Topic Name Description
Course Introduction Page Course Information
Page Course Terms of Use
Unit 1: Analytic Geometry Page Unit 1 Learning Outcomes
1.1: Lines URL Whitman College: David Guichard's "Calculus, Chapter 1: Analytic Geometry, Section 1.1: Lines"

Read Section 1.1 (pages 14-17). Working with lines should be familiar to you, and this section serves as a review of the notions of points, lines, slope, intercepts, and graphing.

URL Whitman College: David Guichard's "Calculus, Chapter 1: Analytic Geometry, Exercises 1.1: Problems 1-18"

Work through problems 1-18. When you are done, check your answers in Appendix A.

1.2: Distance between Two Points, Circles URL Whitman College: David Guichard's "Calculus, Chapter 1: Analytic Geometry, Section 1.2: Distance between Two Points; Circles"

Read Section 1.2 (pages 19 and 20). This reading reviews the notions of distance in the plane and the equations and graphs of circles.

URL Whitman College: David Guichard's "Calculus, Chapter 1: Analytic Geometry, Exercises 1.2, Problems 1, 2, 6"

Please click on the link above, and work through problems 1, 2, and 6 in Exercise 1.2. When you are done, check your answers in Appendix A.

URL Temple University: Gerardo Mendoza and Dan Reich's "Calculus on the Web"

Click the link to Book I, and click on the "Index" button. Click on button 5 (Distance) to launch the first module. Complete problems 1-5. Then, return to the index and click on button 8 (Circles II) to launch the other module. Complete problems 1-5. If at any time the problem set becomes too easy for you, feel free to move on.

1.3: Functions URL Whitman College: David Guichard's "Calculus, Chapter 1: Analytic Geometry, Section 1.3: Functions"

Read Section 1.3 (pages 20-24). This reading reviews the notion of functions, linear functions, domain, range, and dependent and independent variables.

URL Whitman College: David Guichard's "Calculus, Chapter 1: Analytic Geometry, Exercises 1.3, Problems 1-16"

Work through problems 1-16 for Exercise 1.3. When you are done, check your answers in Appendix A.

1.4: Shifts and Dilations URL Temple University: Gerardo Mendoza's and Dan Reich's "Calculus on the Web"

Click the link to Book I, and then click on the "Index" button. Click on button 18 (Transforming Graphs) to launch the module. Complete problems 1-15. If at any time the problem set becomes too easy for you, feel free to move on. 

URL Whitman College: David Guichard's "Calculus, Chapter 1: Analytic Geometry, Section 1.4: Shifts and Dilations"

Read Section 1.4 (pages 25-28). This reading will review the graph transformations of shifts and dilations associated to some basic ways of manipulating functions.

Unit 2: Instantaneous Rate of Change: The Derivative Page Unit 2 Learning Outcomes
2.1: The Slope of a Function URL Whitman College: David Guichard's "Calculus, Chapter 2: Instantaneous Rate of Change: The Derivative, Section 2.1: The Slope of a Function"

Read Section 2.1 (pages 29-33). You will be introduced to the notion of a derivative through studying a specific example. The example will also reveal the necessity of having a precise definition for the limit of a function.

Page Massachusetts Institute of Technology: David Jerison's "Rate of Change"

Watch this video. In this lecture, Jerison introduces the notion of a derivative as the rate of change of a function, or the slope of the tangent line to a function at a point.

URL Whitman College: David Guichard's "Calculus, Chapter 2: Instantaneous Rate of Change: The Derivative, Exercises 2.1, Problems 1 - 6"

Work through problems 1-6 for Exercise 2.1. When you are done, check your answers against Appendix A.

2.2: An Example URL Whitman College: David Guichard's "Calculus, Chapter 2: Instantaneous Rate of Change: The Derivative, Section 2.2: An Example"

Read Section 2.2 (pages 34-36). This reading discusses the derivative in the context of studying the velocity of a falling object. This example again motivates the need for a more rigorous approach to the concept of a limit.

URL Whitman College: David Guichard's "Calculus, Chapter 2: Instantaneous Rate of Change: The Derivative, Exercises 2.2, Problems 1 - 3"

Work through problems 1-3 for Exercise 2.2. When you are done, check your answers against Appendix A.

2.3.1: The Definition and Properties of Limits URL Whitman College: David Guichard's "Calculus, Chapter 2: Instantaneous Rate of Change: The Derivative, Section 2.3: Limits"

Read Section 2.3 (pages 36-45). Read this section carefully, and pay close attention to the definition of the limit and the examples that follow. You should also closely examine the algebraic properties of limits as you will need to take advantage of these in the exercises.

Page Massachusetts Institute of Technology: David Jerison's "Limits"

Watch this video.

URL Whitman College: David Guichard's "Calculus, Chapter 2: Instantaneous Rate of Change: The Derivative, Exercises 2.3, Problems 1 - 18"

Work through problems 1-18 for Exercise 2.3. When you are done, check your answers against Appendix A.

