Study Guides for the Suggested Exercises

**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

**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!

**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.

**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!

**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.

**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

**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

**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

**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

**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)

**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

**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

**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

**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

**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

**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*

**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)

**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

**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.

**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

**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

**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

**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

**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

**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

**Exercises:**Section 2.5 # 9,11,13,15,17,19**Relevant Reading:**Section 2.5 pages 147 - 149 subsections titled*Constant Function Rule*and*Power Rule***Useful Book Examples:**Section 2.5 Examples 1,2,3 (and corresponding Matched Problems)**Useful Lecture Notes:**Class Meeting #11

**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

**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

**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

**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

**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*.

*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."*.

- The author found that
**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.

**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

**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

**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)**Useful Lecture Notes:**Class Meetings #13, 14

**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

**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

**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.

**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

**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

**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

**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

**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

**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

**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

**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

**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.

**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

**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

**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.

**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

**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

**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

**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

**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*)

- 254 - 257 (the subsection titled
**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*

**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

**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

- Class Drill 19 Using a graph of

**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

**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

**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

**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

**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

**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

**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?

**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.

**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.

**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

**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?

**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

**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

**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

**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

**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

**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:**none

**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

**Exercises:**Section 5.5 # 11, 13, 19, 21, 23, 25, 27, 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

**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

**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.

**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

**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

**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

**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

**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

**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

(page maintained by Mark Barsamian, last updated August, 2016