MA101: Single-Variable Calculus I « Saylor.org – Free Online Courses Built by Professors

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Single-Variable Calculus I

Purpose of Course show close

This course is designed to introduce you to the study of calculus. You will learn concrete applications of how calculus is used and, more importantly, why it works. Calculus is not a new discipline; it has been around since the days of Archimedes. However, Isaac Newton and Gottfried Leibniz, two 17^th century European mathematicians concurrently working on the same intellectual discovery hundreds of miles apart, were responsible for developing the field as we know it today. This brings us to our first question, what is calculus today? In its simplest terms, calculus is the study of functions, rates of change, and continuity. While you may have cultivated a basic understanding of functions in previous math courses, in this course you will come to a more advanced understanding of their complexity, learning to take a closer look at their behaviors and nuances. In this course, we will address three major topics: limits, derivatives, and integrals, as well as study their respective foundations and applications. By the end of this course, you will have a solid understanding of the behavior of functions and graphs. Whether you are entirely new to calculus or just looking for a refresher on a particular topic, this course has something to offer, balancing computational proficiency with conceptual depth.

Course Information show close

Welcome to MA101 Single-Variable Calculus. Below, please find general information on this course and its requirements.

Course Designer: Clare Wickman

Primary Resources: This course is comprised of a range of different free, online materials. However, the course makes primary use of the following:

Whitman College: David Guichard’s Calculus
Massachusetts Institute of Technology: David Jerison’s Single Variable Calculus
Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web

Requirements for Completion: In order to complete this course, you will need to work through each unit and all of its assigned materials. Pay close attention to Units 1 and 2, as these lay the groundwork for understanding the more advanced, exploratory material presented in the latter units. Note that for many students Unit 1 will mostly be a review.

Note that you will only receive an official grade on your final exam. However, in order to adequately prepare for this exam, you will need to work through the quizzes and problem sets listed above.

In order to pass this course and earn your Saylor Foundation certificate, you will need to earn a 70% or higher on the final exam. Your score on the exam will be tabulated as soon as you complete it. If you do not pass the exam, you may take it again.

Time Commitment: This course should take you approximately 93.75 hours to complete. At the beginning of each unit, there is a detailed list of time advisories for each subunit. These estimates factor in the time required to watch each lecture, work through each reading thoughtfully, and complete each assignment. However, these should be seen as guidelines, not goals; each learner is different, and you may find that your pace changes throughout the course. Mastery of the material, rather than strict adherence to the time estimates, is the measure of success in this course. It may be useful to take a look at these time advisories and to determine how much time you have over the next few weeks to complete each unit, and then to set a schedule for yourself. For example, Unit 1 should take you 13 hours. Perhaps you can sit down with your calendar and decide to complete subunit 1.1 (a total of 2 hours) on Monday night; subunit 1.2 (a total of 3 hours) on Tuesday night; etc.

Tips/Suggestions: If a video lecture does not make sense to you, pause it—this is a luxury you only have in a course of this nature!—and return to the readings to get up-to-speed on the material. Remember to note down the time at which you paused the video lecture, in case your browser times out. Try to take notes on the resources, writing down any formulas or other information you need to know. These notes will be useful as you study for your final exam.

A version of this course is also available in iTunes U.
Preview the course in your browser or view our entire suite of iTunes U courses.

Learning Outcomes show close

Upon successful completion of this course, the student will be able to:

Define and identify functions.
Define and identify the domain, range, and graph of a function.
Define and identify one-to-one, onto, and linear functions.
Analyze and graph transformations of functions, such as shifts, dilations, and compositions of functions.
Characterize, compute, and graph inverse functions.
Graph and describe exponential and logarithmic functions.
Define and calculate limits and one-sided limits.
Identify vertical asymptotes.
Define continuity and determine whether a function is continuous.
State and apply the Intermediate Value Theorem.
State the Squeeze Theorem, and use it to calculate limits.
Calculate limits at infinity and identify horizontal asymptotes.
Calculate limits of rational and radical functions.
State the epsilon-delta definition of a limit, and use it in simple situations to show a limit exists.
Draw a diagram to explain the tangent-line problem.
State several different versions of the limit definition of the derivative, and use multiple notations for the derivative.
Describe the derivative as a rate of change, and give some examples of its application, such as velocity.
Calculate simple derivatives using the limit definition.
Use the power, product, quotient, and chain rules to calculate derivatives.
Use implicit differentiation to find derivatives.
Find derivatives of inverse functions.
Find derivatives of trigonometric, exponential, logarithmic, and inverse trigonometric functions.
Solve problems involving rectilinear motion using derivatives.
Solve problems involving related rates.
Define local and absolute extrema.
Use critical points to find local extrema.
Use the first and second derivative tests to find intervals of increase and decrease and to find information about concavity and inflection points.
Sketch functions using information from the first and second derivative tests.
Use the first and second derivative tests to solve optimization (maximum/minimum value) problems.
State and apply Rolle’s Theorem and the Mean Value Theorem.
Explain the meaning of linear approximations and differentials with a sketch.
Use linear approximation to solve problems in applications.
State and apply L’Hopital’s Rule for indeterminate forms.
Explain Newton’s method using an illustration.
Execute several steps of Newton’s method and use it to approximate solutions to a root-finding problem.
Define antiderivatives and the indefinite integral.
State the properties of the indefinite integral.
Relate the definite integral to the initial value problem and the area problem.
Set up and calculate a Riemann sum.
Estimate the area under a curve numerically using the Midpoint Rule.
State the Fundamental Theorem of Calculus and use it to calculate definite integrals.
State and apply basic properties of the definite integral.
Use substitution to compute definite integrals.

Course Requirements show close

In order to take this course, you must:

√ Have access to a computer

√ Have continuous broadband Internet access

√ Have the ability/permission to install plug-ins or software (Adobe Reader, Flash, etc.)

√ Have the ability to download and save files and documents to a computer

√ Have the ability to open Microsoft files and documents (.doc, .ppt, .xls, etc.)

√ Be competent in the English language

√ Have read the Saylor Student Handbook.

2 Instantaneous Rate of Change: The Derivative
2.1 The Slope of a Function
2.2 An Example
2.3 Limits
2.3.1 The Definition and Properties of Limits
2.3.2 The Squeeze Theorem
2.4 The Derivative Function
2.5 Adjectives for Functions
2.5.1 Continuous Functions
2.5.2 Differentiable Functions
2.5.3 The Intermediate Value Theorem

3 Rules for Finding Derivatives
3.1 The Power Rule
3.2 Linearity of the Derivative
3.3 The Product Rule
3.4 The Quotient Rule
3.5 The Chain Rule

4 Transcendental Functions
4.1 Trigonometric Functions
4.2 The Derivative of Sine
4.3 A Hard Limit
4.4 The Derivative of Sine, continued
4.5 Derivatives of the Trigonometric Functions
4.6 Exponential and Logarithmic Functions
4.7 Derivatives of the Exponential and Logarithmic Functions
4.8 Implicit Differentiation
4.9 Inverse Trigonometric Functions
4.10 Limits Revisited
4.11 Hyperbolic Functions

5 Curve Sketching
5.1 Maxima and Minima
5.2 The First Derivative Test
5.3 The Second Derivative Test
5.4 Concavity and Inflection Points
5.5 Asymptotes and Other Things to Look For

6 Applications of the Derivative
6.1 Optimization
6.2 Related Rates
6.3 Newton's Method
6.4 Linear Approximations
6.5 The Mean Value Theorem

7 Integration
7.1 Motivation
7.2 The Fundamental Theorem of Calculus
7.3 Some Properties of Integrals
7.4 Integration by Substitution

8 Final Exam

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Unit 1: Analytic Geometry

Most of the material in this unit will be review. However, the notions of points, lines, circles, distance, and functions will be central in everything that follows. Lines are basic geometric objects which will be of great importance in the study of differential calculus in the study of tangent lines and linear approximations.

We will also take a look at the practical uses of mathematical functions. This course will use mathematical models, or structures, that predict practical situations in order to describe and study a number of real-life problems and situations. They are essential to the development of every major business and every scientific field in the modern world.

Unit 1 Time Advisory show close

Unit 1 Learning Outcomes show close

1.1 Lines

Reading: Whitman College: David Guichard’s Calculus: Chapter 1: Analytic Geometry: “Section 1.1: Lines”
Link: Whitman College: David Guichard’s Calculus: Chapter 1: Analytic Geometry: “Section 1.1: Lines” (PDF)

Instructions: Please click on the link above and read Section 1.1 (pages 14-17) in its entirety. 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.

This reading should take you approximately 30 minutes to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike License. This text was originally written by Professor David Guichard. It has since been modified to include edited material from Neal Koblitz of the University of Washington, H. Jerome Keisler of the University of Wisconsin, Albert Schueller, Barry Balof, and Mike Wills. You can access the original version here.
See a broken link? Please let us know!
Assignment: Whitman College: David Guichard’s Calculus: Chapter 1: Analytic Geometry: “Exercises 1.1: Problems 1-18”
Link: Whitman College: David Guichard’s Calculus: Chapter 1: Analytic Geometry: “Exercises 1.1: Problems 1–18” (PDF)

Instructions: Please click on the above link and work through problems 1-18. When you are done, check your answers against “Appendix A: Answers”.

Completing this assignment should take you about two hours to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike License. This text was originally written by Professor David Guichard. It has since been modified to include edited material from Neal Koblitz of the University of Washington, H. Jerome Keisler of the University of Wisconsin, Albert Schueller, Barry Balof, and Mike Wills. You can access the original version here.
See a broken link? Please let us know!

1.2 Distance between Two Points, Circles

Reading: Whitman College: David Guichard’s Calculus: Chapter 1: Analytic Geometry: “Section 1.2: Distance Between Two Points; Circles”
Link: Whitman College: David Guichard’s Calculus: Chapter 1: Analytic Geometry: “Section 1.2: Distance Between Two Points; Circles” (PDF)

Instructions: Please click on the link above and read Section 1.2 (pages 19 and 20) in its entirety. This reading reviews the notions of distance in the plane and the equations and graphs of circles.

This reading should take approximately 30 minutes to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike License. This text was originally written by Professor David Guichard. It has since been modified to include edited material from Neal Koblitz of the University of Washington, H. Jerome Keisler of the University of Wisconsin, Albert Schueller, Barry Balof, and Mike Wills. You can access the original version here.
See a broken link? Please let us know!
Assignment: Whitman College: David Guichard’s Calculus: Chapter 1: Analytic Geometry: “Exercises 1.2: Problems 1, 2, 6”
Link: Whitman College: David Guichard’s Calculus: Chapter 1: Analytic Geometry: “Exercises 1.2: Problems 1, 2, 6” (PDF)

Instructions: Please click on the above link and work through problems 1, 2, and 6 in Exercises 1.2. When you are done, to check your answers against “Appendix A: Answers”.

