 - 359
Useful Book Examples: Section 5.4 all examples (and corresponding Matched Problems)
Useful Lecture Notes: not discussed in class
Useful Class Drills: none

5.5 The Fundamental Theorem of Calculus

Section 5.5 Group 1: Basic Problems Involving the Fundamental Theorem of Calculus

Exercises: Section 5.5 # 11, 13, 19, 21, 23, 25, 27, 30, 31
Relevant Reading: Section 5.5 pages 363 - 365
Useful Book Examples: Section 5.5 Examples 1 and 2 (and corresponding Matched Problems)
Useful Lecture Notes: Class Meeting #47
Useful Class Drills: Class Drill 30: The Fundamental Theorem of Calculus

Section 5.5 Group 2: Harder Problems Involving the Fundamental Theorem of Calculus

Exercises: Section 5.5 # 35, 36, 39, 41, 45
Relevant Reading: Section 5.5 pages 363 - 365
Useful Book Examples: Section 5.5 Examples 1 and 2 (and corresponding Matched Problems)
Useful Lecture Notes: Class Meetings # 47, 48
Useful Class Drill: none

Section 5.5 Group 3: Total Change Problems

Exercises: Section 5.5 # 69, 71, 89
Relevant Reading: Section 5.5 pages 363 - 367
Useful Book Examples: Section 5.5 Example 5 (and corresponding Matched Problem)
Useful Lecture Notes: Class Meeting #48
Useful Class Drill: none
Remark: Total Change Problems were also discussed in Class Meeting #44. Exercises 5.2 #81, 85 are about Total Change.

Section 5.5 Group 4: The Average Value of a Function over an Interval

Exercises: Section 5.5 # 49, 51, 55, 92
Relevant Reading: Section 5.5 pages 369 - 371
Useful Book Examples: Section 5.5 Example 8 (and corresponding Matched Problem)
Useful Lecture Notes: Class Meeting #49
Useful Class Drill: none

6.1 Area between Curves

Section 6.1 Group 1: Area Between Two Curves where One Curve is the X-Axis

Exercises: Section 6.1 # 9,11,17,21,25,27,59,95
Relevant Reading: Section 6.1 pages 382 - 383
Useful Book Examples: Section 6.1 Examples 1 and 2 (and corresponding Matched Problems)
Useful Lecture Notes: Not discussed in class
Useful Class Drill: none

Section 6.1 Group 2: Area Between Two Curves where Neither Curve is the X-Axis

Exercises: Section 6.1 # 39, 49, 55, 57, 59, 65, 67
Relevant Reading: Section 6.1 pages 382 - 385
Useful Book Examples: Section 6.1 Examples 1, 2, 3, 4, 5 (and corresponding Matched Problems)
Useful Lecture Notes: Class Meetings #51, 52
Useful Class Drill: Class Drill 31: Finding the Area Bounded by Curves using Two Different Methods

6.2 Applications in Business and Economics

Section 6.2 Group 1: Total Income Produced by a Continuous Income Stream

Exercises: Section 6.2 # 33, 35, 37, 39
Relevant Reading: Section 6.2 pages middle of 393 - 394 (subsection titled Continuous Income Stream)
Remark: Skip Section 6.2 pages 391 - middle of 393 (subsection titled Probability Density Functions)
Useful Book Examples: Section 6.2 Example 2 (and corresponding Matched Problem)
Useful Lecture Notes: Class Meetings #53
Useful Class Drill: None

Section 6.2 Group 2: Future Value Problems

Exercises: Section 6.2 # 9, 13, 41, 43, 47, 49, 51
Relevant Reading: Section 6.2 pages middle of 393 - 394 (subsection titled Future Value of a Continuous Income Stream)
Useful Book Examples: Section 6.2 Example 3 (and corresponding Matched Problem)
Useful Lecture Notes: Class Meetings #53
Useful Class Drill: None

Section 6.2 Group 3: Consumers' Surplus and Producers' Surplus

Exercises: Section 6.2 # 55, 57, 59, 61, 63
Relevant Reading: Section 6.2 pages middle of 397 - 400 (subsection titled Consumers' and Producers' Surplus
Useful Book Examples: Section 6.2 Examples 4, 5, 6 (and corresponding Matched Problems)
Useful Lecture Notes: Class Meetings # 54, 55
Useful Class Drill: Class Drill 32: Finding Equilibrium Price and Consumers? & Producers? Surplus