Topic  Name  Description 

Course Introduction  Course Syllabus  
Course Terms of Use  
Unit 1: Analytic Geometry  Unit 1 Learning Outcomes  
1.1: Lines  Whitman College: David Guichard's "Calculus, Chapter 1: Analytic Geometry, Section 1.1: Lines"  Read Section 1.1 (pages 1417). 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. 
Whitman College: David Guichard's "Calculus, Chapter 1: Analytic Geometry, Exercises 1.1: Problems 118"  Work through problems 118. When you are done, check your answers in Appendix A. 

1.2: Distance between Two Points, Circles  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. 
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. 

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 15. Then, return to the index and click on button 8 (Circles II) to launch the other module. Complete problems 15. If at any time the problem set becomes too easy for you, feel free to move on. 

1.3: Functions  Whitman College: David Guichard's "Calculus, Chapter 1: Analytic Geometry, Section 1.3: Functions"  Read Section 1.3 (pages 2024). This reading reviews the notion of functions, linear functions, domain, range, and dependent and independent variables. 
Whitman College: David Guichard's "Calculus, Chapter 1: Analytic Geometry, Exercises 1.3, Problems 116"  Work through problems 116 for Exercise 1.3. When you are done, check your answers in Appendix A. 

1.4: Shifts and Dilations  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 115. If at any time the problem set becomes too easy for you, feel free to move on. 
Whitman College: David Guichard's "Calculus, Chapter 1: Analytic Geometry, Section 1.4: Shifts and Dilations"  Read Section 1.4 (pages 2528). 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  Unit 2 Learning Outcomes  
2.1: The Slope of a Function  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 2933). 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. 
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. 

Whitman College: David Guichard's "Calculus, Chapter 2: Instantaneous Rate of Change: The Derivative, Exercises 2.1, Problems 1  6"  Work through problems 16 for Exercise 2.1. When you are done, check your answers against Appendix A. 

2.2: An Example  Whitman College: David Guichard's "Calculus, Chapter 2: Instantaneous Rate of Change: The Derivative, Section 2.2: An Example"  Read Section 2.2 (pages 3436). 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. 
Whitman College: David Guichard's "Calculus, Chapter 2: Instantaneous Rate of Change: The Derivative, Exercises 2.2, Problems 1  3"  Work through problems 13 for Exercise 2.2. When you are done, check your answers against Appendix A. 

2.3.1: The Definition and Properties of Limits  Whitman College: David Guichard's "Calculus, Chapter 2: Instantaneous Rate of Change: The Derivative, Section 2.3: Limits"  Read Section 2.3 (pages 3645). 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. 
Massachusetts Institute of Technology: David Jerison's "Limits"  Watch this video. 

Whitman College: David Guichard's "Calculus, Chapter 2: Instantaneous Rate of Change: The Derivative, Exercises 2.3, Problems 1  18"  Work through problems 118 for Exercise 2.3. When you are done, check your answers against Appendix A. 

University of California, Davis: Duane Kouba's "Precise Limits of Functions as x Approaches a Constant"  Work through problems 110. When you are done, check your solutions against the answers provided. 

2.3.2: The Squeeze Theorem  Whitman College: David Guichard's "Calculus, Chapter 4: Transcendental Functions, Section 4.3: A Hard Limit"  Read section 4.3 (pages 7577). 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. 
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  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 4650). 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. 
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 15 and 811 for Exercise 2.4. When you are done, check your answers against Appendix A. 

2.5: Adjectives for Functions  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 5154). 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 realvalued functions. 
2.5.1: Continuous Functions  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 1526. Return to the "Index," and click on button 27 (Discontinuities of simple piecewise defined functions). Complete problems 110. If at any time the problem set becomes too easy for you, feel free to move on. 
2.5.2: Differentiable Functions  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 110. If at any time the problem set becomes too easy for you, feel free to move on. 
2.5.3: The Intermediate Value Theorem  PatrickJMT: "Intermediate Value Theorem"  Watch this brief video for an explanation on the Intermediate Value Theorem. 
Unit 3: Rules for Finding Derivatives  Unit 3 Learning Outcomes  
3.1: The Power Rule  Whitman College: David Guichard's "Calculus, Chapter 3: Rules for Finding Derivatives, Section 3.1: The Power Rule"  Read Section 3.1 (pages 5557). Here, you will learn a simple rule for finding the derivative of a power function without explicitly computing a limit. 
Whitman College: David Guichard's "Calculus, Chapter 3: Rules for Finding Derivatives, Exercises 3.1, Problems 16"  Work through problems 16 in Exercise 3.1. When you are done, check your answers against Appendix A. 

