**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**

This unit should take you approximately 13.25 hours to complete.

☐ Subunit 6.1: 5 hours

☐ Reading: 3 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.

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.**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:

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.

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.**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:

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:

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.

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.**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:

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.

**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:

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.

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.**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:

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”. Please note that the correct answer for 6.4.4 is actually 32π/25 (highlight to see the correct answer).

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.

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.**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)