# MA241: Real Analysis I

Unit 4: Differentiability and Integration   In Calculus you learned the limit definition of the derivative and connected it to a picture of a sequence of secant lines converging to a tangent line. Similarly, you learned how the integral is really the limit of a series of Riemann Sums. In this unit, you will go through the derivations of those concepts in more detail and with more regard for the technicalities. You will connect the derivative and the integral to what you have learned about sequences and series, and you will learn necessary and sufficient conditions for the exchange of limits. Much of this does in fact have important implications for numerical integration and differentiation, so it is important to understand the meaning of each of the conditions in theorems such as the Weierstrass M-Test.

This unit should take you approximately 34 hours to complete.

☐    Subunit 4.1: 9.5 hours

☐    Subunit 4.2: 9 hours

☐    Subunit 4.3: 15.5 hours

☐    Sub-subunit 4.3.4: 5 hours

Unit4 Learning Outcomes
Upon successful completion of this unit, you will be able to: - define derivatives of real- and extended-real-valued functions; - compute derivatives using the limit definition; - prove basic properties of derivatives; - use L'Hopital's Rule to compute limits; - state the Mean Value Theorem, and use it in proofs; - construct the Riemann Integral, and state its properties; - state the Fundamental Theorem of Calculus, and use it in proofs; - define pointwise and uniform convergence of series of functions; - use the Weierstrass M-Test to check for uniform convergence of series; - construct Taylor Series, and state Taylor's Theorem; - and identify necessary and sufficient conditions for term-by-term differentiation of power series.

4.1 Differentiation   4.1.1 Derivatives of Functions of One Real Variable   - Reading: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 5: Differentiation and Antidifferentiation: Section 1: Derivatives of Functions of One Variable” Link: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 5: Differentiation and Antidifferentiation: Section 1: Derivatives of Functions of One Variable” (PDF)

`````` Instructions: Please read the indicated section on pages 251-257.

the Saylor Foundation’s Open Textbook Challenge. It is attributed to
the Trillia Group and Elias Zakon.
``````

4.1.2 Derivatives of Extended Real-Valued Functions   - Reading: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 5: Differentiation and Antidifferentiation: Section 2: Derivatives of Extended-Real Valued Functions” Link: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 5: Differentiation and Antidifferentiation: Section 2: Derivatives of Extended-Real Valued Functions” (PDF)

`````` Instructions: Please read “Derivatives of Extended-Real Valued
Functions” on pages 259-265.

the Saylor Foundation’s Open Textbook Challenge. It is attributed to
the Trillia Group and Elias Zakon.
``````

4.1.3 L’Hopital’s Rule   - Reading: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 5: Differentiation and Antidifferentiation: Section 3: L’Hopital’s Rule” Link: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 5: Differentiation and Antidifferentiation: Section 3: L’Hopital’s Rule” (PDF)

`````` Instructions: Please read “L’Hopital’s Rule” on pages 266-269.

the Saylor Foundation’s Open Textbook Challenge. It is attributed to
the Trillia Group and Elias Zakon.
``````

4.1.4 Mean Value Theorem   - Reading: University of California, San Diego: Jiri Lebl’s Basic Analysis: Introduction to Real Analysis: “Chapter 4, Section 2: Mean Value Theorem” Link: University of California, San Diego: Jiri Lebl’s Basic Analysis: Introduction to Real Analysis: “Chapter 4, Section 2: Mean Value Theorem” (PDF)

`````` Instructions: Please read section 4.2 on pages 135-139.

attributed to Jiri Lebl and the original version can be found
[here](http://www.jirka.org/ra/realanal.pdf).
``````
• Activity: University of California, San Diego: Jiri Lebl’s Basic Analysis: Introduction to Real Analysis: “Chapter 4, Section 2.6: Mean Value Theorem: Exercises” Link: University of California, San Diego: Jiri Lebl’s Basic Analysis: Introduction to Real Analysis“Chapter 4, Section 2.6: Mean Value Theorem: Exercises” (PDF)

Instructions: Please work through problems 2-7 on page 140. Because this resource is used by some institutions of higher learning for the purposes of assigning grades and credit for classes, complete solutions to the problems are unavailable. However, it is to your benefit to attempt the problems; your solutions should mimic the style of the proofs given in the preceding chapter.