This assignment should take you approximately 30 minutes to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike License. This text was originally written by Professor David Guichard. It has since been modified to include edited material from Neal Koblitz of the University of Washington, H. Jerome Keisler of the University of Wisconsin, Albert Schueller, Barry Balof, and Mike Wills. You can access the original version here.
See a broken link? Please let us know!
Assignment: Temple University: Gerardo Mendoza and Dan Reich’s Calculus on the Web: Calculus Book I: Marc Renault and Molly M. Cow’s “Distance” Module and Dan Reich’s “Equations of Circles II” Module
Link: Temple University: Gerardo Mendoza and Dan Reich’s Calculus on the Web: Calculus Book I: Marc Renault and Molly M. Cow’s “Distance” Module and Dan Reich’s “Equations of Circles II” Module

Instructions: Please click on the link above and click on the number 5 next to “Distance” to launch the first module and complete problems 1-5. Then return to the index and click on the number 8 next to “Circles II” to launch the second module and complete problems 1-5. If at any time the problem set becomes too easy for you, feel free to move forward.

Completing this assignment should take you approximately one hour to complete.

Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
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1.3 Functions

Reading: Whitman College: David Guichard’s Calculus: Chapter 1: Analytic Geometry: “Section 1.3: Functions”
Link: Whitman College: David Guichard’s Calculus: Chapter 1: Analytic Geometry: “Section 1.3: Functions” (PDF)

Instructions: Please click on the link above and read Section 1.3 (pages 20-24) in its entirety. This reading reviews the notion of functions, linear functions, domain, range, and dependent and independent variables.

This reading should take you approximately one hour to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike License. This text was originally written by Professor David Guichard. It has since been modified to include edited material from Neal Koblitz of the University of Washington, H. Jerome Keisler of the University of Wisconsin, Albert Schueller, Barry Balof, and Mike Wills. You can access the original version here.
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Assignment: Whitman College: David Guichard’s Calculus: Chapter 1: Analytic Geometry: “Exercises 1.3: Problems 1-16”
Link: Whitman College: David Guichard’s Calculus: Chapter 1: Analytic Geometry: “Exercises 1.3: Problems 1-16” (PDF)

Instructions: Please click on the link above and work through problems 1-16. When you are done, check your answers against “Appendix A: Answers”.

This assignment should take you approximately two hours to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike License. This text was originally written by Professor David Guichard. It has since been modified to include edited material from Neal Koblitz of the University of Washington, H. Jerome Keisler of the University of Wisconsin, Albert Schueller, Barry Balof, and Mike Wills. You can access the original version here.
See a broken link? Please let us know!

1.4 Shifts and Dilations

Reading: Whitman College: David Guichard’s Calculus: Chapter 1: Analytic Geometry: “Section 1.4: Shifts and Dilations”
Link: Whitman College: David Guichard’s Calculus: Chapter 1: Analytic Geometry: “Section 1.4: Shifts and Dilations” (PDF)

Instructions: Please click on the link above and read Section 1.4 (pages 25-28) in its entirety. This reading will review graph transformations associated with some basic ways of manipulating functions like like shifts and dilations.

This reading should take you approximately 30 minutes to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. This text was originally written by Professor David Guichard. It has since been modified to include edited material from Neal Koblitz of the University of Washington, H. Jerome Keisler of the University of Wisconsin, Albert Schueller, Barry Balof, and Mike Wills. You can access the original version here.
See a broken link? Please let us know!
Assignment: Temple University: Gerardo Mendoza and Dan Reich’s Calculus on the Web: Calculus Book I: Lavanya Myneni and Molly M. Cow’s “Recognizing Algebraic Functions” Module
Link: Temple University: Gerardo Mendoza and Dan Reich’s Calculus on the Web: Calculus Book I: Lavanya Myneni and Molly M. Cow’s “Recognizing Algebraic Functions” Module

Instructions: Please click on the link above and select the “Index” button. Click on the number 18 next to “Transforming Graphs” to launch the module and complete problems 1-15. If at any time the problem set becomes too easy for you, feel free to move forward.

This assignment should take you approximately an hour and 30 minutes to complete.

Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
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Unit 2: Instantaneous Rate of Change: The Derivative

In this unit, you will study the instantaneous rate of change of a function. Motivated by this concept, you will develop the notion of limits, continuity, and the derivative. The limit asks the question, “What does the function do as the independent variable becomes closer and closer to a certain value?” In simpler terms, the limit is the “natural tendency” of a function. The limit is incredibly important due to its relationship to the derivative, the integral, and countless other key mathematical concepts. A strong understanding of limits is essential to the field of mathematics.

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 of an entire function when it is linear. You can also describe the slope of nonlinear functions. The slope, however, will not be constant; it will change as the independent variable changes.

Unit 2 Time Advisory show close

Unit 2 Learning Outcomes show close

2.1 The Slope of a Function

Note: In this section, you will look at the first of two major problems at the heart of calculus: the tangent line problem. This intellectual exercise demonstrates the origins of derivatives for nonlinear functions.

Reading: Whitman College: David Guichard’s Calculus: Chapter 2: Instantaneous Rate of Change: The Derivative: “Section 2.1: The Slope of a Function”
Link: Whitman College: David Guichard’s Calculus: Chapter 2: Instantaneous Rate of Change: The Derivative: “Section 2.1: The Slope of a Function” (PDF)

Instructions: Please click on the link above and read Section 2.1 (pages 29-33) in its entirety. 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.

This reading should take you approximately one hour to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. This text was originally written by Professor David Guichard. It has since been modified to include edited material from Neal Koblitz of the University of Washington, H. Jerome Keisler of the University of Wisconsin, Albert Schueller, Barry Balof, and Mike Wills. You can access the original version here.
See a broken link? Please let us know!
Lecture: Massachusetts Institute of Technology: David Jerison’s Single Variable Calculus: “Lecture 1: Rate of Change”
Link: Massachusetts Institute of Technology: David Jerison’s Single Variable Calculus: “Lecture 1: Rate of Change” (YouTube)

Instructions: Please click on the link above and watch the entire video (51:33). Lecture notes are available here. In this lecture, Professor Jerison introduces the derivative as the rate of change of a function, or the slope of the tangent line to a function at a point.

Viewing this lecture and taking notes should take you approximately one hour to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. It is attributed to David Jerison and MIT's OpenCourseWare. It may be viewed in its original form here.
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Assignment: Whitman College: David Guichard’s Calculus: Chapter 2: Instantaneous Rate of Change: The Derivative: “Exercises 2.1: Problems 1-6”
Link: Whitman College: David Guichard’s Calculus: Chapter 2: Instantaneous Rate of Change: The Derivative: “Exercises 2.1: Problems 1-6” (PDF)

Instructions: Please click on the link above and work through problems 1-6. When you are done, check your answers against “Appendix A: Answers”.

This assignment should take you approximately 30 minutes to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. This text was originally written by Professor David Guichard. It has since been modified to include edited material from Neal Koblitz of the University of Washington, H. Jerome Keisler of the University of Wisconsin, Albert Schueller, Barry Balof, and Mike Wills. You can access the original version here.
See a broken link? Please let us know!

2.2 An Example

Reading: Whitman College: David Guichard’s Calculus: Chapter 2: Instantaneous Rate of Change: The Derivative: “Section 2.2: An Example”
Link: Whitman College: David Guichard’s Calculus: Chapter 2: Instantaneous Rate of Change: The Derivative: “Section 2.2: An Example” (PDF)

Instructions: Please click on the link above and read Section 2.2 (pages 34-36) in its entirety. This reading discusses the derivative in the context of studying the velocity of a falling object.

This reading should take you approximately 30 minutes to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. This text was originally written by Professor David Guichard. It has since been modified to include edited material from Neal Koblitz of the University of Washington, H. Jerome Keisler of the University of Wisconsin, Albert Schueller, Barry Balof, and Mike Wills. You can access the original version here.
See a broken link? Please let us know!
Assignment: Whitman College: David Guichard’s Calculus: Chapter 2: Instantaneous Rate of Change: The Derivative: “Exercises 2.2: Problems 1-3”
Link: Whitman College: David Guichard’s Calculus: Chapter 2: Instantaneous Rate of Change: The Derivative: “Exercises 2.2: Problems 1-3” (PDF)

Instructions: Please click on the link above and work through problems 1-3. When you are done, check your answers against “Appendix A: Answers”.

This assignment should take you approximately 30 minutes to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. This text was originally written by Professor David Guichard. It has since been modified to include edited material from Neal Koblitz of the University of Washington, H. Jerome Keisler of the University of Wisconsin, Albert Schueller, Barry Balof, and Mike Wills. You can access the original version here.
See a broken link? Please let us know!

2.3 Limits

In this section, you will take a close look at a concept that you have used intuitively for several years: the limit. The limit asks the question, “What does the function do as the independent variable gets closer and closer to a certain value?” In simpler terms, the limit is the “natural tendency” of a function. The limit is incredibly important due to its relationship to the derivative, the integral, and countless other key mathematical concepts. A strong understanding of the limit is essential to the field of mathematics.

2.3.1 The Definition and Properties of Limits

Reading: Whitman College: David Guichard’s Calculus: Chapter 2: Instantaneous Rate of Change: The Derivative: “Section 2.3: Limits”
Link: Whitman College: David Guichard’s Calculus: Chapter 2: Instantaneous Rate of Change: The Derivative: “Section 2.3: Limits” (PDF)

Instructions: Please click on the link above and read Section 2.3 (pages 36-45) in its entirety. 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.

This reading should take you approximately two hours to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. This text was originally written by Professor David Guichard. It has since been modified to include edited material from Neal Koblitz of the University of Washington, H. Jerome Keisler of the University of Wisconsin, Albert Schueller, Barry Balof, and Mike Wills. You can access the original version here.
See a broken link? Please let us know!
Lecture: Massachusetts Institute of Technology: David Jerison’s Single Variable Calculus: “Lecture 2: Limits”
Link: Massachusetts Institute of Technology: David Jerison’s Single Variable Calculus: “Lecture 2: Limits" (YouTube)

Instructions: Please click on the link above and watch the entire video (52:47). Lecture notes are available here.

Viewing this lecture and taking notes should take you approximately one hour to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. It is attributed to David Jerison and MIT's OpenCourseWare. It may be viewed in its original form here.
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Assignment: Whitman College: David Guichard’s Calculus: Chapter 2: Instantaneous Rate of Change: The Derivative: “Exercises 2.3: Problems 1-18”
Link: Whitman College: David Guichard’s Calculus: Chapter 2: Instantaneous Rate of Change: The Derivative: “Exercises 2.3: Problems 1-18” (PDF)

Instructions: Please click on the link above and work through problems 1-18. When you are done, check your answers against “Appendix A: Answers”.