3.2: Linearity of the Derivative  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. 
Whitman College: David Guichard's "Calculus, Chapter 3: Rules for Finding Derivatives, Exercises 3.2, Problems 1  9, 11, 12"  Work through problems 19, 11, and 12 for Exercise 3.2. When you are done, check your answers against Appendix A. 

3.3: The Product Rule  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. 
Whitman College: David Guichard's "Calculus, Chapter 3: Rules for Finding Derivatives: Exercises 3.3, Problems 15"  Work through problems 15 for Exercises 3.3. When you are done, check your answers against Appendix A. 

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 110. If at any time the problem set becomes too easy for you, feel free to move on. 

3.4: The Quotient Rule  Whitman College: David Guichard's "Calculus, Chapter 3: Rules for Finding Derivatives: Section 3.4: The Quotient Rule"  Read Section 3.4 (pages 6265). 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. 
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. 

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 110. If at any time the problem set becomes too easy for you, feel free to move on. 

3.5: The Chain Rule  Whitman College: David Guichard's "Calculus, Chapter 3: Rules for Finding Derivatives, Section 3.5: The Chain Rule"  Read Section 3.5 (pages 6569). 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. 
Massachusetts Institute of Technology: David Jerison's "Chain Rule"  Watch this video. 

Whitman College: David Guichard's "Calculus, Chapter 3: Rules for Finding Derivatives, Exercises 3.5, Problems 120, 3639"  Work through problems 120 and 3639 for Exercise 3.5. When you are done, check your answers against Appendix A. 

Unit 4: Transcendental Functions  Unit 4 Learning Outcomes  
4.1: Trigonometric Functions  Whitman College: David Guichard's "Calculus, Chapter 4: Transcendental Functions, Section 4.1: Trigonometric Functions"  Read Section 4.1 (pages 7174). This reading will review with you the definition of trigonometric functions. 
Whitman College: David Guichard's "Calculus, Chapter 4: Transcendental Functions, Exercises 4.1: Problems 14, 11"  Work through problems 14 and 11 for Exercise 4.1. When you are done, check your answers against Appendix A. 

4.2: The Derivative of Sine  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  Whitman College: David Guichard's "Calculus, Chapter 4: Transcendental Functions, Section 4.3: A Hard Limit"  Read Section 4.3 (pages 7577). 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. 
Whitman College: David Guichard's Calculus: "Chapter 4: Transcendental Functions, Exercises 4.3: Problems 17"  Work through problems 17 for Exercise 4.3. When you are done, check your answers against Appendix A. 

4.4: The Derivative of Sine, Continued  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. 
Whitman College: David Guichard's Calculus: "Chapter 4: Transcendental Functions, Exercises 4.4: Problems 15"  Work through problems 15 for Exercise 4.4. When you are done, check your answers against Appendix A. 

4.5: Derivatives of the Trigonometric Functions  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. 
Whitman College: David Guichard's "Calculus, Chapter 4: Transcendental Functions, Exercises 4.5: Problems 118"  Work through problems 118 for Exercise 4.5. When you are done, check your answers against Appendix A. 

4.6: Exponential and Logarithmic Functions  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  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 8286). 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. 
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. 

Whitman College: David Guichard's "Calculus, Chapter 4: Transcendental Functions, Exercises 4.7: Problems 115, 20"  Work through problems 115 and 20 for Exercise 4.7. When you are done, check your answers against Appendix A. 

4.8: Implicit Differentiation  Whitman College: David Guichard's "Calculus, Chapter 4: Transcendental Functions, Section 4.8: Implicit Differentiation"  Read Section 4.8 (pages 8790). 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. 
Massachusetts Institute of Technology: David Jerison's "Implicit Differentiation"  Watch this video. 

Whitman College: David Guichard's "Calculus, Chapter 4: Transcendental Functions, Exercises 4.8: Problems 19, 1116"  Work through problems 19 and 1116 for Exercise 4.8. When you are done, check your answers against Appendix A. 