URL University of California, Davis: Duane Kouba's "Precise Limits of Functions as x Approaches a Constant"

Work through problems 1-10. When you are done, check your solutions against the answers provided.

2.3.2: The Squeeze Theorem URL Whitman College: David Guichard's "Calculus, Chapter 4: Transcendental Functions, Section 4.3: A Hard Limit"

Read section 4.3 (pages 75-77). The Squeeze Theorem is an important application of the limit concept and is useful in many limit computations. This reading teaches you a useful trick for calculating limits of functions where at first glance you might seem to be dividing by zero.

Page PatrickJMT: "The Squeeze Theorem for Limits"

Watch this brief video. The creator of this video describes and illustrates the Squeeze Theorem by using specific examples.

2.4: The Derivative Function URL Whitman College: David Guichard's "Calculus, Chapter 2: Instantaneous Rate of Change: The Derivative, Section 2.4: The Derivative Function"

Read Section 2.4 (pages 46-50). In this reading, you will see how limits are used to compute derivatives. A derivative is a description of how a function changes as its input varies. In the case of a straight line, this description is the same at every point, which is why we can describe the slope (another word for the derivative) of an entire function with one slope when it is linear. You will learn that we can do the same for nonlinear functions. The slope, however, will not be constant; it will change as the independent variable changes.

URL Whitman College: David Guichard's "Calculus, Chapter 2: Instantaneous Rate of Change: The Derivative: Exercises 2.4, Problems 1 - 5, 8 - 11"

Work through problems 1-5 and 8-11 for Exercise 2.4. When you are done, check your answers against Appendix A.

2.5: Adjectives for Functions URL Whitman College: David Guichard's "Calculus, Chapter 2: Instantaneous Rate of Change: The Derivative, Section 2.5: Adjectives for Functions"

Read Section 2.5 (pages 51-54). The intuitive notion of a continuous function is made precise using limits. In addition, you will be introduced to the Intermediate Value Theorem, which rigorously captures the intuitive behavior of continuous real-valued functions.

2.5.1: Continuous Functions URL Temple University: Gerardo Mendoza's and Dan Reich's "Calculus on the Web"

Click the link to Book I, and click on the "Index" button. Click on button 26 (A missing value). Work on problems 15-26. Return to the "Index," and click on button 27 (Discontinuities of simple piecewise defined functions). Complete problems 1-10. If at any time the problem set becomes too easy for you, feel free to move on. 

2.5.2: Differentiable Functions URL Temple University: Gerardo Mendoza's and Dan Reich's "Calculus on the Web"

Click the link to Book I, and click on the "Index" button. Click on button 39 (Differentiability). Complete problems 1-10. If at any time the problem set becomes too easy for you, feel free to move on.

2.5.3: The Intermediate Value Theorem Page PatrickJMT: "Intermediate Value Theorem"

Watch this brief video for an explanation on the Intermediate Value Theorem.

Unit 3: Rules for Finding Derivates Page Unit 3 Learning Outcomes
3.1: The Power Rule URL Whitman College: David Guichard's "Calculus, Chapter 3: Rules for Finding Derivatives, Section 3.1: The Power Rule"

Read Section 3.1 (pages 55-57). Here, you will learn a simple rule for finding the derivative of a power function without explicitly computing a limit.

URL Whitman College: David Guichard's "Calculus, Chapter 3: Rules for Finding Derivatives, Exercises 3.1, Problems 1-6"

Work through problems 1-6 in Exercise 3.1. When you are done, check your answers against Appendix A.

3.2: Linearity of the Derivative URL Whitman College: David Guichard's "Calculus, Chapter 3: Rules for Finding Derivatives, Section 3.2: Linearity of the Derivative"

Read Section 3.2 (pages 58 and 59). In this reading, you will see how the derivative behaves with regard to addition and subtraction of functions and with scalar multiplication. That is, you will see that the derivative is a linear operation.

URL Whitman College: David Guichard's "Calculus, Chapter 3: Rules for Finding Derivatives, Exercises 3.2, Problems 1 - 9, 11, 12"

Work through problems 1-9, 11, and 12 for Exercise 3.2. When you are done, check your answers against Appendix A.

3.3: The Product Rule URL Whitman College: David Guichard's "Calculus, Chapter 3: Rules for Finding Derivatives, Section 3.3: The Product Rule"

Read Section 3.3 (pages 60 and 61). The naïve assumption is that the derivative of a product of two functions is the product of the derivatives of the two functions. This assumption is false. In this reading, you will see that the derivative of a product is slightly more complicated, but that it follows a definite rule, called the product rule.

URL Whitman College: David Guichard's "Calculus, Chapter 3: Rules for Finding Derivatives: Exercises 3.3, Problems 1-5"

Work through problems 1-5 for Exercises 3.3. When you are done, check your answers against Appendix A.