This assignment should take you approximately two hours to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. This text was originally written by Professor David Guichard. It has since been modified to include edited material from Neal Koblitz of the University of Washington, H. Jerome Keisler of the University of Wisconsin, Albert Schueller, Barry Balof, and Mike Wills. You can access the original version here.
See a broken link? Please let us know!
Assignment: University of California at Davis: Duane Kouba’s “Precise Limits of Functions as X Approaches a Constant: Problems 1-10”
Link: University of California at Davis: Duane Kouba’s “Precise Limits of Functions as X Approaches a Constant: Problems 1-10” (HTML)

Instructions: Please click on the link above and work through problems 1-10. When you are done, select the “click HERE” beneath each problem to check your solution.

This assignment should take you approximately one hour to complete.

Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
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2.3.2 The Squeeze Theorem

Reading: Whitman College: David Guichard’s Calculus: Chapter 4: Transcendental Functions: “Section 4.3: A Hard Limit”
Link: Whitman College: David Guichard’s Calculus: Chapter 4: Transcendental Functions: “Section 4.3: A Hard Limit” (PDF)

Instructions: Please click on the link above and read Section 4.3 (pages 75-77) in its entirety. The Squeeze Theorem is an important application of the limit and is useful in many limit computations.

This reading should take you approximately 30 minute to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. This text was originally written by Professor David Guichard. It has since been modified to include edited material from Neal Koblitz of the University of Washington, H. Jerome Keisler of the University of Wisconsin, Albert Schueller, Barry Balof, and Mike Wills. You can access the original version here.
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Web Media: PatrickJMT’s “The Squeeze Theorem for Limits”
Link: PatrickJMT’s “The Squeeze Theorem for Limits” (YouTube)

Instructions: Please click on the link above and watch the entire video (7:13), which illustrates the Squeeze Theorem using specific examples.

Viewing this video and taking notes should take you approximately 15 minutes to complete.

Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
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2.4 The Derivative Function

Reading: Whitman College: David Guichard’s Calculus: Chapter 2: Instantaneous Rate of Change: The Derivative: “Section 2.4: The Derivative Function”
Link: Whitman College: David Guichard’s Calculus: Chapter 2: Instantaneous Rate of Change: The Derivative: “Section 2.4: The Derivative Function” (PDF)

Instructions: Please click on the link above and read Section 2.4 (pages 46-50) in its entirety. 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 of an entire function when it is linear. You can also describe the slope of nonlinear functions. The slope, however, will not be constant; it will change as the independent variable changes.

This reading should take you approximately one hour to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. This text was originally written by Professor David Guichard. It has since been modified to include edited material from Neal Koblitz of the University of Washington, H. Jerome Keisler of the University of Wisconsin, Albert Schueller, Barry Balof, and Mike Wills. You can access the original version here.
See a broken link? Please let us know!
Assignment: Whitman College: David Guichard’s Calculus: Chapter 2: Instantaneous Rate of Change: The Derivative: “Exercises 2.4: Problems 1-5 and 8-11”
Link: Whitman College: David Guichard’s Calculus: Chapter 2: Instantaneous Rate of Change: The Derivative: “Exercises 2.4: Problems 1-5 and 8-11” (PDF)

Instructions: Please click on the link above and work through problems 1-5 and 8-11. When you are done, check your answers against “Appendix A: Answers”.

This assignment should take you approximately one hour to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. This text was originally written by Professor David Guichard. It has since been modified to include edited material from Neal Koblitz of the University of Washington, H. Jerome Keisler of the University of Wisconsin, Albert Schueller, Barry Balof, and Mike Wills. You can access the original version here.
See a broken link? Please let us know!

2.5 Adjectives for Functions

Reading: Whitman College: David Guichard’s Calculus: Chapter 2: Instantaneous Rate of Change: The Derivative: “Section 2.5: Adjectives for Functions”
Link: Whitman College: David Guichard’s Calculus: Chapter 2: Instantaneous Rate of Change: The Derivative: “Section 2.5: Adjectives for Functions” (PDF)

Instructions: Please click on the link above and read Section 2.5 (pages 51-54) in its entirety. This reading covers the topics outlined in sub-subunits 2.5.1 through 2.5.3. The intuitive notion of a continuous function is made precise using limits. Additionally, you will be introduced to the Intermediate Value Theorem, which rigorously captures the intuitive behavior of continuous real-valued functions.

This reading should take you approximately one hour to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. This text was originally written by Professor David Guichard. It has since been modified to include edited material from Neal Koblitz of the University of Washington, H. Jerome Keisler of the University of Wisconsin, Albert Schueller, Barry Balof, and Mike Wills. You can access the original version here.
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2.5.1 Continuous Functions

Assignment: Temple University: Gerardo Mendoza and Dan Reich’s Calculus on the Web: Calculus Book I: James Palermo and Molly M. Cow’s “Extending Continuity at a Missing Point” Module and Gerardo Mendoza’s “Discontinuities of Simple Piecewise Defined Functions” Module
Link: Temple University: Gerardo Mendoza and Dan Reich’s Calculus on the Web: Calculus Book I: James Palermo and Molly M. Cow’s “Extending Continuity at a Missing Point” Module and Gerardo Mendoza’s “Discontinuities of Simple Piecewise Defined Functions” Module (HTML)

Instructions: Please click on the link above select the “Index.” Click on the number 26 next to “A Missing Value” to launch the first module and complete problems 15-26. Then return to the index and click on the number 27 next to “Discontinuities of simple piecewise defined functions” to launch the second module and complete problems 1-10. If at any time the problem set becomes too easy for you, feel free to move forward.

Completing this assignment should take you approximately two hours to complete.

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2.5.2 Differentiable Functions

Assignment: Temple University: Gerardo Mendoza and Dan Reich’s Calculus on the Web: Calculus Book I: Dan Reich’s “Differentiability of Simple Piecewise Functions” Module
Link: Temple University: Gerardo Mendoza and Dan Reich’s Calculus on the Web: Calculus Book I: Dan Reich’s “Differentiability of Simple Piecewise Functions” Module (HTML)

Instructions: Please click on the link above and select the “Index.” Click on the number 39 next to “Differentiability” to launch the module and complete problems 1-10. If at any time the problem set becomes too easy for you, feel free to move forward.

This assignment should take you approximately one hour to complete.

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2.5.3 The Intermediate Value Theorem

Web Media: PatrickJMT’s “Intermediate Value Theorem”
Link: PatrickJMT’s “Intermediate Value Theorem” (YouTube)

Instructions: Click on the link above and watch the entire video (7:53) for an explanation of the Intermediate Value Theorem.

Viewing this video and taking notes should take you approximately 15 minutes to complete.

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Unit 3: Rules for Finding Derivatives

Computing a derivative requires computing a limit. Because limit computations can be rather involved, we like to minimize the amount of work we have to do in practice. In this unit, you will build your skill using some rules for differentiation which will speed up your calculations of derivatives. In particular, you will see how to differentiate the sum, difference, product, quotient, and composition of two or more functions. You will also learn rules for differentiating power functions, including polynomial and root functions.

Unit 3 Time Advisory show close

Unit 3 Learning Outcomes show close

3.1 The Power Rule

Reading: Whitman College: David Guichard’s Calculus: Chapter 3: Rules for Finding Derivatives: “Section 3.1: The Power Rule”
Link: Whitman College: David Guichard’s Calculus: Chapter 3: Rules for Finding Derivatives: “Section 3.1: The Power Rule” (PDF)

Instructions: Please click on the link above and read Section 3.1 (pages 55-57) in its entirety. This section will show you a simple rule for how to find the derivative of a power function without explicitly computing a limit.

This reading should take you approximately 30 minutes to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. This text was originally written by Professor David Guichard. It has since been modified to include edited material from Neal Koblitz of the University of Washington, H. Jerome Keisler of the University of Wisconsin, Albert Schueller, Barry Balof, and Mike Wills. You can access the original version here.
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Assignment: Whitman College: David Guichard’s Calculus: Chapter 3: Rules for Finding Derivatives: “Exercises 3.1: Problems 1-6”
Link: Whitman College: David Guichard’s Calculus: Chapter 3: Rules for Finding Derivatives: “Exercises 3.1: Problems 1-6” (PDF)

Instructions: Please click on the above link and work through problems 1-6. When you are done, check your answers against “Appendix A: Answers”.

This assignment should take you approximately 30 minutes to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. This text was originally written by Professor David Guichard. It has since been modified to include edited material from Neal Koblitz of the University of Washington, H. Jerome Keisler of the University of Wisconsin, Albert Schueller, Barry Balof, and Mike Wills. You can access the original version here. (PDF)
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3.2 Linearity of the Derivative

Reading: Whitman College: David Guichard’s Calculus: Chapter 3: Rules for Finding Derivatives: “Section 3.2: Linearity of the Derivative”
Link: Whitman College: David Guichard’s Calculus: Chapter 3: Rules for Finding Derivatives: “Section 3.2: Linearity of the Derivative” (PDF)

Instructions: Please click on the link above and read Section 3.2 (pages 58-59) in its entirety. In this reading, you will see how the derivative behaves with regards to addition and subtraction of functions and with scalar multiplication. That is, you will see that the derivative is a linear operation.

This reading should take you approximately 30 minutes to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. This text was originally written by Professor David Guichard. It has since been modified to include edited material from Neal Koblitz of the University of Washington, H. Jerome Keisler of the University of Wisconsin, Albert Schueller, Barry Balof, and Mike Wills. You can access the original version here. (PDF)
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Assignment: Whitman College: David Guichard’s Calculus: Chapter 3: Rules for Finding Derivatives: “Exercises 3.2: Problems 1-9, 11, and 12”
Link: Whitman College: David Guichard’s Calculus: Chapter 3: Rules for Finding Derivatives: “Exercises 3.2: Problems 1-9, 11, and 12” (PDF)

Instructions: Please click on the link above and work through problems 1-9, 11, and 12. When you are done, check your answers “Appendix A: Answers”.

This assignment should take you approximately one hour to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. This text was originally written by Professor David Guichard. It has since been modified to include edited material from Neal Koblitz of the University of Washington, H. Jerome Keisler of the University of Wisconsin, Albert Schueller, Barry Balof, and Mike Wills. You can access the original version here.
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3.3 The Product Rule

Reading: Whitman College: David Guichard’s Calculus: Chapter 3: Rules for Finding Derivatives: “Section 3.3: The Product Rule”
Link: Whitman College: David Guichard’s Calculus: Chapter 3: Rules for Finding Derivatives: “Section 3.3: The Product Rule” (PDF)

Instructions: Please click on the link above and read Section 3.3 (pages 60-61) in its entirety. 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.

