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
This unit should take you approximately 16.5 hours to complete.
☐ Subunit 2.1: 2.5 hours
☐ Subunit 2.2: 1 hour
☐ Subunit 2.3: 6.75 hours
☐ Sub-subunit 2.3.1: 6 hours
☐ Sub-subunit 2.3.2: 0.75 hours
☐ Subunit 2.4: 2 hours
☐ Subunit 2.5: 4.25 hours
☐ Introduction: 1 hour
☐ Sub-subunit 2.5.1: 2 hours
☐ Sub-subunit 2.5.2: 1 hour
☐ Sub-subunit 2.5.3: 0.25 hours
Unit2 Learning Outcomes
Upon the successful completion of this unit, the student will be able
to:
- 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.
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.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.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.
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.
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.
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.
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.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.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.
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.
- 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.
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.
- 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.
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.
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.
Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.
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.
Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.
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.
Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.