URL Temple University: Gerardo Mendoza and Dan Reich's "Calculus on the Web"

Click the link to Book I, and click on the "Index" button. Click on button 44 (Product Rule). Complete problems 1-10. If at any time the problem set becomes too easy for you, feel free to move on.

3.4: The Quotient Rule URL Whitman College: David Guichard's "Calculus, Chapter 3: Rules for Finding Derivatives: Section 3.4: The Quotient Rule"

Read Section 3.4 (pages 62-65). As with products, the derivative of a quotient of two functions is not simply the quotient of the two derivatives. In this reading, you will see the quotient rule for differentiating a quotient of two functions. In particular, this will allow you to find the derivative of any rational function.

URL Whitman College: David Guichard's "Calculus, Chapter 3: Rules for Finding Derivatives: Exercises 3.4, Problems 5, 6, 8, 9"

Work through problems 5, 6, 8, and 9 for Exercise 3.4. When you are done, check your answers against Appendix A.

URL Temple University: Gerardo Mendoza's and Dan Reich's "Calculus on the Web"

Click the link to Book I, and click on the "Index" button. Click on button 45 (Quotient Rule) to launch the module. Complete problems 1-10. If at any time the problem set becomes too easy for you, feel free to move on.

3.5: The Chain Rule URL Whitman College: David Guichard's "Calculus, Chapter 3: Rules for Finding Derivatives, Section 3.5: The Chain Rule"

Read Section 3.5 (pages 65-69). The chain rule explains how the derivative applies to the composition of functions. Pay particular attention to Example 3.11, as it works through a derivative computation where all of the differentiation rules of this unit are applied in finding the derivative of one function.

Page Massachusetts Institute of Technology: David Jerison's "Chain Rule"

Watch this video.

URL Whitman College: David Guichard's "Calculus, Chapter 3: Rules for Finding Derivatives, Exercises 3.5, Problems 1-20, 36-39"

Work through problems 1-20 and 36-39 for Exercise 3.5. When you are done, check your answers against Appendix A.

Unit 4: Transcendental Functions Page Unit 4 Learning Outcomes
4.1: Trigonometric Functions URL Whitman College: David Guichard's "Calculus, Chapter 4: Transcendental Functions, Section 4.1: Trigonometric Functions"

Read Section 4.1 (pages 71-74). This reading will review with you the definition of trigonometric functions.

URL Whitman College: David Guichard's "Calculus, Chapter 4: Transcendental Functions, Exercises 4.1: Problems 1-4, 11"

Work through problems 1-4 and 11 for Exercise 4.1. When you are done, check your answers against Appendix A.

4.2: The Derivative of Sine URL Whitman College: David Guichard's "Calculus, Chapter 4: Transcendental Functions, Section 4.2: The Derivative of sin x"

Read Section 4.2 (pages 74 and 75). This reading begins the computation of the derivative of the sine function. Two specific limits will need to be evaluated in order to complete the computation. These limits are addressed in the following section.

4.3: A Hard Limit URL Whitman College: David Guichard's "Calculus, Chapter 4: Transcendental Functions, Section 4.3: A Hard Limit"

Read Section 4.3 (pages 75-77). You read this section before to become acquainted with the Squeeze Theorem. When you read the section again, pay particular attention to the geometric argument used to set up the application of the Squeeze Theorem.

URL Whitman College: David Guichard's Calculus: "Chapter 4: Transcendental Functions, Exercises 4.3: Problems 1-7"

Work through problems 1-7 for Exercise 4.3. When you are done, check your answers against Appendix A.

4.4: The Derivative of Sine, Continued URL Whitman College: David Guichard's "Calculus, Chapter 4: Transcendental Functions, Section 4.4: The Derivative of sin x, Continued"
Read Section 4.4 (pages 77 and 78). This reading completes the computation of the derivative of the sine function. Be sure to review all of the concepts which are involved in this computation.
URL Whitman College: David Guichard's Calculus: "Chapter 4: Transcendental Functions, Exercises 4.4: Problems 1-5"

Work through problems 1-5 for Exercise 4.4. When you are done, check your answers against Appendix A.

4.5: Derivatives of the Trigonometric Functions URL Whitman College: David Guichard's "Calculus, Chapter 4: Transcendental Functions, Section 4.5: Derivatives of the Trigonometric Functions"

Read Section 4.5 (pages 78 and 79). Building on the work done to compute the derivative of the sine function and the rules of differentiation from previous readings, the derivatives of the remaining trigonometric functions are computed.

URL Whitman College: David Guichard's "Calculus, Chapter 4: Transcendental Functions, Exercises 4.5: Problems 1-18"

Work through problems 1-18 for Exercise 4.5. When you are done, check your answers against Appendix A.