This reading should take you approximately 30 minutes to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. This text was originally written by Professor David Guichard. It has since been modified to include edited material from Neal Koblitz of the University of Washington, H. Jerome Keisler of the University of Wisconsin, Albert Schueller, Barry Balof, and Mike Wills. You can access the original version here. (PDF)
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Assignment: Whitman College: David Guichard’s Calculus: Chapter 3: Rules for Finding Derivatives: “Exercises 3.3: Problems 1-5”
Link: Whitman College: David Guichard’s Calculus: Chapter 3: Rules for Finding Derivatives: “Exercises 3.3: Problems 1-5” (PDF)

Instructions: Please click on the link above and work through problems 1-5. When you are done, check your answers against “Appendix A: Answers”.

This assignment should take you approximately 30 minutes to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. This text was originally written by Professor David Guichard. It has since been modified to include edited material from Neal Koblitz of the University of Washington, H. Jerome Keisler of the University of Wisconsin, Albert Schueller, Barry Balof, and Mike Wills. You can access the original version here.
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Assignment: Temple University: Gerardo Mendoza and Dan Reich’s Calculus on the Web: Calculus Book I: Dan Reich’s “Derivatives – The Product Rule” Module
Link: Temple University: Gerardo Mendoza and Dan Reich’s Calculus on the Web: Calculus Book I: Dan Reich’s “Derivatives – The Product Rule” Module (HTML)

Instructions: Please click on the link above and select the “Index.” Click on the number 44 next to “Product Rule” to launch the module and complete problems 1-10. If at any time the problem set becomes too easy for you, feel free to move forward.

This assignment should take you approximately one hour to complete.

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3.4 The Quotient Rule

Assignment: Temple University: Gerardo Mendoza and Dan Reich's Calculus on the Web: Calculus Book I: Dan Reich's "Derivatives - The Quotient Rule" Module
Link: Temple University: Gerardo Mendoza and Dan Reich's Calculus on the Web: Calculus Book I: Dan Reich's "Derivatives - The Quotient Rule" Module (HTML)

Instructions: Please click on the link above and select the "Index." Click on the number 45 next to "The Quotient Rule" to launch the module and complete problems 1-10. If at any time the problem set becomes too easy for you, feel free to move forward.

This assignment should take you approximately one hour to complete.

Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
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Reading: Whitman College: David Guichard’s Calculus: Chapter 3: Rules for Finding Derivatives: “Section 3.4: The Quotient Rule”
Link: Whitman College: David Guichard’s Calculus: Chapter 3: Rules for Finding Derivatives: “Section 3.4: The Quotient Rule” (PDF)

Instructions: Please click on the link above and read Section 3.4 (pages 62-65) in its entirety. As with product of two functions, the derivative of a quotient of two functions is not simply the quotient of the two derivatives. This reading will introduce you to the quotient rule for differentiating a quotient of two functions. In particular, it will allow you to find the derivative of any rational function.

This reading should take you approximately one hour to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. This text was originally written by Professor David Guichard. It has since been modified to include edited material from Neal Koblitz of the University of Washington, H. Jerome Keisler of the University of Wisconsin, Albert Schueller, Barry Balof, and Mike Wills. You can access the original version here.
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Assignment: Whitman College: David Guichard’s Calculus: Chapter 3: Rules for Finding Derivatives: “Exercises 3.4: Problems 5, 6, 8, and 9”
Link: Whitman College: David Guichard’s Calculus: Chapter 3: Rules for Finding Derivatives: “Exercises 3.4: Problems 5, 6, 8, and 9” (PDF)

Instructions: Please click on the link above and work through problems 5, 6, 8, and 9. When you are done, check your answers against “Appendix A: Answers”.

This assignment should take you approximately 30 minutes to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. This text was originally written by Professor David Guichard. It has since been modified to include edited material from Neal Koblitz of the University of Washington, H. Jerome Keisler of the University of Wisconsin, Albert Schueller, Barry Balof, and Mike Wills. You can access the original version here.
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3.5 The Chain Rule

Reading: Whitman College: David Guichard’s Calculus: Chapter 3: Rules for Finding Derivatives: “Section 3.5: The Chain Rule”
Link: Whitman College: David Guichard’s Calculus: Chapter 3: Rules for Finding Derivatives: “Section 3.5: The Chain Rule” (PDF)

Instructions: Please click on the link above and read Section 3.5 (pages 65-69) in its entirety. The chain rule explains how the derivative applies to the composition of functions. Pay particular attention to Example 3.11, which works through a derivative computation where all of the differentiation rules of this unit are applied in finding the derivative of one function.

This reading should take you approximately one hour to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. This text was originally written by Professor David Guichard. It has since been modified to include edited material from Neal Koblitz of the University of Washington, H. Jerome Keisler of the University of Wisconsin, Albert Schueller, Barry Balof, and Mike Wills. You can access the original version here.
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Lecture: Massachusetts Institute of Technology: David Jerison’s Single Variable Calculus: “Lecture 4: Chain Rule”
Link: Massachusetts Institute of Technology: David Jerison’s Single Variable Calculus: “Lecture 4: Chain Rule" (YouTube)

Instructions: Please click on the link above and watch the entire video (46:03). Lecture notes are available here.

Viewing this lecture and taking notes should take you approximately one hour to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. It is attributed to David Jerison and MIT's OpenCourseWare. It may be viewed in its original form here.
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Assignment: Whitman College: David Guichard’s Calculus: Chapter 3: Rules for Finding Derivatives: “Exercises 3.5: Problems 1-20 and 36-39”
Link: Whitman College: David Guichard’s Calculus: Chapter 3: Rules for Finding Derivatives: “Exercises 3.5: Problems 1-20 and 36-39” (PDF)

Instructions: Please click on the link above and work through problems 1-20 and 36-39. When you are done, check your answers against “Appendix A: Answers”.

This assignment should take you approximately three hours to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. This text was originally written by Professor David Guichard. It has since been modified to include edited material from Neal Koblitz of the University of Washington, H. Jerome Keisler of the University of Wisconsin, Albert Schueller, Barry Balof, and Mike Wills. You can access the original version here.
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Unit 4: Transcendental Functions

In this unit, you will investigate the derivatives of trigonometric, inverse trigonometric, exponential, and logarithmic functions. Along the way, you will develop a technique of differentiation called implicit differentiation. Aside from allowing you to compute derivatives of inverse function, implicit differentiation will also be important in studying related rates problems later on.

Unit 4 Time Advisory show close

Unit 4 Learning Outcomes show close

4.1 Trigonometric Functions

Reading: Whitman College: David Guichard’s Calculus: Chapter 4: Transcendental Functions: “Section 4.1: Trigonometric Functions”
Link: Whitman College: David Guichard’s Calculus: Chapter 4: Transcendental Functions: “Section 4.1: Trigonometric Functions” (PDF)

Instructions: Please click on the link above and read Section 4.1 (pages 71-74) in its entirety. This reading will review the definition of trigonometric functions.

This reading should take you approximately one hour to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. This text was originally written by Professor David Guichard. It has since been modified to include edited material from Neal Koblitz of the University of Washington, H. Jerome Keisler of the University of Wisconsin, Albert Schueller, Barry Balof, and Mike Wills. You can access the original version here.
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Assignment: Whitman College: David Guichard’s Calculus: Chapter 4: Transcendental Functions: “Exercises 4.1: Problems 1-4 and 11”
Link Whitman College: David Guichard’s Calculus: Chapter 4: Transcendental Functions: “Exercises 4.1: Problems 1-4 and 11” (PDF)

Instructions: Please click on the link above and work through problems 1-4 and 11. When you are done, check your answers against “Appendix A: Answers”.

This assignment should take you approximately 30 minutes to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. This text was originally written by Professor David Guichard. It has since been modified to include edited material from Neal Koblitz of the University of Washington, H. Jerome Keisler of the University of Wisconsin, Albert Schueller, Barry Balof, and Mike Wills. You can access the original version here.
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4.2 The Derivative of Sine

Reading: Whitman College: David Guichard’s Calculus: Chapter 4: Transcendental Functions: “Section 4.2: The Derivative of sin x”
Link: Whitman College: Professor David Guichard’s Calculus: Chapter 4: Transcendental Functions: “Section 4.2: The Derivative of sin x” (PDF)

Instructions: Please click on the link above and read Section 4.2 (pages 74-75) in its entirety. This reading begins the computation of the derivative of the sine function. Two specific limits will need to be evaluated in order to complete this computation. These limits are addressed in the following section.

This reading should take you approximately 15 minutes to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. This text was originally written by Professor David Guichard. It has since been modified to include edited material from Neal Koblitz of the University of Washington, H. Jerome Keisler of the University of Wisconsin, Albert Schueller, Barry Balof, and Mike Wills. You can access the original version here.
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4.3 A Hard Limit

Reading: Whitman College: David Guichard’s Calculus: Chapter 4: Transcendental Functions: “Section 4.3: A Hard Limit”
Link: Whitman College: David Guichard’s Calculus: Chapter 4: Transcendental Functions: “Section 4.3: A Hard Limit” (PDF)

Instructions: Please click on the link above and read Section 4.3 (pages 75-77) in its entirety. You have read this section previously to become acquainted with the Squeeze Theorem. When you read the section the second time, pay particular attention to the geometric argument used to set up the application of the Squeeze Theorem.

This reading should take you approximately 30 minutes to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. This text was originally written by Professor David Guichard. It has since been modified to include edited material from Neal Koblitz of the University of Washington, H. Jerome Keisler of the University of Wisconsin, Albert Schueller, Barry Balof, and Mike Wills. You can access the original version here.
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Assignment: Whitman College: David Guichard’s Calculus: Chapter 4: Transcendental Functions: “Exercises 4.3: Problems 1-7”
Link: Professor David Guichard’s Calculus: Chapter 4: Transcendental Functions: “Exercises 4.3: Problems 1-7” (PDF)

Instructions: Please click on the link above and work through problems 1-7. When you are done, check your answers against “Appendix A: Answers”.

This assignment should take you approximately 30 minutes to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. This text was originally written by Professor David Guichard. It has since been modified to include edited material from Neal Koblitz of the University of Washington, H. Jerome Keisler of the University of Wisconsin, Albert Schueller, Barry Balof, and Mike Wills. You can access the original version here. You can access the original version here.
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4.4 The Derivative of Sine, continued

Reading: Whitman College: David Guichard’s Calculus: Chapter 4: Transcendental Functions: “Section 4.4: The Derivative of sin x, continued”
Link: Whitman College: David Guichard’s Calculus: Chapter 4: Transcendental Functions: “Section 4.4: The Derivative of sin x, continued” (PDF)

Instructions: Please click on the link above and read Section 4.4 (pages 77-78) in its entirety. This reading completes the computation of the derivative of the sine function. Be sure to review all of the concepts involved in this computation.