Terms of Use: The article above is released under Creative Commons Attribution-NonCommercial-ShareAlike 3.0. It is attributed to Jiri Lebl and the original version can be found here.

• Lecture: YouTube: Harvey Mudd College: Professor Francis Su’s Real Analysis: “Lecture 24: The Derivative and the Mean Value Theorem” Link: YouTube: Harvey Mudd College: Professor Francis Su’s Real Analysis: “Lecture 24: The Derivative and the Mean Value Theorem” (YouTube)

Instructions: Please note that these video lectures address topics outlined in sub-subunits 4.1.1 and 4.1.4 of this course. Watch this lecture, in which Professor Su defines the derivative and shows how to derive some of the standard rules of differentiation. He proves the existence of continuous, nowhere-differentiable functions. He states and uses the Mean Value Theorem and the Generalized Mean Value Theorem.

Watching this video and pausing to take notes should take approximately 1 hour and 30 minutes.

4.2 The Riemann Integral   4.2.1 The Riemann Integral   - Reading: University of California, San Diego: Jiri Lebl’s Basic Analysis: Introduction to Real Analysis: “Chapter 5, Section 1: The Riemann Integral” Link: University of California, San Diego: Jiri Lebl’s Basic Analysis: Introduction to Real Analysis: “Chapter 5, Section 1: The Riemann Integral” (PDF)

`````` Instructions: Read “The Riemann Integral” on pages 147-154.

attributed to Jiri Lebl and the original version can be found
[here](http://www.jirka.org/ra/realanal.pdf).
``````

4.2.2 Properties of the Integral   - Reading: University of California, San Diego: Jiri Lebl’s Basic Analysis: Introduction to Real Analysis: “Chapter 5, Section 2: Properties of the Integral” Link: University of California, San Diego: Jiri Lebl’s Basic Analysis: Introduction to Real Analysis“Chapter 5, Section 2: Properties of the Integral” (PDF)

`````` Instructions: Read “Properties of the Integral” on pages 156-161.

attributed to Jiri Lebl and the original version can be found
[here](http://www.jirka.org/ra/realanal.pdf).
``````

4.2.3 The Fundamental Theorem of Calculus   - Reading: University of California, San Diego: Jiri Lebl’s Basic Analysis: Introduction to Real Analysis: “Chapter 5, Section 3: Fundamental Theorem of Calculus” Link: University of California, San Diego: Jiri Lebl’s Basic Analysis: Introduction to Real Analysis“Chapter 5, Section 3: Fundamental Theorem of Calculus” (PDF)

`````` Instructions: Read “Fundamental Theorem of Calculus” on pages
164-168.

attributed to Jiri Lebl and the original version can be found
[here](http://www.jirka.org/ra/realanal.pdf).
``````
• Lecture: YouTube: University of Nottingham: Dr. Joel Feinstein’s “Mathematical Analysis: An Introduction to Riemann Integration” Link: YouTube: University of Nottingham: Dr. Joel Feinstein’s “Mathematical Analysis: An Introduction to Riemann Integration” (YouTube)

Also available on:
iTunes U (#13)

Instructions: Please note these videos cover the topics outlined in sub-subunits 4.2.1-4.2.3 of this course. Please watch this lecture, in which Professor Feinstein defines characteristic functions and partitions, explains the Riemann integral, and states the fundamental theorem of calculus.

Watching this lecture and pausing to take notes should take approximately 1 hour and 15 minutes.