4.6: Exponential and Logarithmic Functions URL Whitman College: David Guichard's "Calculus, Chapter 4: Transcendental Functions, Section 4.6: Exponential and Logarithmic Functions"

Read Section 4.6 (pages 80 and 81). This reading reviews the exponential and logarithmic functions, their properties, and their graphs.

4.7: Derivatives of the Exponential and Logarithmic Functions URL Whitman College: David Guichard's "Calculus, Chapter 4: Transcendental Functions, Section 4.7: Derivatives of the Exponential and Logarithmic Functions"

Read Section 4.7 (pages 82-86). In this reading, the derivatives of the exponential and logarithmic functions are computed. otice that along the way the number e is defined in terms of a particular limit.

Page Massachusetts Institute of Technology: David Jerison's "Exponential and Log"

Watch this video. In this video, Jerison makes use of implicit differentiation at times during this lecture.

URL Whitman College: David Guichard's "Calculus, Chapter 4: Transcendental Functions, Exercises 4.7: Problems 1-15, 20"

Work through problems 1-15 and 20 for Exercise 4.7. When you are done, check your answers against Appendix A.

4.8: Implicit Differentiation URL Whitman College: David Guichard's "Calculus, Chapter 4: Transcendental Functions, Section 4.8: Implicit Differentiation"

Read Section 4.8 (pages 87-90). As a result of the chain rule, we have a method for computing derivatives of curves which are not explicitly described by a function. This method, called implicit differentiation, allows us to find tangent lines to such curves.

Page Massachusetts Institute of Technology: David Jerison's "Implicit Differentiation"

Watch this video.

URL Whitman College: David Guichard's "Calculus, Chapter 4: Transcendental Functions, Exercises 4.8: Problems 1-9, 11-16"

Work through problems 1-9 and 11-16 for Exercise 4.8. When you are done, check your answers against Appendix A.

4.9: Inverse Trigonometric Functions URL Whitman College: David Guichard's "Calculus, Chapter 4: Transcendental Functions, Section 4.9: Inverse Trigonometric Functions"

Read Section 4.9 (pages 91-94). In this reading, implicit differentiation and the Pythagorean identity are used to compute the derivatives of inverse trigonometric functions. You should notice that the same techniques can be used to find derivatives of other inverse functions as well.

URL Whitman College: David Guichard's "Calculus, Chapter 4: Transcendental Functions, Exercises 4.9: Problems 3-11"

Work through problems 3-11 for Exercise 4.9. When you are done, check your answers against Appendix A.

4.10: Limits Revisited URL Whitman College: David Guichard's "Calculus, Chapter 4: Transcendental Functions, Section 4.10: Limits Revisited"

Read Section 4.10 (pages 94-97). You will learn how derivatives relate back to limits. Limits of Indeterminate Forms (or limits of functions that, when evaluated, tend to 0/0 or ∞/∞) have previously been beyond our grasp. Using L'Hopital's Rule, you will find that these limits are attainable with derivatives.

URL Whitman College: David Guichard's "Calculus, Chapter 4: Transcendental Functions, Exercises 4.10: Problems 1-10, 21-24"

Work through problems 1-10 and 21-24 for Exercise 4.10. When you are done, check your answers against Appendix A.

4.11: Hyperbolic Functions URL Whitman College: David Guichard's "Calculus, Chapter 4: Transcendental Functions, Section 4.11: Hyperbolic Functions"

Read Section 4.11 (pages 99-102). In this reading, you are introduced to the hyperbolic trigonometric functions. These functions, which appear in many engineering and physics applications, are specific combinations of exponential functions which have properties similar to those that the ordinary trigonometric functions have.

Unit 5: Curve Sketching Page Unit 5 Learning Outcomes
5.1: Maxima and Minima URL Whitman College: David Guichard's "Calculus, Chapter 5: Curve Sketching, Section 5.1: Maxima and Minima"

Read Section 5.1 (pages 103-106). Fermat's Theorem indicates how derivatives can be used to find where a function attains its highest or lowest points.

Page Massachusetts Institute of Technology: David Jerison's "Curve Sketching"

Watch this video from 30:00 to the end. The lecture will make use of the first and second derivative tests, which you will read about below.

URL Whitman College: David Guichard's "Calculus, Chapter 5: Curve Sketching, Exercises 5.1: Problems 1-12, 15"

Work through problems 1-12 and 15 for Exercise 5.1. When you are done, check your answers against Appendix A.

5.2: The First Derivative Test URL Whitman College: David Guichard's "Calculus, Chapter 5: Curve Sketching, Section 5.2: The First Derivative Test"

Read Section 5.2 (page 107). In this reading, you will see how to use information about the derivative of a function to find local maxima and minima.

URL Whitman College: David Guichard's "Calculus, Chapter 5: Curve Sketching, Exercises 5.2: Problems 1-15"

Work through problems 1-15 for Exercise 5.2. When you are done, check your answers against Appendix A.