This reading should take you approximately 30 minutes to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. This text was originally written by Professor David Guichard. It has since been modified to include edited material from Neal Koblitz of the University of Washington, H. Jerome Keisler of the University of Wisconsin, Albert Schueller, Barry Balof, and Mike Wills. You can access the original version here.
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Assignment: Whitman College: David Guichard’s Calculus: Chapter 4: Transcendental Functions: “Exercises 4.4: Problems 1-5”
Link: Whitman College: David Guichard’s Calculus: Chapter 4: Transcendental Functions: “Exercises 4.4, Problems 1-5” (PDF)

Instructions: Please click on the link above and work through problems 1-5. When you are done, check your answers against “Appendix A: Answers”.

This assignment should take you approximately 30 minutes to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. This text was originally written by Professor David Guichard. It has since been modified to include edited material from Neal Koblitz of the University of Washington, H. Jerome Keisler of the University of Wisconsin, Albert Schueller, Barry Balof, and Mike Wills. You can access the original version here.
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4.5 Derivatives of the Trigonometric Functions

Reading: Whitman College: David Guichard’s Calculus: Chapter 4: Transcendental Functions: “Section 4.5: Derivatives of the Trigonometric Functions”
Link: Whitman College: David Guichard’s Calculus: Chapter 4: Transcendental Functions: “Section 4.5: Derivatives of the Trigonometric Functions” (PDF)

Instructions: Please click on the link above and read Section 4.5 (pages 78 and 79) in its entirety. 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.

This reading should take you approximately 15 minutes to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. This text was originally written by Professor David Guichard. It has since been modified to include edited material from Neal Koblitz of the University of Washington, H. Jerome Keisler of the University of Wisconsin, Albert Schueller, Barry Balof, and Mike Wills. You can access the original version here.
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Assignment: Whitman College: David Guichard’s Calculus: Chapter 4: Transcendental Functions: “Exercises 4.5: Problems 1-18”
Link: Whitman College: David Guichard’s Calculus: Chapter 4: Transcendental Functions: “Exercises 4.5: Problems 1-18” (PDF)

Instructions: Please click on the link above and work through problems 1-18. When you are done, check your answers against “Appendix A: Answers”.

This assignment should take you approximately one hour to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. This text was originally written by Professor David Guichard. It has since been modified to include edited material from Neal Koblitz of the University of Washington, H. Jerome Keisler of the University of Wisconsin, Albert Schueller, Barry Balof, and Mike Wills. You can access the original version here.
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4.6 Exponential and Logarithmic Functions

Reading: Whitman College: David Guichard’s Calculus: Chapter 4: Transcendental Functions: “Section 4.6: Exponential and Logarithmic Functions”
Link: Whitman College: David Guichard’s Calculus: Chapter 4: Transcendental Functions: “Section 4.6: Exponential and Logarithmic Functions” (PDF)

Instructions: Please click on the link above and read Section 4.6 (pages 80-81) in its entirety. This reading reviews the exponential and logarithmic functions, their properties, and their graphs.

This reading should take you approximately 30 minutes to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. This text was originally written by Professor David Guichard. It has since been modified to include edited material from Neal Koblitz of the University of Washington, H. Jerome Keisler of the University of Wisconsin, Albert Schueller, Barry Balof, and Mike Wills. You can access the original version here.
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4.7 Derivatives of the Exponential and Logarithmic Functions

Reading: Whitman College: David Guichard’s Calculus: Chapter 4: Transcendental Functions: “Section 4.7: Derivatives of the Exponential and Logarithmic Functions”
Link: Whitman College: David Guichard’s Calculus: Chapter 4: Transcendental Functions: “Section 4.7: Derivatives of the Exponential and Logarithmic Functions” (PDF)

Instructions: Please click on the link above and read Section 4.7 (pages 82-86) in its entirety. In this reading, the derivatives of the exponential and logarithmic functions are computed. Notice that the number e is defined in terms of a particular limit.

This reading should take you approximately one hour to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. This text was originally written by Professor David Guichard. It has since been modified to include edited material from Neal Koblitz of the University of Washington, H. Jerome Keisler of the University of Wisconsin, Albert Schueller, Barry Balof, and Mike Wills. You can access the original version here.
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Lecture: Massachusetts Institute of Technology: David Jerison’s Single Variable Calculus: “Lecture 6: Exponential and Log”
Link: Massachusetts Institute of Technology: David Jerison’s Single Variable Calculus: “Lecture 6: Exponential and Log” (YouTube)

Instructions: Please click on the link above and watch the entire video (47:57). Lecture notes are available here. Professor Jerison makes use of implicit differentiation at times during this lecture. You should take note of this and re-watch those portions of the video after completing subunit 4.8 below.

Viewing this video and taking notes should take you approximately one hour to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. It is attributed to David Jerison and MIT's OpenCourseWare. It may be viewed in its original form here.
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Assignment: Whitman College: David Guichard’s Calculus: Chapter 4: Transcendental Functions: “Exercises 4.7: Problems 1-15 and 20”
Link: Whitman College: David Guichard’s Calculus: Chapter 4: Transcendental Functions: “Exercises 4.7: Problems 1-15 and 20” (PDF)

Instructions: Please click on the link above and work through problems 1-15 and 20 for Exercise 4.7. When you are done, check your answers against “Appendix A: Answers”.

This assignment should take you approximately one hour to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. This text was originally written by Professor David Guichard. It has since been modified to include edited material from Neal Koblitz of the University of Washington, H. Jerome Keisler of the University of Wisconsin, Albert Schueller, Barry Balof, and Mike Wills. You can access the original version here.
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4.8 Implicit Differentiation

Reading: Whitman College: David Guichard’s Calculus: Chapter 4: Transcendental Functions: “Section 4.8: Implicit Differentiation”
Link: Whitman College: David Guichard’s Calculus: Chapter 4: Transcendental Functions: “Section 4.8: Implicit Differentiation” (PDF)

Instructions: Please click on the link above and read Section 4.8 (pages 87-90) in its entirety. 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.

This reading should take you approximately one hour to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. This text was originally written by Professor David Guichard. It has since been modified to include edited material from Neal Koblitz of the University of Washington, H. Jerome Keisler of the University of Wisconsin, Albert Schueller, Barry Balof, and Mike Wills. You can access the original version here.
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Lecture: Massachusetts Institute of Technology: David Jerison’s Single Variable Calculus: “Lecture 5: Implicit Differentiation”
Link: Massachusetts Institute of Technology: David Jerison’s Single Variable Calculus: “Lecture 5: Implicit Differentiation” (YouTube)

Instructions: Please click on the link above and watch the entire video (49:01). Lecture notes are available here.

Viewing this lecture and taking notes should take you approximately one hour to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. It is attributed to David Jerison and MIT's OpenCourseWare. It may be viewed in its original form here.
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Assignment: Whitman College: David Guichard’s Calculus: Chapter 4: Transcendental Functions: “Exercises 4.8: Problems 1-9 and 11-16”
Link: Whitman College: David Guichard’s Calculus: Chapter 4: Transcendental Functions: “Exercises 4.8: Problems 1-9 and 11-16” (PDF)

Instructions: Please click on the link above and work through problems 1-9 and 11-16 for Exercises 4.8. When you are done, check your answers against “Appendix A: Answers”.

This assignment should take you approximately one hour to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. This text was originally written by Professor David Guichard. It has since been modified to include edited material from Neal Koblitz of the University of Washington, H. Jerome Keisler of the University of Wisconsin, Albert Schueller, Barry Balof, and Mike Wills. You can access the original version here.
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4.9 Inverse Trigonometric Functions

Reading: Whitman College: David Guichard’s Calculus: Chapter 4: Transcendental Functions: “Section 4.9: Inverse Trigonometric Functions”
Link: Whitman College: David Guichard’s Calculus: Chapter 4: Transcendental Functions: “Section 4.9: Inverse Trigonometric Functions” (PDF)

Instructions: Please click on the link above and read Section 4.9 (pages 91-94) in its entirety. 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.

This reading should take you approximately one hour to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. This text was originally written by Professor David Guichard. It has since been modified to include edited material from Neal Koblitz of the University of Washington, H. Jerome Keisler of the University of Wisconsin, Albert Schueller, Barry Balof, and Mike Wills. You can access the original version here.
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Assignment: Whitman College: David Guichard’s Calculus: Chapter 4: Transcendental Functions: “Exercises 4.9: Problems 3-11”
Link: Whitman College: David Guichard’s Calculus: Chapter 4: Transcendental Functions: “Exercises 4.9: Problems 3-11” (PDF)

Instructions: Please click on the link above and work through problems 3-11 for Exercises 4.9. When you are done, check your answers against “Appendix A: Answers”.

This assignment should take you approximately one hour to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. This text was originally written by Professor David Guichard. It has since been modified to include edited material from Neal Koblitz of the University of Washington, H. Jerome Keisler of the University of Wisconsin, Albert Schueller, Barry Balof, and Mike Wills. You can access the original version here.
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4.10 Limits Revisited

Reading: Whitman College: David Guichard’s Calculus: Chapter 4: Transcendental Functions: “Section 4.10: Limits Revisited”
Link: Whitman College: David Guichard’s Calculus: Chapter 4: Transcendental Functions: “Section 4.10: Limits Revisited” (PDF)

Instructions: Please click on the link above and read Section 4.10 (pages 94-97) in its entirety. In this section, 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.

This reading should take you approximately one hour to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. This text was originally written by Professor David Guichard. It has since been modified to include edited material from Neal Koblitz of the University of Washington, H. Jerome Keisler of the University of Wisconsin, Albert Schueller, Barry Balof, and Mike Wills. You can access the original version here.
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Assignment: Whitman College: David Guichard’s Calculus: Chapter 4: Transcendental Functions: “Exercises 4.10: Problems 1-10 and 21-24”
Link: Whitman College: David Guichard’s Calculus: Chapter 4: Transcendental Functions: “Exercises 4.10: Problems 1-10 and 21-24” (PDF)

Instructions: Please click on the link above link and work through problems 1-10 and 21-24 for Exercise 4.10. When you are done, check your answers against “Appendix A: Answers”.

This assignment should take you approximately one hour and 30 minutes to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. This text was originally written by Professor David Guichard. It has since been modified to include edited material from Neal Koblitz of the University of Washington, H. Jerome Keisler of the University of Wisconsin, Albert Schueller, Barry Balof, and Mike Wills. You can access the original version here.
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4.11 Hyperbolic Functions

Reading: Whitman College: David Guichard’s Calculus: Chapter 4: Transcendental Functions: “Section 4.11: Hyperbolic Functions”
Link: Whitman College: David Guichard’s Calculus: Chapter 4: Transcendental Functions: “Section 4.11: Hyperbolic Functions” (PDF)

Instructions: Please click on the link above and read Section 4.11 (pages 99-102) in its entirety. 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.

