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
This unit should take you approximately 17 hours to complete.
☐ Subunit 4.1: 1.5 hours
☐ Subunit 4.2: 0.25 hourd
☐ Subunit 4.3: 1 hour
☐ Subunit 4.4: 1 hour
☐ Subunit 4.5: 1.25 hours
☐ Subunit 4.6: 0.5 hours
☐ Subunit 4.7: 3 hours
☐ Subunit 4.8: 3 hours
☐ Subunit 4.9: 2 hours
☐ Subunit 4.10: 2.5 hours
☐ Subunit 4.11: 1 hours
Unit4 Learning Outcomes
Upon successful completion of this unit, the student will be able to:
- Use implicit differentiation to find derivatives.
- Find derivatives of inverse functions.
- Find derivatives of trigonometric, exponential, logarithmic, and
inverse trigonometric functions.
- State and apply L’Hopital’s Rule for indeterminate forms.
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.
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 aCreative 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.
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.
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](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 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.
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.
- 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.
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.
- 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.
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.
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.
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.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.
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.
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.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.
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.
- 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.
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.
- 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.
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.