4.3 Interchange of Limits and Series of Functions   4.3.1 Interchange of Limits   - Reading: University of California, San Diego: Jiri Lebl’s Basic Analysis: Introduction to Real Analysis: “Chapter 6, Section 2: Interchange of Limits” Link: University of California, San Diego: Jiri Lebl’s Basic Analysis: Introduction to Real Analysis“Chapter 6, Section 2: Interchange of Limits” (PDF)

`````` Instructions: Read “Interchange of Limits” on pages 195-199.

attributed to Jiri Lebl and the original version can be found
[here](http://www.jirka.org/ra/realanal.pdf).
``````

4.3.2 Series of Functions   - Reading: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 4: Function Limits and Continuity: Section 12: Sequences and Series of Functions” Link: The Trillia Group: Elias Zakon’s Mathematical Analysis I“Chapter 4: Function Limits and Continuity: Section 12: Sequences and Series of Functions” (PDF)

`````` Instructions: Please read “Sequences and Series of Functions” on
pages 227-232.

the Saylor Foundation’s Open Textbook Challenge. It is attributed to
the Trillia Group and Elias Zakon.
``````
• Lecture: University of Colorado at Colorado Springs: Professor Rinaldo Schinazi’s “Modern Analysis II, Lecture 16: Series of Functions” Link: University of Colorado at Colorado Springs: Professor Rinaldo Schinazi’s “Modern Analysis II, Lecture 16: Series of Functions” (Flash)

Instructions: To access this lecture you will need to create a free account. Once logged in, click on the link which says “Go to the UCCS Math Video Archive.” Under “Spring Semester 2008,” click on the link to “Math 432/532,” then scroll down to Lecture 21 and click the icon on the left-hand side.

In this lecture, Professor Schinazi discusses the convergence or lack thereof of the derivatives of a sequence of functions. (In Lecture 14, not required by this course, he discussed the convergence of the integrals of a uniformly convergent sequence of functions.) He then introduces series of functions and proves the Weierstrass M-Test, which is the most important result in this lecture. You need not watch the section on the Weierstrass Approximation Theorem, although it is very interesting.

Watching this lecture and pausing to take notes should take approximately 1 hour.

4.3.3 Absolutely Convergent Series   - Reading: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 4: Function Limits and Continuity: Section 13: Absolutely Convergent Series and Power Series” Link: The Trillia Group: Elias Zakon’s Mathematical Analysis I“Chapter 4: Function Limits and Continuity: Section 13: Absolutely Convergent Series and Power Series” (PDF)

`````` Instructions: Please read the indicated section on pages 237-244.

the Saylor Foundation’s Open Textbook Challenge. It is attributed to
the Trillia Group and Elias Zakon.
``````
• Lecture: University of Colorado at Colorado Springs: Professor Rinaldo Schinazi’s “Modern Analysis II, Lecture 21: Power Series and Ratio Rule” Link: University of Colorado at Colorado Springs: Professor Rinaldo Schinazi’s “Modern Analysis II, Lecture 21: Power Series and Ratio Rule” (Flash)

Instructions: To access this lecture you will need to create a free account. Once logged in, click on the link which says “Go to the UCCS Math Video Archive.” Under “Spring Semester 2008,” click on the link to “Math 432/532,” then scroll down to Lecture 21 and click the icon on the left-hand side.

In this lecture, Professor Schinazi discusses power series and justifies the various convergence tests (such as the root test and the ratio test). In the second half (which is optional), he proves a result about the derivative of the pointwise limit of a sequence of differentiable functions. He later discusses term-by-term differentiation (which is really an interchange of limits).

Watching this lecture and pausing to take notes should take approximately 1 hour.

4.3.4 Taylor’s Theorem   - Reading: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 5: Differentiation and Antidifferentiation: Section 6: Differentials; Taylor’s Theorem and Taylor Series” Link: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 5: Differentiation and Antidifferentiation: Section 6: Differentials; Taylor’s Theorem and Taylor Series” (PDF)

`````` Instructions: Please read the indicated section on pages 288-296.