This reading should take you approximately one hour to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. This text was originally written by Professor David Guichard. It has since been modified to include edited material from Neal Koblitz of the University of Washington, H. Jerome Keisler of the University of Wisconsin, Albert Schueller, Barry Balof, and Mike Wills. You can access the original version here.
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Unit 5: Curve Sketching

This section will ask you to apply a little critical thinking to the topics this course has covered thus far. To properly sketch a curve, you must analyze the function and its first and second derivatives in order to obtain information about how the function behaves, taking into account its intercepts, asymptotes (vertical and horizontal), maximum values, minimum values, points of inflection, and the respective intervals between each of these. After collecting this information, you will need to piece it all together in order to sketch an approximation of the original function.

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5.1 Maxima and Minima

Reading: Whitman College: David Guichard’s Calculus: Chapter 5: Curve Sketching: “Section 5.1: Maxima and Minima”
Link: Whitman College: David Guichard’s Calculus: Chapter 5: Curve Sketching: “Section 5.1: Maxima and Minima” (PDF)

Instructions: Please click on the link above and read Section 5.1 (pages 103-106) in its entirety. Fermat's Theorem indicates how derivatives can be used to find where a function reaches its highest or lowest points.

This reading should take you approximately one hour to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. This text was originally written by Professor David Guichard. It has since been modified to include edited material from Neal Koblitz of the University of Washington, H. Jerome Keisler of the University of Wisconsin, Albert Schueller, Barry Balof, and Mike Wills. You can access the original version here.
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Lecture: Massachusetts Institute of Technology: David Jerison’s Single Variable Calculus: “Lecture 10: Curve Sketching”
Link: Massachusetts Institute of Technology: David Jerison’s Single Variable Calculus: “Lecture 10: Curve Sketching” (YouTube)

Instructions: Please click on the link above and watch the video from the 30:00 minute mark to the end. Lecture notes are available here. The lecture will make use of the first and second derivative tests, which you will read about below.

Viewing this lecture and taking notes should take you approximately one hour to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. It is attributed to David Jerison and MIT's OpenCourseWare. It may be viewed in its original form here.
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Assignment: Whitman College: David Guichard’s Calculus: Chapter 5: Curve Sketching: “Exercises 5.1: Problems 1-12 and 15”
Link: David Guichard’s Calculus: Chapter 5: Curve Sketching: “Exercises 5.1: Problems 1-12 and 15” (PDF)

Instructions: Please click on the link above and work through problems 1-12 and 15 for Exercises 5.1. When you are done, check your answers against “Appendix A: Answers”.

This assignment should take you approximately one hour to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. This text was originally written by Professor David Guichard. It has since been modified to include edited material from Neal Koblitz of the University of Washington, H. Jerome Keisler of the University of Wisconsin, Albert Schueller, Barry Balof, and Mike Wills. You can access the original version here.
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5.2 The First Derivative Test

Reading: Whitman College: David Guichard’s Calculus: Chapter 5: Curve Sketching: “Section 5.2: The First Derivative Test”
Link: Whitman College: David Guichard’s Calculus: Chapter 5: Curve Sketching: “Section 5.2: The First Derivative Test” (PDF)

Instructions: Please click on the link above and read Section 5.2 (page 107) in its entirety. In this reading, you will see how to use information about the derivative of a function to find local maxima and minima.

This reading should take you approximately 15 minutes to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. This text was originally written by Professor David Guichard. It has since been modified to include edited material from Neal Koblitz of the University of Washington, H. Jerome Keisler of the University of Wisconsin, Albert Schueller, Barry Balof, and Mike Wills. You can access the original version here.
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Assignment: Whitman College: David Guichard’s Calculus: Chapter 5: Curve Sketching: “Exercises 5.2: Problems 1-15”
Link: Whitman College: David Guichard’s Calculus: Chapter 5: Curve Sketching: “Exercises 5.2: Problems 1-15” (PDF)

Instructions: Please click on the link above and work through problems 1-15 for Exercises 5.2. When you are done, check your answers against “Appendix A: Answers”.

This assignment should take you approximately one hour and 30 minutes to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. This text was originally written by Professor David Guichard. It has since been modified to include edited material from Neal Koblitz of the University of Washington, H. Jerome Keisler of the University of Wisconsin, Albert Schueller, Barry Balof, and Mike Wills. You can access the original version here.
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5.3 The Second Derivative Test

Reading: Whitman College: David Guichard’s Calculus: Chapter 5: Curve Sketching: “Section 5.3: The Second Derivative Test”
Link: Whitman College: David Guichard’s Calculus: Chapter 5: Curve Sketching: “Section 5.3: The Second Derivative Test” (PDF)

Instructions: Please click on the link above and read Section 5.3 (pages 108-109) in its entirety. 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.

This reading should take you approximately 30 minutes to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. This text was originally written by Professor David Guichard. It has since been modified to include edited material from Neal Koblitz of the University of Washington, H. Jerome Keisler of the University of Wisconsin, Albert Schueller, Barry Balof, and Mike Wills. You can access the original version here.
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Assignment: Whitman College: David Guichard’s Calculus: Chapter 5: Curve Sketching: “Exercises 5.3: Problems 1-10”
Link: Whitman College: David Guichard’s Calculus: Chapter 5: Curve Sketching: “Exercises 5.3: Problems 1-10” (PDF)

Instructions: Please click on the link above and work through problems 1-10 for Exercises 5.3. When you are done, check your answers against “Appendix A: Answers”.

This assignment should take you approximately one hour to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. This text was originally written by Professor David Guichard. It has since been modified to include edited material from Neal Koblitz of the University of Washington, H. Jerome Keisler of the University of Wisconsin, Albert Schueller, Barry Balof, and Mike Wills. You can access the original version here.
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5.4 Concavity and Inflection Points

Reading: Whitman College: David Guichard’s Calculus: Chapter 5: Curve Sketching: “Section 5.4: Concavity and Inflection Points”
Link: Whitman College: David Guichard’s Calculus: Chapter 5: Curve Sketching: “Section 5.4: Concavity and Inflection Points” (PDF)

Instructions: Please click on the link above and read Section 5.4 (pages 109-110) in its entirety. In this reading, you will see how the second derivative relates to the 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.

This reading should take you approximately 30 minutes to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. This text was originally written by Professor David Guichard. It has since been modified to include edited material from Neal Koblitz of the University of Washington, H. Jerome Keisler of the University of Wisconsin, Albert Schueller, Barry Balof, and Mike Wills. You can access the original version here.
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Assignment: Whitman College: David Guichard’s Calculus: Chapter 5: Curve Sketching: “Exercises 5.4: Problems 1-9 and 19”
Link: Whitman College: David Guichard’s Calculus: Chapter 5: Curve Sketching: “Exercises 5.4: Problems 1-9 and 19” (PDF)

Instructions: Please click on the link above and work through problems 1-9 and 19 for Exercises 5.4. When you are done, check your answers against “Appendix A: Answers”.

This assignment should take you approximately one hour to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. This text was originally written by Professor David Guichard. It has since been modified to include edited material from Neal Koblitz of the University of Washington, H. Jerome Keisler of the University of Wisconsin, Albert Schueller, Barry Balof, and Mike Wills. You can access the original version here.
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5.5 Asymptotes and Other Things to Look For

Reading: Whitman College: David Guichard’s Calculus: Chapter 5: Curve Sketching: “Section 5.5: Asymptotes and Other Things to Look For”
Link: Whitman College: David Guichard’s Calculus: Chapter 5: Curve Sketching: “Section 5.5: Asymptotes and Other Things to Look For” (PDF)

Instructions: Please click on the link above and read Section 5.5 (pages 111-112) in its entirety. In this reading, you will see how limits can be used to find any asymptotes the graph of a function may have.

This reading should take you approximately 30 minutes to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. This text was originally written by Professor David Guichard. It has since been modified to include edited material from Neal Koblitz of the University of Washington, H. Jerome Keisler of the University of Wisconsin, Albert Schueller, Barry Balof, and Mike Wills. You can access the original version here.
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Lecture: Massachusetts Institute of Technology: David Jerison’s Single Variable Calculus: “Lecture 11: Max-min”
Link: Massachusetts Institute of Technology: David Jerison’s Single Variable Calculus: “Lecture 11: Max-min” (YouTube)

Instructions: Please click on the link above and watch the video from the beginning to the 45:00 minute mark. Lecture notes are available here. The majority of the video lecture is about curve sketching, despite the title of the video.

Viewing this lecture and taking notes should take you approximately one hour to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. It is attributed to David Jerison and MIT's OpenCourseWare. It may be viewed in its original form here.
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Assignment: Whitman College: David Guichard’s Calculus: Chapter 5: Curve Sketching: “Exercises 5.5: Problems 1-5 and 15-19”
Link: Whitman College: David Guichard’s Calculus: Chapter 5: Curve Sketching: “Exercises 5.5: Problems 1-5 and 15-19” (PDF)

Instructions: Please click on the link above and work through problems 1-5 and 15-19 for Exercises 5.5. When you are done, graph the curves using Wolfram Alpha to check your answers.

This assignment should take you approximately one hour to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. This text was originally written by Professor David Guichard. It has since been modified to include edited material from Neal Koblitz of the University of Washington, H. Jerome Keisler of the University of Wisconsin, Albert Schueller, Barry Balof, and Mike Wills. You can access the original version here.
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Unit 6: Applications of the Derivative

With a sufficient amount of sophisticated machinery under your belt, you will now start to look at how differentiation can be used to solve problems in various applied settings. Optimization is an important notion in fields like biology, economics, and physics when we want to know when growth is maximized, for example. In addition to methods we use to solve problems directly, we can also use the derivative to find approximate solutions to problems. You will explore two such methods in this section: Newton's method and differentials.

Unit 6 Time Advisory show close

Unit 6 Learning Outcomes show close

6.1 Optimization

Reading: Whitman College: David Guichard’s Calculus: Chapter 6: Applications of the Derivative: “Section 6.1: Optimization”
Link: Whitman College: David Guichard’s Calculus: Chapter 6: Applications of the Derivative: “Section 6.1: Optimization” (PDF)

Instructions: Please click on the link above and read Section 6.1 (pages 115-124) in its entirety. An important application of the derivative is to find the global maximum and global minimum of a function. The Extreme Value Theorem indicates how to approach this problem. Pay particular attention to the summary at the end of the section.