4.9: Inverse Trigonometric Functions  Whitman College: David Guichard's "Calculus, Chapter 4: Transcendental Functions, Section 4.9: Inverse Trigonometric Functions"  Read Section 4.9 (pages 9194). 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. 
Whitman College: David Guichard's "Calculus, Chapter 4: Transcendental Functions, Exercises 4.9: Problems 311"  Work through problems 311 for Exercise 4.9. When you are done, check your answers against Appendix A. 

4.10: Limits Revisited  Whitman College: David Guichard's "Calculus, Chapter 4: Transcendental Functions, Section 4.10: Limits Revisited"  Read Section 4.10 (pages 9497). 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. 
Whitman College: David Guichard's "Calculus, Chapter 4: Transcendental Functions, Exercises 4.10: Problems 110, 2124"  Work through problems 110 and 2124 for Exercise 4.10. When you are done, check your answers against Appendix A. 

4.11: Hyperbolic Functions  Whitman College: David Guichard's "Calculus, Chapter 4: Transcendental Functions, Section 4.11: Hyperbolic Functions"  Read Section 4.11 (pages 99102). 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  Unit 5 Learning Outcomes  
5.1: Maxima and Minima  Whitman College: David Guichard's "Calculus, Chapter 5: Curve Sketching, Section 5.1: Maxima and Minima"  Read Section 5.1 (pages 103106). Fermat's Theorem indicates how derivatives can be used to find where a function attains its highest or lowest points. 
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. 

Whitman College: David Guichard's "Calculus, Chapter 5: Curve Sketching, Exercises 5.1: Problems 112, 15"  Work through problems 112 and 15 for Exercise 5.1. When you are done, check your answers against Appendix A. 

5.2: The First Derivative Test  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. 
Whitman College: David Guichard's "Calculus, Chapter 5: Curve Sketching, Exercises 5.2: Problems 115"  Work through problems 115 for Exercise 5.2. When you are done, check your answers against Appendix A. 

5.3: The Second Derivative Test  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. 
Whitman College: David Guichard's "Calculus, Chapter 5: Curve Sketching, Exercises 5.3: Problems 110"  Work through problems 110 for Exercise 5.3. When you are done, check your answers against Appendix A. 

5.4: Concavity and Inflection Points  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. 
Whitman College: David Guichard's "Calculus, Chapter 5: Curve Sketching, Exercises 5.4: Problems 19, 19"  Work through problems 19 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  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. 
Massachusetts Institute of Technology: David Jerison's "MaxMin"  Watch this lecture until 45:00. The majority of the video lecture is about curve sketching, despite the title of the video. 

Whitman College: David Guichard's "Calculus, Chapter 5: Curve Sketching, Exercises 5.5, Problems 15, 1519"  Work through problems 15 and 1519 for Exercise 5.5. When you are done, graph the curves using Wolfram Alpha to check your answers. 

Unit 6: Applications of the Derivative  Unit 6 Learning Outcomes  
6.1: Optimization  Whitman College: David Guichard's "Calculus, Chapter 6: Applications of the Derivative, Section 6.1: Optimization"  Read Section 6.1 (pages 115124). 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. 
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. 

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  Whitman College: David Guichard's "Calculus, Chapter 6: Applications of the Derivative, Section 6.2: Related Rates"  Read Section 6.2 (pages 127132). 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. 
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. 

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, 1921, and 25 for Exercise 6.2. When you are done, check your answers against Appendix A. 

6.3: Newton's Method  Whitman College: David Guichard's "Calculus, Chapter 6: Applications of the Derivative, Section 6.3: Newton's Method"  Read Section 6.3 (pages 135138). 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. 
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. 

Whitman College: David Guichard's "Calculus, Chapter 6: Applications of the Derivative, Exercises 6.3: Problems 14"  Work through problems 14 for Exercise 6.3. When you are done, check your answers against Appendix A. 

6.4: Linear Approximations  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. 
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. 

Whitman College: David Guichard's "Calculus, Chapter 6: Applications of the Derivative, Exercises 6.4: Problems 14"  Work through problems 14 for Exercise 6.4. When you are done, check your answers against Appendix A. 

6.5: The Mean Value Theorem  Whitman College: David Guichard's "Calculus, Chapter 6: Applications of the Derivative, Section 6.5: The Mean Value Theorem"  Read Section 6.5 (pages 141144). 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. 
Massachusetts Institute of Technology: David Jerison's "The Mean Value Theorem"  Watch this video from 15:10 to the end. 

