# Math 140B: Elementary Analysis (Fall 2019)

• ### Lectures

Lectures are by Jeff Viaclovsky:

• Lecture 44783 meets on MWF at 1:00-1:50 PM in RH114.

Some important information:

• I will have office hours on Mondays and Wednesdays in Rowland Hall 440G from 2:00-2:50 PM.
• If you email me, please always put the course number 140B in the subject line, or I might miss your email.
• If anyone is spotted browsing the internet or sending text messages during lectures, their computer or cell phone will be confiscated for the remainder of the lecture. However, taking notes on a tablet PC, or briefly using a computer to look up a formula is acceptable (except on an exam of course).
• It is acceptable to form study groups and discuss homework with other students, but you must write up your own solutions individually. Do NOT just copy solutions from someone else or from the internet (we have access to the internet too, and can easily spot if a solution was just copied from an internet source).
• Cheating is not allowed, and will result in very serious consequences if you are caught.

• ### Textbook

Elementary Analysis: The Theory of Calculus by Kenneth A. Ross, second edition.

• ### Prerequisites

MATH 140A ( min grade = C- ) AND NO REPEATS ALLOWED IF GRADE = C OR BETTER.

• ### Discussion Section and TA

 SECTION LECTURE TIME LOCATION TEACHING ASSISTANT 10 A TuTh 04:00--04:50PM SE2 1306 Alberto Takase

Alberto Takase may be reached at atakase "at" uci.edu. His office hours are on Tuesdays and Thursdays from 5:00PM -6:30PM in 440T Rowland Hall.

The homework grader for this course is Dajun Xu, who may be reached at dajunx1 "at" uci.edu.

• ### Examinations, Homework, Quizzes, and Final Grade

• There will be weekly homework assignments, 2 in-class exams, weekly quizzes in discussion, and a final exam.
• Quizzes: 15%, Homework: 15%, Exam I: 20%, Exam II: 20%, Final: 30%. Cutoff for A = 90, cutoff for B = 80, cutoff for C = 70, cutoff for D = 60, and below 60 is failing.
• The in-class exams are scheduled for October 25, November 22. The final exam is on December 11. There are no make-up exams. Exams may not be missed or rescheduled, except with a note from the dean. (With a valid note, the weight of the exam missed will be added to the weight of the final exam.)
• Homework is due at the beginning of discussion on Thursdays, unless otherwise noted. No late homework will be accepted.
• Quizzes will be in discussion on Tuesdays, except for the first week and the 2 weeks with an exam. There are no make-up quizzes.

• HW #1: Due Thursday, October 10:
1. Problems 20.2, 20.6, 20.14, 23.1aceg, 23.8, 23.9, 24.1, 24.2, 24.10a.
• HW #2: Due Thursday, October 17:
1. Problems 24.11, 24.12, 24.13, 25.5, 25.6, 25.10, 26.2, 26.3, 26.6.
• HW #3: Due Thursday, October 24:
1. Problems 28.3, 28.4, 28.6, 28.8, 28.14.
• HW #4: Due Thursday, October 31:
1. Problems 29.1, 29.2, 29.5, 29.7, 29.9.
• HW #5: Due Thursday, November 7:
1. Problems 29.11, 29.12, 29.13, 29.14, 29.15, 29.16, 29.17, 29.18, 30.1, 30.2.
• HW #6: Due Thursday, November 14:
1. Problems 30.3, 30.5, 31.1, 31.2, 31.4, 31.5.
• HW #7: Due Thursday, November 21:
1. Problems 32.1, 32.3, 32.6, 32.7, 32.8.
• HW #8: Due Thursday, December 5:
1. Problems 33.1, 33.4, 33.7, 33.8, 33.9, 34.2, 34.3, 34.6, 34.7, 34.8.

• ### Topics covered

This course is an introduction to real analysis including convergence of sequences, infinite series, differentiation and integration, and sequences of functions. Students are expected to do proofs. Most of Chapters 4-6 will be covered. The following lecture outline will describe exactly which sections will be covered, and will be updated very frequently.

