Math 140B

• Office: APM 6-101
• Office hour: Wednesday 4:10-5:10pm, Friday 10-11am
• Email: doprea at math dot ucsd dot edu

• Discussion: APM 5402, Thursday 12-12:50pm and 1:00-1:50pm
• Office: APM 5760
• Office hour: Mon 9-11, Wed 10-11, Fri 12-1 (Week 1 only: Friday 1-1:55pm).
• Email: cwho at ucsd dot edu.

Differentiation. Riemann integral. Sequences and series of functions. Special functions. Fourier series. This corresponds to chapters 5-8 in Rudin's book.

Math 140A or permission of instructor. Students will not receive credit for both Math 140 and Math 142. Math 140 is a difficult and time consuming course, so enroll only if your course load allows it.

• Homework 20%, Midterm I 20%, Midterm II 20%, Final Exam 40%
• Homework 20%, Midterm I 15%, Midterm II 15%, Final Exam 50%

W. Rudin, Principles of Mathematical Analysis, Third Edition.

Reading the sections of the textbook corresponding to the assigned homework exercises is considered part of the homework assignment. You are responsible for material in the assigned reading whether or not it is discussed in the lecture. It will be expected that you read the assigned material in advance of each lecture.

Homework problems will be assigned on the course homework page. There will be 7 problem sets, typically due on Friday at 4:30PM in the TA's mailbox. The due date of some of the problem sets may change during the quarter depending on the pace at which we cover the relevant topics. The best six problem sets will be used to compute the final grade.

You may work together with your classmates on your homework and/or ask the TA (or myself) for help on assigned homework problems. However, the work you turn in must be your own. No late homework assignments will be accepted.

There will be two midterm exams given on April 26 and May 19. There will be no makeup exams.

Regrading policy: graded exams will be handed back in section. Regrading is not possible after the exam leaves the room.

The final examination will be held on Friday, June 16, 3:00-6:00PM. There is no make up final examination. It is your responsability to ensure that you do not have a schedule conflict during the final examination; you should not enroll in this class if you cannot sit for the final examination at its scheduled time.

• Monday, April 3: First lecture
• Wednesday, April 26: Midterm I
• Friday, May 19: Midterm II
• Friday, May 29: Memorial Day, no class
• Friday, June 9: Last Lecture
• Friday, June 16: FINAL EXAM, 3:00-6:00pm.

• Preparation for Midterm 1:
• Practice midterm from Spring 2013 - PDF and Solutions
• Practice midterm from Winter 2015 - PDF and Solutions
• Midterm 1 and Solutions
• Preparation for Midterm 2:
• Practice midterm from Spring 2013 - PDF and Solutions
• Practice midterm from Winter 2015 - PDF and Solutions
• Midterm 2 and Solutions
• Preparation for Final:
• Practice final from Spring 2013 - PDF and Solutions
• Practice final from Winter 2015 - PDF and Solutions
• Final Exam and Solutions

• Lecture 1: Introduction. Derivatives. Differentiability and continuity. Properties of derivatives.
• Lecture 2: Chain rule. Local minima and maxima and derivatives. Rolle functions. Rolle's theorem.
• Lecture 3: Mean value theorems. Monotonic functions and derivatives. Intermediate value property and derivatives.
• Lecture 4: L'Hospital's rule and proof first in a particular case, then in the general case.
• Lecture 5: Taylor's theorem. Motivation and examples. Proof of Taylor.
• Lecture 6: Integration. Lower and upper Riemann sums. Lower and upper integrals. Integrable functions.
• Lecture 7: Riemann's criterion for integrability. Refinements of partitions and effects on upper and lower sums. Monotonic functions are integrable.
• Lecture 8: Continuous functions are integrable. Functions with finitely many discontinuities are integrable. Composition of integrable functions. Composition of continuous and integrable functions.
• Lecture 9: Properties of the integral: sums and products of integrable functions. Inequalities.
• Lecture 10: Relationship between integration and differentiation. Fundamental theorem of calculus.
• Lecture 11: Integration by parts. Change of variables formula and proof.
• Lecture 12: Sequences of functions. Pointwise limits do not behave well with respect to continuity, derivatives, integration.
• Lecture 13: Uniform convergence. Uniformly Cauchy sequences and Cauchy's criterion. Uniform convergence and integration.
• Lecture 14: Uniform limit of continuous functions is continuous. The space of continuous bounded functions in complete.
• Lecture 15: Uniform convergence and differentiability. Series. Weierstrass M-test.
• Lecture 16: Example of a function which is continuous but nowhere differentiable.
• Lecture 17: Uniformly bounded sequences. Equicontinuity. Examples of equicontinuous families. Uniform convergence implies equicontinuity.
• Lecture 18: Uniformly bounded and equicontinuous sequences admit convergent subsequences.
• Lecture 19: Weierstrass approximation theorem and proof.
• Lecture 20: Algebras of functions. Examples. Algebras that separate points and vanish nowhere. Construction of functions with prescribed values.
• Lecture 21: Proof of Stone-Weierstrass.
• Lecture 22: Power series. Radius of convergence. Power series can be differentiated term by term. Behaviour at the endpoints and Abel's theorem.
• Lecture 23: Applications of power series: the exponential function. Definition and properties.
• Lecture 24: Applications of power series: sine and cosine. Agreement with the geometric definition. Onto Fourier analysis: trigonometric polynomials.
• Lecture 25: Fourier coefficients and Fourier series. Examples. L^2-convergence defined. Parseval's theorem stated.
• Lecture 26: Geometry of L^2 spaces and intuition behind Parseval's theorem. The complex exponentials form an orthogonal system for L^2. Least square properties of Fourier partial sums.
• Lecture 27: Proof of Parseval's theorem first for continuous functions, then in general. L^2-approximation of integrable functions by continuous functions. Review.