5.3: The Second Derivative Test URL Whitman College: David Guichard's "Calculus, Chapter 5: Curve Sketching, Section 5.3: The Second Derivative Test"

Read Section 5.3 (pages 108 and 109). In this reading, you will see how to use information about the second derivative (that is, the derivative of the derivative) of a function to find local maxima and minima.

URL Whitman College: David Guichard's "Calculus, Chapter 5: Curve Sketching, Exercises 5.3: Problems 1-10"

Work through problems 1-10 for Exercise 5.3. When you are done, check your answers against Appendix A.

5.4: Concavity and Inflection Points URL Whitman College: David Guichard's "Calculus, Chapter 5: Curve Sketching, Section 5.4: Concavity and Inflection Points"

Read Section 5.4 (pages 109 and 110). In this reading, you will see how the second derivative relates to concavity of the graph of a function and use this information to find the points where the concavity changes, i.e. the inflection points of the graph.

URL Whitman College: David Guichard's "Calculus, Chapter 5: Curve Sketching, Exercises 5.4: Problems 1-9, 19"

Work through problems 1-9 and 19 for Exercise 5.4. When you are done, check your answers against Appendix A.

5.5: Asymptotes and Other Things to Look For URL Whitman College: David Guichard's "Calculus, Chapter 5: Curve Sketching, Section 5.5: Asymptotes and Other Things to Look For"

Read Section 5.5 (pages 111 and 112). In this reading, you will see how limits can be used to find any asymptotes the graph of a function may have.

Page Massachusetts Institute of Technology: David Jerison's "Max-Min"

Watch this lecture until 45:00. The majority of the video lecture is about curve sketching, despite the title of the video.

URL Whitman College: David Guichard's "Calculus, Chapter 5: Curve Sketching, Exercises 5.5, Problems 1-5, 15-19"

Work through problems 1-5 and 15-19 for Exercise 5.5. When you are done, graph the curves using Wolfram Alpha to check your answers.

Unit 6: Applications of the Derivative Page Unit 6 Learning Outcomes
6.1: Optimization URL Whitman College: David Guichard's "Calculus, Chapter 6: Applications of the Derivative, Section 6.1: Optimization"

Read Section 6.1 (pages 115-124). An important application of the derivative is to find where a function takes its global maximum and its global minimum. The Extreme Value Theorem indicates how to approach this problem. Pay particular attention to the summary at the end of the section.

Page Massachusetts Institute of Technology: David Jerison's "Related Rates"

Watch this video until 45:00. The majority of the lecture is about optimization, despite its title.

URL Whitman College: David Guichard's "Calculus, Chapter 6: Applications of the Derivative, Exercises 6.1: Problems 5, 7, 9, 10, 14, 16, 22, 26, 28, 33"

Work through problems 5, 7, 9, 10, 14, 16, 22, 26, 28, and 33 for Exercise 6.1. When you are done, check your answers against Appendix A.

6.2: Related Rates URL Whitman College: David Guichard's "Calculus, Chapter 6: Applications of the Derivative, Section 6.2: Related Rates"

Read Section 6.2 (pages 127-132). Another application of the chain rule, related rates problems apply to situations where multiple dependent variables are changing with respect to the same independent variable. Make note of the summary in the middle of page 128.

Page Massachusetts Institute of Technology: David Jerison's "Newton's Method"

Watch this video until 40:30. The majority of the lecture is about related rates, despite its title.

URL Whitman College: David Guichard's "Calculus, Chapter 6: Applications of the Derivative, Exercises 6.2: Problems 1, 3, 5, 11, 14, 16, 19, 20, 21, 25"

Work through problems 1, 3, 5, 11, 14, 16, 19-21, and 25 for Exercise 6.2. When you are done, check your answers against Appendix A.

6.3: Newton's Method URL Whitman College: David Guichard's "Calculus, Chapter 6: Applications of the Derivative, Section 6.3: Newton's Method"

Read Section 6.3 (pages 135-138). In this reading, you will be introduced to a numerical approximation technique called Newton's Method. This method is useful for finding approximate solutions to equations which cannot be solved exactly.

Page Massachusetts Institute of Technology: David Jerison's "The Mean Value Theorem"

Watch this video until 15:10. This portion of the video is about Newton's Method, despite the title.

URL Whitman College: David Guichard's "Calculus, Chapter 6: Applications of the Derivative, Exercises 6.3: Problems 1-4"

Work through problems 1-4 for Exercise 6.3. When you are done, check your answers against Appendix A.

6.4: Linear Approximations URL Whitman College: David Guichard's "Calculus, Chapter 6: Applications of the Derivative, Section 6.4: Linear Approximations"

Read Section 6.4 (pages 139 and 140). In this reading, you will see how tangent lines can be used to locally approximate functions.