Whitman College: David Guichard's "Calculus, Chapter 6: Applications of the Derivative": "Exercises 6.5, Problems 1, 2, 69"  Work through problems 1, 2, and 69 for Exercise 6.5. When you are done, check your answers against Appendix A. 

Unit 7: Integration  Unit 7 Learning Outcomes  
7.1: Motivation  Whitman College: David Guichard's "Calculus, Chapter 7: Integration, Section 7.1: Two Examples"  Read Section 7.1 (pages 145149). 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. 
Massachusetts Institute of Technology: David Jerison's "Lecture 18: Definite Integrals"  Watch this video. 

Whitman College: David Guichard's "Calculus, Chapter 7: Integration, Exercises 7.1: Problems 18"  Work through problems 18 for Exercise 7.1. When you are done, check your answers against Appendix A. 

7.2: The Fundamental Theorem of Calculus  Whitman College: David Guichard's "Calculus, Chapter 7: Integration, Section 7.2: The Fundamental Theorem of Calculus"  Read Section 7.2 (pages 149155). 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. 
Massachusetts Institute of Technology: David Jerison's "The First Fundamental Theorem" and "The Second Fundamental Theorem"  Watch these videos. 

Whitman College: David Guichard's "Calculus, Chapter 7: Integration, Exercises 7.2: Problems 722"  Work through problems 722 for Exercise 7.2. When you are done, check your answers against Appendix A. 

7.3: Some Properties of Integrals  Whitman College: David Guichard's "Calculus, Chapter 7: Integration, Section 7.3: Some Properties of Integrals"  Read Section 7.3 (pages 156160). 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. 
Massachusetts Institute of Technology: David Jerison's "Antiderivatives"  Watch this video until 30:00. 

Whitman College: David Guichard's "Calculus, Chapter 7: Integration, Exercises 7.3: Problems 16"  Work through problems 16 for Exercises 7.3. When you are done, check your answers against Appendix A. 

7.4: Integration by Substitution  Whitman College: David Guichard's "Calculus, Chapter 8: Techniques of Integration, Section 8.1: Substitution"  Read Section 8.1 (pages 161166). 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. 
Massachusetts Institute of Technology: David Jerison's "Antiderivatives"  Watch this video from 30:00 to the end. 

Whitman College: David Guichard's "Calculus, Chapter 8: Techniques of Integration, Exercises 8.1: Problems 519"  Work through problems 519 for Exercise 8.1. When you are done, check your answers against Appendix A. 

7.5.1: Exponential Functions  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. 
University of California, Davis: Duane Kouba's "The Integration of Exponential Functions: Problems 112"  Work through problems 112. When you are done, check your solutions against the answers provided. 

7.5.2: Natural Logarithmic Functions  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. 
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. 

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 110. If at any time a problem set seems too easy for you, feel free to move on. 

7.5.3: Hyperbolic Functions  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." 
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. 

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  Unit 8 Learning Outcomes  
8.1: The Area between Curves  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). 
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. 

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 613. Next, choose button 116 (Area between Curves II), and complete problems 410. If at any time a problem set seems too easy for you, feel free to move on. 

Indiana University Southeast: Margaret Ehringe's "Section 5.3 Area between Two Curves"  Work on problems 13 and 69. When you have finished, scroll down to the bottom of the page; the answers to the problems are printed upsidedown 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  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. 
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  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). 
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. 

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 112. If at any time a problem set seems too easy for you, feel free to move on. 

8.3.2: Cylindrical Shells  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 517. If at any time a problem set seems too easy for you, feel free to move on. 
8.4: Lengths of Curves  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 parametricallydefined curves. 
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. 

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 (19). If at any time a problem set seems too easy for you, feel free to move on. 

8.5: Surface Areas of Solids  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 rigorouslydefined 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. 
Massachusetts Institute of Technology: David Jerison's "Parametric Equations, Arclength, Surface Area"  Watch this video from 26:10 to 40:35. 

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  University of Wisconsin: H. Jerome Keisler's "Elementary Calculus, Chapter 6: Applications of the Integral, Section 6.5: Averages"  Read Section 6.5 (pages 336340). 
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. 

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 311. If at any time a problem set seems too easy for you, feel free to move on. 

8.7.1: Distance  Whitman College: David Guichard's "Calculus, Chapter 9: Applications of Integration, Section 9.2: Distance, Velocity, Acceleration"  Read Section 9.2 (pages 192194). 
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  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 341351). 
8.7.3: Moments  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  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. 
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. 