• ### Lecture outline

• Friday, September 27
1. Introduction and outline.
2. Review of limits and limit theorems.
• Monday, September 30
1. Review of infinite series.
2. Review of continuity.
• Wednesday, October 2
1. Review of uniform continuity.
• Friday, October 4
1. Section 19, limits of functions.
2. Left-handed and right-handed limits.
• Monday, October 7
1. Limit exists iff and only both handed limits exists and are equal.
2. Section 23: Power series.
• Wednesday, October 9
1. Section 24: Pointwise convergence of sequences of function.
2. Uniform convergence of sequences of functions.
• Friday, October 11
1. Uniform limit of continuous functions is continuous.
2. Remark 24.4 about uniform convergence.
3. Section 25: Integration commutes with uniform limits.
• Monday, October 14
1. Section 25: Uniformly Cauchy sequences of functions.
2. Cauchy criterion for series of functions.
3. Weierstrass M-test.
• Wednesday, October 16
1. Section 26: Differentiation and integration of power series.
• Friday, October 18
1. ln(2) = value of alternating harmonic series.
2. Start on Chapter 5: Definition of the derivative.
3. Derivative of f(x) = x^n.
4. Differentiable implies continuous, but converse not true in general.
• Monday, October 21
1. Derivatives of sums, products, quotients.
2. The chain rule.
• Wednesday, October 23
1. Begin Section 29: The mean value theorem.
2. Exam review.
• Friday, October 25
1. Exam I, in-class
• Monday, October 28
1. Rolle's Theorem.
2. Mean Value Theorem.
• Wednesday, October 30
1. Increasing and decreasing functions.
2. Review of inverse functions.
• Friday, November 1
1. Review of continuity of an inverse function.
2. The derivative of an inverse function.
• Monday, November 4
1. More examples of derivatives of inverse functions.
2. Begin section 30: L'Hopital's rule.
3. Easy proof of L'Hopital's if g'(a) is non-zero.
4. Mean value theorem for 2 functions.
• Wednesday, November 6
1. Proof of full L'Hopital's rule.
2. Applications of L'Hopital's rule.
• Friday, November 8
1. Begin on Section 31: Taylor's Theorem.
2. Proof of Taylor's Theorem with Lagrange form of remainder.
• Monday, November 11
1. Veterans's Day: no class.
• Wednesday, November 13
1. If all derivatives uniformly bounded on an interval, then real analytic.
2. Taylor series examples of e^x, sin(x).
3. Example of infinitely differentiable function which is not real analytic.
• Friday, November 15
1. Proof of Taylor's theorem with Peano form of remainder.
2. Begin Chapter 6: The Riemann integral.
3. Partitions, Upper and lower Darboux sums.
4. Upper and lower Darboux integrals.
• Monday, November 18
1. Proof that L(f) is less that or equal to U(f).
• Wednesday, November 20
1. Cauchy criterion for Darboux integrability.
2. The mesh of a partition and Riemann integrability.
3. Exam review.
• Friday, November 22
1. Exam II, in-class
• Monday, November 25
1. Section 33: Properties of the Riemann integral.
2. Monotonic functions on are integrable on closed intervals.
3. Continuous functions are inegrable on closed intervals.
4. Linearity of the integral.
• Wednesday, November 27
1. Course evaluations.
• Friday, November 29
1. Thanksgiving Break: no class.
• Monday, December 2
1. Sum of integrable functions is integrable.
2. Triangle inequality for integrals.
3. Integral of piecewise monotonic and piecewise continuous functions.
4. Intermediate value theorem for integrals.
• Wednesday, December 4
1. Section 34: Fundamental theorem of calculus I.
2. Integration by parts.
3. Fundamental theorem of calculus II.
4. Change of variables formula for integrals.
• Friday, December 6
1. Review.
• Wednesday, December 11
1. Final Exam: 1:30 - 2:30 PM.

#### Contact Information

Dr. Jeff Viaclovsky
Department of Mathematics
University of California
340 Rowland Hall (Bldg.#400)
Irvine, CA 92697

Office: Rowland Hall 440G
Office phone: 949-824-5508
e-mail: jviaclov@uci.edu