Page Massachusetts Institute of Technology: David Jerison's "Linear and Quadratic Approximations"

Watch this video until 39:00. Beyond 39:00, Jerison discusses quadratic approximations to functions, which are in a certain sense one step beyond linear approximations. If you are interested, please continue viewing the lecture to the end.

URL Whitman College: David Guichard's "Calculus, Chapter 6: Applications of the Derivative, Exercises 6.4: Problems 1-4"

Work through problems 1-4 for Exercise 6.4. When you are done, check your answers against Appendix A.

6.5: The Mean Value Theorem URL Whitman College: David Guichard's "Calculus, Chapter 6: Applications of the Derivative, Section 6.5: The Mean Value Theorem"

Read Section 6.5 (pages 141-144). The Mean Value Theorem is an important application of the derivative which most often is used in further developing mathematical theories. A special case of the Mean Value Theorem, called Rolle's Theorem, leads to a characterization of antiderivatives.

Page Massachusetts Institute of Technology: David Jerison's "The Mean Value Theorem"

Watch this video from 15:10 to the end.

URL Whitman College: David Guichard's "Calculus, Chapter 6: Applications of the Derivative": "Exercises 6.5, Problems 1, 2, 6-9"

Work through problems 1, 2, and 6-9 for Exercise 6.5. When you are done, check your answers against Appendix A.

Unit 7: Integration Page Unit 7 Learning Outcomes
7.1: Motivation URL Whitman College: David Guichard's "Calculus, Chapter 7: Integration, Section 7.1: Two Examples"

Read Section 7.1 (pages 145-149). This reading motivates the integral through two examples. The first addresses the question of how to determine the distance traveled based only on information about velocity. The second addresses the question of how to determine the area under the graph of a function. Surprisingly, these two questions are closely related to each other and to the derivative.

Page Massachusetts Institute of Technology: David Jerison's "Lecture 18: Definite Integrals"

Watch this video.

URL Whitman College: David Guichard's "Calculus, Chapter 7: Integration, Exercises 7.1: Problems 1-8"
Work through problems 1-8 for Exercise 7.1. When you are done, check your answers against Appendix A.
7.2: The Fundamental Theorem of Calculus URL Whitman College: David Guichard's "Calculus, Chapter 7: Integration, Section 7.2: The Fundamental Theorem of Calculus"

Read Section 7.2 (pages 149-155). Pay close attention to the treatment of Riemann sums, which lead to the definite integral. The Fundamental Theorem of Calculus explicitly describes the relationship between integrals and derivatives.

Page Massachusetts Institute of Technology: David Jerison's "The First Fundamental Theorem" and "The Second Fundamental Theorem"

Watch these videos.

URL Whitman College: David Guichard's "Calculus, Chapter 7: Integration, Exercises 7.2: Problems 7-22"

Work through problems 7-22 for Exercise 7.2. When you are done, check your answers against Appendix A.

7.3: Some Properties of Integrals URL Whitman College: David Guichard's "Calculus, Chapter 7: Integration, Section 7.3: Some Properties of Integrals"

Read Section 7.3 (pages 156-160). In particular, note that the definite integral enjoys the same linearity properties that the derivative does, in addition to some others. In the applications with velocity functions, pay particular attention to the distinction between distance traveled and net distance traveled.

Page Massachusetts Institute of Technology: David Jerison's "Antiderivatives"

Watch this video until 30:00.

URL Whitman College: David Guichard's "Calculus, Chapter 7: Integration, Exercises 7.3: Problems 1-6"

Work through problems 1-6 for Exercises 7.3. When you are done, check your answers against Appendix A.

7.4: Integration by Substitution URL Whitman College: David Guichard's "Calculus, Chapter 8: Techniques of Integration, Section 8.1: Substitution"

Read Section 8.1 (pages 161-166). This section explains the process of taking the integral of slightly more complicated functions. We do so by implementing a "change of variables," or rewriting a complicated integral in terms of elementary functions that we already know how to integrate. Simply speaking, integration by substitution is merely the act of taking the chain rule in reverse.

Page Massachusetts Institute of Technology: David Jerison's "Antiderivatives"

Watch this video from 30:00 to the end.

URL Whitman College: David Guichard's "Calculus, Chapter 8: Techniques of Integration, Exercises 8.1: Problems 5-19"

Work through problems 5-19 for Exercise 8.1. When you are done, check your answers against Appendix A.

7.5.1: Exponential Functions URL University of Wisconsin: H. Jerome Keisler's "Elementary Calculus, Chapter 8: Exponential and Logarithmic Functions, Section 8.3: Derivatives of Exponential Functions and the Number e"

Read Section 8.3 (pages 441 through 447). This chapter recaps the definition of the number e and the exponential function and its behavior under differentiation and integration.

URL University of California, Davis: Duane Kouba's "The Integration of Exponential Functions: Problems 1-12"

Work through problems 1-12. When you are done, check your solutions against the answers provided.

