Math 140B

• Office: APM 6-101
• Office hour: M 2:30 - 3:30, W 3:30 - 4:30. Office hours will be hybrid, but in-person participants will have some priority. The zoom link is posted in canvas.
• Email: doprea at math dot ucsd dot edu

• Discussion: CENTR 217A, Tuesday 4-4:50pm and 5:00-5:50pm
• Office: APM 6414
• Office hour: TH 9-11AM, F 2-4PM.
• Email: fmcglade 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%, Best Midterm 20%, Final Exam 60%

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 via Gradescope. There are several ways to upload your homework in Gradescope. A possible method is described here.

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 6 problem sets will be used to compute the final grade. No late homework assignments will be accepted.

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. At the top of each homework assignment, you must specify all outside resources that you consulted, or write "None" if none were used. You do not need to report the use of the textbook or class notes, or discussions with me or the TA. You do need to report other resources including the names of other students you collaborated with.

There will be two midterm exams given on April 22 and May 18. 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 Monday, June 6, 8:00-11:00AM. 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, March 28: First lecture
• Friday, April 22: Midterm I
• Wednesday, May 18: Midterm II
• Monday, May 30: Memorial Day, no class
• Friday, June 3: Last Lecture
• Monday, June 6: FINAL EXAM, 8:00-11:00am.

• Preparation for Midterm 1:
• Practice midterm from Winter 2015 - PDF and Solutions
• Practice Midterm from Spring 2017 - PDF and Solutions
• Midterm 1 and Solutions
• Preparation for Midterm 2:
• Practice midterm from Winter 2015 - PDF and Solutions
• Practice Midterm from 2017 - PDF and Solutions
• Midterm 2 and Solutions
• Preparation for Final:
• Practice final from Winter 2015 - PDF and Solutions
• Practice final from Spring 2017 - PDF and Solutions
• Final and Solutions

• Lecture 1: Introduction. Derivatives. Differentiability and continuity. Properties of derivatives.
• Lecture 2: Chain rule and its proof. Local maxima and local minima. Rolle functions and Rolle's theorem.
• Lecture 3: Proof of Rolle's theorem. Cauchy's theorem. Derivatives and monotonic functions. Intermediate value property.
• Lecture 4: Proof of the intermediate value property for derivatives. L'Hospital's rule and proof first in a particular case, then in the general case.
• Lecture 5: Taylor's theorem and its proof. Applications.
• Lecture 6: Riemann-Stieltjes integral. Lower and upper Riemann sums. Lower and upper integrals. Integrable functions.
• Lecture 7: Refinements of partitions and effects on upper and lower sums. Riemann's criterion for integrability. Monotonic functions are integrable.
• Lecture 8: Continuous functions are integrable. Functions with finitely many discontinuities are integrable. Composition of continuous and integrable functions.
• Lecture 9: Properties of the integral: sums and products of integrable functions. Inequalities.
• Lecture 10: Two cases of the Riemann-Stieltjes integral. The case of a step function. Connection between the Riemann and Riemann-Stieltjes integral.
• Lecture 11: Relationship between integration and differentiation. Fundamental theorem of calculus. Integration by parts.
• Lecture 12: Change of variables. Sequences and series 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. Completeness of the space of continuous functions.
• Lecture 15: Uniform convergence and differentiablity. Uniform convergence of series. The Weierstrass M-test.
• Lecture 16: Example of a function which is continuous but nowhere differentiable.
• Lecture 17: Pointwise and uniformly bounded sequences. Equicontinuity. Examples of equicontinuous families. Uniform convergence implies equicontinuity.
• Lecture 18: Proof of Arzela-Ascoli: uniformly bounded and equicontinuous sequences admit convergent subsequences.
• Lecture 19: Weierstrass approximation theorem and its proof.
• Lecture 20: Algebras of functions. Examples. Algebras that separate points and vanish nowhere. Construction of functions with prescribed values.
• Lecture 21: Stone-Weierstrass theorem and its proof.
• 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. The number pi defined. Agreement with geometry.
• Lecture 25: Fourier coefficients and Fourier series. Examples. L^2-convergence defined. Parseval's theorem stated - PDF
• Lecture 26: Orthonormal systems and generalized Fourier series. Least square properties of Fourier partial sums.
• Lecture 27: Proof of Parseval's theorem. L^2-approximation of integrable functions by trigonometric polynomials. Review.