# MA101: Single-Variable Calculus I

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.

This unit should take you approximately 13.25 hours to complete.

☐    Subunit 6.1: 5 hours

☐    Lecture: 1 hour

☐    Assignments: 1 hour

☐    Subunit 6.2: 3 hours

☐    Subunit 6.3: 1.75 hours

☐    Subunit 6.4: 1.75 hours

☐    Subunit 6.5: 1.75 hours

Unit6 Learning Outcomes
Upon successful completion of this unit, the student will be able to:
- Solve problems involving rectilinear motion using derivatives. - Solve problems involving related rates. - 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.

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.

• 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.

• 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.

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.

• 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.

• 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.

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.

• 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.

• 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)

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.

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.

• 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.

• 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)

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.

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.

• 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.