7.5.2: Natural Logarithmic Functions URL University of Wisconsin: H. Jerome Keisler's "Elementary Calculus, Chapter 8: Exponential and Logarithmic Functions, Section 8.3: Derivatives of Exponential Functions and the Number e"

Read Section 8.5 (pages 454 through 459). This chapter reintroduces the natural logarithm (the logarithm with base e) and discusses its derivative and antiderivative. Recall that you can use these properties of the natural log to extrapolate the same properties for logarithms with arbitrary bases by using the change of base formula.

URL University of Houston: Dr. Selwyn Hollis's "Video Calculus: The Natural Logarithmic Function" and "The Exponential Function"

Watch "Video 31: The Natural Logarithmic Function" through the 5th slide (marked 5 of 8). Then watch "Video 32: The Exponential Function."

The first short video gives one definition of the natural logarithm and derives all the properties of the natural log from that definition. It shows examples of limits, curve sketching, differentiation, and integration using the natural log. We will return to this video later to watch the last three slides. The second video explains the number e, the exponential function and its derivative and antiderivative, curve sketching using the exponential function, and how to perform similar operations on power functions with other bases using the change of base formula.

URL Temple University: Gerardo Mendoza's and Dan Reich's "Calculus on the Web"

Click "Index" button for Book II. Scroll down to "3. Transcendental Functions," and click button 137 (Logarithm, Definite Integrals). Do problems 1-10. If at any time a problem set seems too easy for you, feel free to move on.

7.5.3: Hyperbolic Functions Page Donny Lee's "Hyperbolic Functions" and "Hyperbolic Functions - Derivatives"

Watch these videos. The creator of the video pronounces "sinh" as "chingk." The more usual pronunciation is "sinch."

URL University of Wisconsin: H. Jerome Keisler's "Elementary Calculus, Chapter 8: Exponential and Logarithmic Functions, Section 8.4: Some Uses of Exponential Functions"

Read Section 8.4 (pages 449 through 453). In this chapter, you will learn the definitions of the hyperbolic trig functions and how to differentiate and integrate them. The chapter also introduces the concept of capital accumulation.

URL Clinton Community College: Elizabeth Wood's "Supplemental Notes for Calculus II: Hyperbolic Functions"

Work through each of the sixteen examples on the page. As in any assignment, solve the problem on your own first. Solutions are given beneath each example.

Unit 8: Applications of Integration Page Unit 8 Learning Outcomes
8.1: The Area between Curves URL University of Wisconsin: H. Jerome Keisler's "Elementary Calculus, Chapter 4: Integration, Section 4.5: Areas between Two Curves"

Read Section 4.5 (pages 218 through 222).

Page Massachusetts Institute of Technology: David Jerison's "Applications of Logarithms and Geometry"

Watch this lecture from 21:30 to the end. In this lecture, Dr. Jerison will explain how to calculate the area between two curves.

URL Temple University: Gerardo Mendoza and Dan Reich's "Calculus on the Web"

Click the "Index" button for Book II. Scroll down to "2. Applications of Integration," and click button 115 (Area between Curves I). Work on problems 6-13. Next, choose button 116 (Area between Curves II), and complete problems 4-10. If at any time a problem set seems too easy for you, feel free to move on.

URL Indiana University Southeast: Margaret Ehringe's "Section 5.3 Area between Two Curves"

Work on problems 1-3 and 6-9. When you have finished, scroll down to the bottom of the page; the answers to the problems are printed upside-down at the end of the exercises.

The point of this third assignment is for you to practice setting up and completing these problems without the graphical aids provided by the Temple University media; you will have to graph these curves for yourself in order to begin the problems.

8.2: Volumes of Solids URL University of Houston: Dr. Selwyn Hollis's "Video Calculus: Volumes I"

Watch "Video 27: Volumes I". This video explains how to use integral calculus to calculate the volumes of general solids. 

URL Clinton Community College: Elizabeth Wood's "Supplemental Notes for Calculus I: Finding Volumes by Slicing"

Work through each of the three examples on the page. As in any assignment, solve the problem on your own first. Solutions are given beneath each example.

8.3.1: Disks and Washers URL University of Wisconsin: H. Jerome Keisler's "Elementary Calculus, Chapter 6: Applications of the Integral, Section 6.2: Volumes of Solids of Revolution"

Read Section 6.2 (pages 308 through 318).

Page Massachusetts Institute of Technology: David Jerison's "Lecture 22: Volumes by Disks and Shells"

Watch this video. Dr. Jerison elaborates on some tangential material for a few minutes in the middle, but returns to the essential material very quickly.

URL Temple University: Gerardo Mendoza and Dan Reich's "Calculus on the Web"

Click on the "Index" for Book II. Scroll down to "2. Applications of Integration," and click button 119 (Solid of Revolution - Washers). Work on problems 1-12. If at any time a problem set seems too easy for you, feel free to move on.