This reading should take you approximately three hours to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. This text was originally written by Professor David Guichard. It has since been modified to include edited material from Neal Koblitz of the University of Washington, H. Jerome Keisler of the University of Wisconsin, Albert Schueller, Barry Balof, and Mike Wills. You can access the original version here.
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Lecture: Massachusetts Institute of Technology: David Jerison’s Single Variable Calculus: “Lecture 12: Related Rates”
Link: Massachusetts Institute of Technology: David Jerison’s Single Variable Calculus: "Lecture 12: Related Rates" (YouTube)

Instructions: Please click on the link above and watch the video from the beginning to the 45:00 minute mark. Lecture notes are available here. The majority of the video lecture is about optimization, despite the title of the video.

Viewing this lecture and taking notes should take you approximately one hour to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. It is attributed to David Jerison and MIT's OpenCourseWare. It may be viewed in its original form here.
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Assignment: 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, and 33”
Link: Whitman College: Professor David Guichard’s Calculus: “Chapter 6: Applications of the Derivative:” “Exercises 6.1, Problems 5, 7, 9, 10, 14, 16, 22, 26, 28, 33” (PDF)

Instructions: Please click on the link above link and work through problems 5, 7, 9, 10, 14, 16, 22, 26, 28, and 33 for Exercises 6.1. When you are done, to check your answers against “Appendix A: Answers”.

This assignment should take you approximately one hour to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. This text was originally written by Professor David Guichard. It has since been modified to include edited material from Neal Koblitz of the University of Washington, H. Jerome Keisler of the University of Wisconsin, Albert Schueller, Barry Balof, and Mike Wills. You can access the original version here.
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6.2 Related Rates

Note: You now know how to take the derivative with respect to the independent variable. In other words, you know how to determine a function’s rate of change when given the input’s rate of change. But what if the independent variable was itself a function? What if, for example, the input was a function of time? How do we identify how the function changes as time changes? This subunit will explore the answers to these questions.

Reading: Whitman College: David Guichard’s Calculus: Chapter 6: Applications of the Derivative: “Section 6.2: Related Rates”
Link: Whitman College: David Guichard’s Calculus: Chapter 6: Applications of the Derivative: “Section 6.2: Related Rates” (PDF)

Instructions: Please click on the link above and read Section 6.2 (pages 127-132) in its entirety. 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.

This reading should take you approximately one hour to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. This text was originally written by Professor David Guichard. It has since been modified to include edited material from Neal Koblitz of the University of Washington, H. Jerome Keisler of the University of Wisconsin, Albert Schueller, Barry Balof, and Mike Wills. You can access the original version here.
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Lecture: Massachusetts Institute of Technology: David Jerison’s Single Variable Calculus: “Lecture 13: Newton's Method”
Link: Massachusetts Institute of Technology: David Jerison’s Single Variable Calculus: “Lecture 13: Newton's Method” (YouTube)

Instructions: Please click on the link above and watch the video from the beginning to the 40:30 minute mark. Lecture notes are available here. The majority of the video lecture is about related rates, despite the title of the video.

Viewing this lecture and taking notes should take you approximately one hour to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. It is attributed to David Jerison and MIT's OpenCourseWare. It may be viewed in its original form here.
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Assignment: Whitman College: David Guichard’s Calculus: Chapter 6: Applications of the Derivative: “Exercises 6.2: Problems 1, 3, 5, 11, 14, 16, 19-21, and 25”
Link: Whitman College: David Guichard’s Calculus: Chapter 6: Applications of the Derivative: “Exercises 6.2: Problems 1, 3, 5, 11, 14, 16, 19-21, and 25” (PDF)

Solutions: Ibid: “Appendix A: Answers” (PDF)

Instructions: Please click on the above link, and work through problems 1, 3, 5, 11, 14, 16, 19-21, and 25 for Exercises 6.2. When you are done, check your answers against “Appendix A: Answers”.

This assignment should take you approximately one hour to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. This text was originally written by Professor David Guichard. It has since been modified to include edited material from Neal Koblitz of the University of Washington, H. Jerome Keisler of the University of Wisconsin, Albert Schueller, Barry Balof, and Mike Wills. You can access the original version here.
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6.3 Newton's Method

Note: Newton's Method is a process by which we estimate the roots of a real-valued function. You may remember the bisection method, whereby we find a root by creating smaller and smaller intervals. Newton's Method uses the derivative in order to account for both the speed at which the function changes and its actual position. This creates an algorithm that can help us identify the location of roots even more quickly.

Newton's Method requires that you start “sufficiently close” (a somewhat arbitrary specification that varies from problem to problem) to the actual root in order to estimate it with accuracy. If you start too far from the root, an algorithm can be led awry in certain situations.

Reading: Whitman College: David Guichard’s Calculus: Chapter 6: Applications of the Derivative: “Section 6.3: Newton's Method”
Link: Whitman College: David Guichard’s Calculus: Chapter 6: Applications of the Derivative: “Section 6.3: Newton's Method” (PDF)

Instructions: Please click on the link above and read Section 6.3 (pages 135-138) in its entirety. In this section, 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.

This reading should take you approximately one hour to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. This text was originally written by Professor David Guichard. It has since been modified to include edited material from Neal Koblitz of the University of Washington, H. Jerome Keisler of the University of Wisconsin, Albert Schueller, Barry Balof, and Mike Wills. You can access the original version here.
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Lecture: Massachusetts Institute of Technology: David Jerison’s Single Variable Calculus: “Lecture 14: The Mean Value Theorem”
Link: Massachusetts Institute of Technology: David Jerison’s Single Variable Calculus: “Lecture 14: The Mean Value Theorem” (YouTube)

Instructions: Please click on the link above watch the video from the beginning to the 15:10 minute mark. Lecture notes are available here. This portion of the video is about Newton's Method, despite the title of the video.

Viewing this lecture and taking notes should take you approximately 15-20 minutes to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. It is attributed to David Jerison and MIT's OpenCourseWare. It may be viewed in its original form here.
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Assignment: Whitman College: David Guichard’s Calculus: Chapter 6: Applications of the Derivative: “Exercises 6.3: Problems 1-4”
Link: Whitman College: David Guichard’s Calculus: Chapter 6: Applications of the Derivative: “Exercises 6.3: Problems 1-4” (PDF)

Instructions: Please click on the link above link and work through problems 1-4 for Exercises 6.3. When you are done, check your answers “Appendix A: Answers”.

This assignment should take you approximately 30 minutes to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. This text was originally written by Professor David Guichard. It has since been modified to include edited material from Neal Koblitz of the University of Washington, H. Jerome Keisler of the University of Wisconsin, Albert Schueller, Barry Balof, and Mike Wills. You can access the original version here.
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6.4 Linear Approximations

Note: In this subunit, you will learn how to estimate future data points based on what you know about a previous data point and how it changed at that particular moment. This concept is extremely useful in the field of economics.

Reading: Whitman College: David Guichard’s Calculus: Chapter 6: Applications of the Derivative: “Section 6.4: Linear Approximations”
Link: Whitman College: David Guichard’s Calculus: Chapter 6: Applications of the Derivative: “Section 6.4: Linear Approximations” (PDF)

Instructions: Please click on the link above and read Section 6.4 (pages 139-140) in its entirety. In this reading, you will see how tangent lines can be used to locally approximate functions.

This reading should take you approximately 30 minutes to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. This text was originally written by Professor David Guichard. It has since been modified to include edited material from Neal Koblitz of the University of Washington, H. Jerome Keisler of the University of Wisconsin, Albert Schueller, Barry Balof, and Mike Wills. You can access the original version here.
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Lecture: Massachusetts Institute of Technology: David Jerison’s Single Variable Calculus: “Lecture 9: Linear and Quadratic Approximations”
Link: Massachusetts Institute of Technology: David Jerison’s Single Variable Calculus: “Lecture 9: Linear and Quadratic Approximations” (YouTube)

Instructions: Please click on the link above and watch the video up to the 39:00 minute mark. At the 39:00 mark Professor Jerison begins to discuss 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. Lecture notes are available here.

Viewing this lecture and taking notes should take you approximately 45 minutes to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. It is attributed to David Jerison and MIT's OpenCourseWare. It may be viewed in its original form here.
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Assignment: Whitman College: David Guichard’s Calculus: Chapter 6: Applications of the Derivative: “Exercises 6.4: Problems 1-4”
Link: Whitman College: David Guichard’s Calculus: Chapter 6: Applications of the Derivative: “Exercises 6.4: Problems 1-4” (PDF)

Instructions: Please click on the link above link and work through problems 1-4 for Exercises 6.4. When you are done, check your answers against “Appendix A: Answers”.

This assignment should take you approximately 30 minutes to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. This text was originally written by Professor David Guichard. It has since been modified to include edited material from Neal Koblitz of the University of Washington, H. Jerome Keisler of the University of Wisconsin, Albert Schueller, Barry Balof, and Mike Wills. You can access the original version here.
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6.5 The Mean Value Theorem

Reading: Whitman College: David Guichard’s Calculus: Chapter 6: Applications of the Derivative: “Section 6.5: The Mean Value Theorem”
Link: Whitman College: David Guichard’s Calculus: Chapter 6: Applications of the Derivative: “Section 6.5: The Mean Value Theorem” (PDF)

Instructions: Please click on the link above and read Section 6.5 (pages 141-144) in its entirety. The Mean Value Theorem is an important application of the derivative which is used most often in developing further mathematical theories. A special case of the Mean Value Theorem, called Rolle's Theorem, leads to a characterization of antiderivatives.

This reading should take you approximately 30 minutes to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. This text was originally written by Professor David Guichard. It has since been modified to include edited material from Neal Koblitz of the University of Washington, H. Jerome Keisler of the University of Wisconsin, Albert Schueller, Barry Balof, and Mike Wills. You can access the original version here.
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Lecture: Massachusetts Institute of Technology: David Jerison’s Single Variable Calculus: “Lecture 14: The Mean Value Theorem”
Link: Massachusetts Institute of Technology: David Jerison’s Single Variable Calculus: “Lecture 14: The Mean Value Theorem” (YouTube)

Instructions: Please click on the link above and watch the video from the 15:10 minute mark to the end. Lecture notes are available here.

Viewing this lecture and taking notes should take you approximately 45 minutes to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. It is attributed to David Jerison and MIT's OpenCourseWare. It may be viewed in its original form here.
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Assignment: Whitman College: Professor David Guichard’s Calculus: “Chapter 6: Applications of the Derivative:” “Exercises 6.5, Problems 1, 2, 6-9”
Link: Whitman College: Professor David Guichard’s Calculus: “Chapter 6: Applications of the Derivative”: “Exercises 6.5, Problems 1, 2, 6-9” (PDF)

Solutions: Ibid: “Appendix A: Answers” (PDF)

Instructions: Please click on the above link, and work through problems 1, 2, and 6-9 for Exercise 6.5. When you are done, click the second link to check your answers.

