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.
Unit 5 Time Advisory
This unit should take approximately 10.25 hours to complete.
☐ Subunit 5.1: 3 hours
☐ Subunit 5.2: 1.75 hours
☐ Subunit 5.3: 1.5 hours
☐ Subunit 5.4: 1.5 hours
☐ Subunit 5.5: 2.5 hours
Unit5 Learning Outcomes
Upon successful completion of this unit, the student will be able to:
- 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.
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.
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.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.
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](http://creativecommons.org/licenses/by-nc-sa/3.0/). 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](http://www.whitman.edu/mathematics/calculus/).
- 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.
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.
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.
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.
- 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.
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.
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.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.