8.3.2: Cylindrical Shells URL Temple University: Gerardo Mendoza and Dan Reich's "Calculus on the Web"

Click on the "Index" for Book II. Scroll down to "2. Applications of Integration," and click button 120 (Solid of Revolution - Shells). Work on problems 5-17. If at any time a problem set seems too easy for you, feel free to move on.

URL Math Centre: "Volume Exercises"

Work through the exercises using the method you feel is most appropriate.

8.4: Lengths of Curves URL University of Wisconsin H. Jerome Keisler's "Elementary Calculus, Chapter 6: Applications of the Integral, Section 6.3: Length of a Curve"

Read Section 6.3 (pages 319 through 325). This reading discusses how to calculate the length of a curve, also known as arc length. This includes calculating arc length for parametrically-defined curves.

Page Massachusetts Institute of Technology: David Jerison's "Parametric Equations, Arclength, Surface Area"

Watch this lecture until 26:10. Lecture notes are available in PDF; the link is on the same page as the lecture.

URL Temple University: Gerardo Mendoza and Dan Reich's "Calculus on the Web"

Click on the "Index" for Book II. Scroll down to "2. Applications of Integration," and click button 125 (Arc Length). Work on all of the problems (1-9). If at any time a problem set seems too easy for you, feel free to move on.

8.5: Surface Areas of Solids URL University of Wisconsin: H. Jerome Keisler's "Elementary Calculus, Chapter 6: Applications of the Integral, Section 6.4: Area of a Surface of Revolution"

Read Section 6.4 (pages 327 through 335). In this beautiful presentation of areas of surfaces of revolution, the author again makes use of rigorously-defined infinitesimals, as opposed to limits. Recall that the approaches are equivalent; using an infinitesimal is the same as using a variable and then taking the limit as that variable tends to zero.

Page Massachusetts Institute of Technology: David Jerison's "Parametric Equations, Arclength, Surface Area"

Watch this video from 26:10 to 40:35.

URL Clinton Community College: Elizabeth Wood's "Supplemental Notes for Calculus II: Areas of Surfaces of Revolution"

Work through each of the three examples on the page. As in any assignment, solve the problem on your own first. Solutions are given beneath each example.

8.6: Average Value of Functions URL University of Wisconsin: H. Jerome Keisler's "Elementary Calculus, Chapter 6: Applications of the Integral, Section 6.5: Averages"

Read Section 6.5 (pages 336-340).

Page Massachusetts Institute of Technology: David Jerison's "Work, Average Value, Probability"

Watch this video until 30:00. In this lecture, Jerison will explain how to calculate average values and weighted average values.

URL Temple University: Gerardo Mendoza and Dan Reich's "Calculus on the Web"

Click on the "Index" for Book II. Scroll down to "4. Assorted Application," and click button 124 (Average Value). Work on problems 3-11. If at any time a problem set seems too easy for you, feel free to move on.

8.7.1: Distance URL Whitman College: David Guichard's "Calculus, Chapter 9: Applications of Integration, Section 9.2: Distance, Velocity, Acceleration"

Read Section 9.2 (pages 192-194).

URL Clinton Community College: Elizabeth Wood's "Supplemental Notes for Calculus I: Displacement vs. Distance Traveled"

Work through each of the three examples on the page. As in any assignment, solve the problem on your own first. Solutions are given beneath each example.

8.7.2: Mass and Density URL University of Wisconsin: H. Jerome Keisler's "Elementary Calculus, Chapter 6: Applications of the Integral, Section 6.6: Some Applications to Physics"

Read this section (pages 341-351).

8.7.3: Moments URL Clinton Community College: Elizabeth Wood's "Supplemental Notes for Calculus I: Moments and Centers of Mass"

Work through each of the four examples on the page. As in any assignment, solve the problem on your own first. Solutions are given beneath each example.

8.7.4: Work URL Whitman College: David Guichard's "Calculus, Chapter 9: Applications of Integration, Section 9.5: Work"

Read Section 9.5 (pages 205 through 208). Work is a fundamental concept from physics roughly corresponding to the distance traveled by an object multiplied by the force required to move it that distance.

URL Clinton Community College: Elizabeth Wood's "Supplemental Notes for Calculus I: Work, Fluid Pressures, and Forces"

Work through each of the seven examples on the page. As in any assignment, solve the problem on your own first. Solutions are given beneath each example.

Optional Course Evaluation Survey URL Optional Course Evaluation Survey

Please take a few moments to provide some feedback about this course. Consider completing the survey whether you have completed the course, you are nearly at that point, or you have just come to study one unit or a few units of this course.

Your feedback will focus our efforts to continually improve our course design, content, technology, and general ease-of-use. Additionally, your input will be considered alongside our consulting professors' evaluation of the course during its next round of peer review. As always, please report urgent course experience concerns to contact@saylor.org and/or our discussion forums.