This assignment should take approximately 30 minutes to complete.

Terms of Use: This PDF is licensed under a Creative Commons Attribution-Noncommercial-ShareAlike License (HTML). This text was originally written by Professor David Guichard. Since then, it has been modified to include edited material from Neal Koblitz at the University of Washington, H. Jerome Keisler at the University of Wisconsin, Albert Schueller, Barry Balof, and Mike Wills. You can access the original version here. (PDF)
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Unit 7: Integration

In the last unit of this course, you will learn about “integral calculus,” a subfield of calculus that studies the area formed under the curve of a function. Although its relationship with the derivative is not necessarily intuitive, integral calculus is closely linked to the derivative, which you will revisit in this unit.

Unit 7 Time Advisory show close

Unit 7 Learning Outcomes show close

7.1 Motivation

Reading: Whitman College: David Guichard’s Calculus: Chapter 7: Integration: “Section 7.1: Two Examples”
Link: Whitman College: David Guichard’s Calculus: Chapter 7: Integration: “Section 7.1: Two Examples” (PDF)

Instructions: Please click on the link above and read Section 7.1 (pages 145-149) in its entirety. This reading introduces the integral through two examples. The first example addresses the question of how to determine the distance traveled based only on information about velocity. The second example 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.

This reading should take you approximately one hour to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. This text was originally written by Professor David Guichard. It has since been modified to include edited material from Neal Koblitz of the University of Washington, H. Jerome Keisler of the University of Wisconsin, Albert Schueller, Barry Balof, and Mike Wills. You can access the original version here.
See a broken link? Please let us know!
Lecture: Massachusetts Institute of Technology: David Jerison’s Single Variable Calculus: “Lecture 18: Definite Integrals”
Link: Massachusetts Institute of Technology: David Jerison’s Single Variable Calculus: “Lecture 18: Definite Integrals” (YouTube)

Instructions: Please click on the link above and watch the entire video (47:14). Lecture notes are available here.

Viewing this lecture and taking notes should take you approximately one hour to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. It is attributed to David Jerison and MIT's OpenCourseWare. It may be viewed in its original form here.
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Assignment: Whitman College: David Guichard’s Calculus: Chapter 7: Integration: “Exercises 7.1: Problems 1-8”
Link: Whitman College: David Guichard’s Calculus: Chapter 7: Integration: “Exercises 7.1: Problems 1-8” (PDF)

Instructions: Please click on the link above and work through problems 1-8 for Exercises 7.1. When you are done, check your answers against “Appendix A: Answers”.

This assignment should take you approximately one hour to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. This text was originally written by Professor David Guichard. It has since been modified to include edited material from Neal Koblitz of the University of Washington, H. Jerome Keisler of the University of Wisconsin, Albert Schueller, Barry Balof, and Mike Wills. You can access the original version here.
See a broken link? Please let us know!

7.2 The Fundamental Theorem of Calculus

Note: The Fundamental Theorem of Calculus is the apex of our course. It explains the relationship between the derivative and the integral, tying the two major facets of this course together. In the previous section, you learned the definition of the definite integral as a limit of a Riemann Sum. The computations were long and involved. In this subunit, you will learn about the Fundamental Theorem of Calculus, which makes the computation of definite integrals significantly easier.

Reading: Whitman College: David Guichard’s Calculus: Chapter 7: Integration: “Section 7.2: The Fundamental Theorem of Calculus”
Link: Whitman College: David Guichard’s Calculus: Chapter 7: Integration: “Section 7.2: The Fundamental Theorem of Calculus” (PDF)

Instructions: Please click on the link above and read Section 7.2 (pages 149-155) in its entirety. 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.

This reading should take you approximately one hour and 30 minutes to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. This text was originally written by Professor David Guichard. It has since been modified to include edited material from Neal Koblitz of the University of Washington, H. Jerome Keisler of the University of Wisconsin, Albert Schueller, Barry Balof, and Mike Wills. You can access the original version here.
See a broken link? Please let us know!
Lecture: Massachusetts Institute of Technology: David Jerison’s Single Variable Calculus: “Lecture 19: The First Fundamental Theorem”
Link: Massachusetts Institute of Technology: David Jerison’s Single Variable Calculus: “Lecture 19: The First Fundamental Theorem” (YouTube)

Instructions: Please click on the link above and watch the entire video (48:53). Lecture notes are available here.

Viewing this lecture and taking notes should take you approximately one hour to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. It is attributed to David Jerison and MIT's OpenCourseWare. It may be viewed in its original form here.
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Lecture: Massachusetts Institute of Technology: David Jerison’s Single Variable Calculus: “Lecture 20: The Second Fundamental Theorem”
Link: Massachusetts Institute of Technology: David Jerison’s Single Variable Calculus: “Lecture 20: The Second Fundamental Theorem” (YouTube)

Instructions: Please click on the link above and watch the entire video (49:30). Lecture notes are available here.

Viewing this lecture and taking notes should take you approximately one hour to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. It is attributed to David Jerison and MIT's OpenCourseWare. It may be viewed in its original form here.
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Assignment: Whitman College: David Guichard’s Calculus: Chapter 7: Integration: “Exercises 7.2: Problems 7-22”
Link: Whitman College: David Guichard’s Calculus: Chapter 7: Integration: “Exercises 7.2: Problems 7-22” (PDF)

Instructions: Please click on the link above and work through problems 7-22 for Exercises 7.2. When you are done, check your answers against “Appendix A: Answers”.

This assignment should take you approximately two hours to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. This text was originally written by Professor David Guichard. It has since been modified to include edited material from Neal Koblitz of the University of Washington, H. Jerome Keisler of the University of Wisconsin, Albert Schueller, Barry Balof, and Mike Wills. You can access the original version here.
See a broken link? Please let us know!

7.3 Some Properties of Integrals

Reading: Whitman College: David Guichard’s Calculus: Chapter 7: Integration: “Section 7.3: Some Properties of Integrals”
Link: Whitman College: David Guichard’s Calculus: Chapter 7: Integration: “Section 7.3: Some Properties of Integrals” (PDF)

Instructions: Please click on the link above and read Section 7.3 (pages 156-160) in its entirety. In particular, note that the definite integral enjoys the same linearity properties that the derivative does, in addition to some others. In its application to velocity functions, pay particular attention to the distinction between distance traveled and net distance traveled.

This reading should take you approximately one hour to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. This text was originally written by Professor David Guichard. It has since been modified to include edited material from Neal Koblitz of the University of Washington, H. Jerome Keisler of the University of Wisconsin, Albert Schueller, Barry Balof, and Mike Wills. You can access the original version here.
See a broken link? Please let us know!
Lecture: Massachusetts Institute of Technology: David Jerison’s Single Variable Calculus: “Lecture 15: Antiderivatives”
Link: Massachusetts Institute of Technology: David Jerison’s Single Variable Calculus: “Lecture 15: Antiderivatives” (YouTube)

Instructions: Please click on the link above and watch the video from the beginning to the 30:00 minute mark. Lecture notes are available here.

Viewing this lecture and taking notes should take you approximately 45 minutes to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. It is attributed to David Jerison and MIT's OpenCourseWare. It may be viewed in its original form here.
See a broken link? Please let us know!
Assignment: Whitman College: David Guichard’s Calculus: Chapter 7: Integration: “Exercises 7.3: Problems 1-6”
Link: Whitman College: David Guichard’s Calculus: Chapter 7: Integration: “Exercises 7.3: Problems 1-6” (PDF)

Instructions: Please click on the link above link and work through problems 1-6 for Exercises 7.3. When you are done, check your answers against “Appendix A: Answers”.

This assignment should take you approximately 45 minutes to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. This text was originally written by Professor David Guichard. It has since been modified to include edited material from Neal Koblitz of the University of Washington, H. Jerome Keisler of the University of Wisconsin, Albert Schueller, Barry Balof, and Mike Wills. You can access the original version here.
See a broken link? Please let us know!

7.4 Integration by Substitution

Reading: Whitman College: David Guichard’s Calculus: Chapter 8: Techniques of Integration: “Section 8.1: Substitution”
Link: Whitman College: David Guichard’s Calculus: Chapter 8: Techniques of Integration: “Section 8.1: Substitution” (PDF)

Instructions: Please click on the link above and read Section 8.1 (pages 161-166) in its entirety. This section explains the process of taking the integral of slightly more complicated functions. We do this by implementing a “change of variables,” or rewriting a complicated integral in terms of elementary functions that we already know how to integrate. Simply put, integration by substitution is merely the act of taking the chain rule in reverse.

This reading should take approximately one hour and 30 minutes to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. This text was originally written by Professor David Guichard. It has since been modified to include edited material from Neal Koblitz of the University of Washington, H. Jerome Keisler of the University of Wisconsin, Albert Schueller, Barry Balof, and Mike Wills. You can access the original version here.
See a broken link? Please let us know!
Lecture: Massachusetts Institute of Technology: David Jerison’s Single Variable Calculus: “Lecture 15: Antiderivatives”
Link: Massachusetts Institute of Technology: David Jerison’s Single Variable Calculus: “Lecture 15: Antiderivatives” (YouTube)

Instructions: Please click on the link above and watch the video from the 30:00 minute mark to the end. Lecture notes are available here.

Viewing this lecture and taking notes should take you approximately 45 minutes to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. It is attributed to David Jerison and MIT's OpenCourseWare. It may be viewed in its original form here.
See a broken link? Please let us know!
Assignment: Whitman College: David Guichard’s Calculus: Chapter 8: Techniques of Integration: “Exercises 8.1: Problems 5-19”
Link: Whitman College: David Guichard’s Calculus: Chapter 8: Techniques of Integration: “Exercises 8.1: Problems 5-19” (PDF)

Instructions: Please click on the link above link and work through problems 5-19 for Exercises 8.1. When you are done, check your answers against “Appendix A: Answers”.

This assignment should take you approximately two hours to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. This text was originally written by Professor David Guichard. It has since been modified to include edited material from Neal Koblitz of the University of Washington, H. Jerome Keisler of the University of Wisconsin, Albert Schueller, Barry Balof, and Mike Wills. You can access the original version here.
See a broken link? Please let us know!

Final Exam

Final Exam: The Saylor Foundation's “MA101 Final Exam”
Link: The Saylor Foundation's “MA101 Final Exam” (HTML)

Instructions: Please click on the link above and complete the final exam. You must be logged into your Saylor Foundation School account in order to access this exam. If you do not yet have an account, you will be able to create one, free of charge, after clicking the link.

Terms of Use: This resource is licensed under a Creative Commons Attribution 3.0 License. It is attributed to The Saylor Foundation.
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