Last modified: February 21, 2018.
Homework assignments will be available on this webpage throughout the term. All homework assignments must be submitted to the drop boxes in the basement of AP&M by 4:00 PM on the deadline.
Notice that $\frac{d}{dt}\int_0^t f(x)\,dx \neq f(t)$.
Suppose $f$ is defined as follows: $$\begin{align}f : [4, 12] &\longrightarrow \R\\ t &\longmapsto \begin{cases} 3 & \text{ if } 5 \lt t \leq 10 \\ 1 & \text{ otherwise.}\end{cases}\end{align}$$ Show that $f$ does not have an antiderivative.
(Hint: There are several ways of solving this problem. One may take a direct approach: first, show that if $f$ does have an antiderivative, then it must be a piecewise linear function with different slopes on the different pieces; but there is no way such a function could be differentiable where the slope changes. In fact, it is possible to prove a much more general statement: any function with a jump discontinuity has no antiderivative.)
Suppose that $f : [a, b]\to\R$ is continuous, and that $F : [a,b] \to \R$ is continuous, differentiable on $(a,b)$, and $F'(z) = f(z)$ for $z \in (a,b)$. Use the second fundamental theorem to prove the first fundamental theorem by showing that $$\frac{d}{dt}\paren{F(t)  \int_a^t f(x)\,dx} = 0.$$
First, show the identity above. Then, use this to prove the first fundamental theorem: if $F : [a, b] \to \R$ is continuous, differentiable on $(a, b)$, and $F'$ is continuous and bounded on $(a, b)$ then $$\int_a^b F'(x)\,dx = F(b)  F(a).$$
A set $S \subset \R$ is said to be null if for any $\epsilon \gt 0$ there is a sequence of open intervales $\paren{(a_n, b_n)}_n$ such that their union contains $S$ and their combined length is less than $\epsilon$: $$S\subseteq \bigcup_{n=1}^\infty (a_n, b_n) \qquad\qquad\text{ and }\qquad\qquad \sum_{n=1}^\infty b_na_n \lt \epsilon.$$
As it turns out, there are null sets which are not countable; the Cantor set is a famous example. Somewhat more surprisingly, the function which is $1$ on the Cantor set and $0$ on its complement is discontinuous precisely on the Cantor set, and therefore integrable by this problem. On the last assignment's bonus problem we showed that there was an integrable function with a countable set of discontinuities; now there is even one with an uncountable set of discontinuities.
Suppose that $R \gt 0$ and $f : (R, R) \to \R$ is infinitely differentiable. Suppose further that for any $0 \lt r \lt R$ there is a sequence $(M_{r,k})_k$ such that for all $x \in (r, r)$, $$\abs{f^{(k)}(x)} \leq M_{r,k} \qquad\qquad\text{and}\qquad\qquad \lim_{k\to\infty} M_{r,k}\frac{r^k}{k!} = 0.$$ Show that for every $x \in (\color{red}{R}, \color{red}{R})$, $$\int_0^x f(t)\,dt = \sum_{k=1}^\infty f^{(k1)}(0)\frac{x^k}{k!}.$$
(Hint: first show that our assumptions bounding the derivatives of $f$ mean that the Taylor polynomials converge uniformly to $f$ in the sense of this 142A assignment; next show that if a sequence of functions $(f_n)$ converges uniformly to $f$ then the sequence of integrals $\int_a^b f_n$ converges to $\int_a^b f$; then put these two statements together.)
The following two problems are thematically related  both relate to when continuous things cannot be approximated by polynomials  but you should not assume that either will be useful for proving the other. The second problem is a result usually proved using sophisticated techniques from complex analysis which are not within the scope of this course. However, the result is possible to establish (though not stratightforward) using techniques available to us.
A complex number is something of the form $\alpha + \beta i$ where $\alpha, \beta \in \R$; the set of complex numbers is denoted $\C$, and one identifies $\R$ with the set $\set{x+0i : x \in \R} \subset \C$. Addition is performed by components (so $(a + bi) + (c + di) = (a+c) + (b+d)i$), while multiplication is distributed and it is understood that $i^2 = 1$ (so $(a + bi)(c + di) = (ac  bd) + (bc+ad)i$). If $z = a+bi$, its complex conjugate, denoted $\bar{z}$, is $\bar{z} = abi$. The norm of $z = a+bi$ is the quantity $z = \sqrt{a^2+b^2}$. It follows that $z\bar{z} = z^2 + 0i \in \R$. In turn it follows that if $z = 1$ then $z\bar{z} = 1$.
A complex polynomial is a function $p : \C \to \C$ of the form $p(z) = a_0 + a_1z + a_2z^2 + \ldots + a_nz^n$, where $a_0, \ldots, a_n \in \C$.
Let $\mathbb{D}$ be the set $\set{z \in \C : z \leq 1}$. Show that the function $f : \mathbb{D} \to \C$ given by $f(z) = \bar{z}$ is not uniformly approximated by polynomials: there is some $\epsilon \gt 0$ so that for any polynomial $p$ there is some $z \in \mathbb{D}$ so that $f(z)  \bar{z} \gt \epsilon$.
You may wish to type your homework; for example, this makes it much easier for others to read, and makes it easier to edit and produce a coherent final argument. Most modern mathematics papers are typeset using a system called \(\mathrm{\LaTeX}\) (pronounced "lahtech" or "laytech"; see the Wikipedia entry on Pronouncing and writing "LaTeX"). Although it has a steep learning curve, it is extremely useful for typesetting complicated mathematical expressions. There are many resources available online, such as this reference by Oetiker, Partl, Hyna, and Schlegl. I have also made available an assignment template here, which produces this output when compiled correctly.
Working in groups on assignments is encouraged in this course. However, you should write your answers individually.
Name  Role  Office  
Ian Charlesworth  Instructor  AP&M 5880C  ilc@math.ucsd.edu 
Woonam Lim  Teaching Assistant  AP&M 6414  w9lim@ucsd.edu 
Yucheng Tu  Teaching Assistant  AP&M 5720  y7tu@ucsd.edu 
My office hours are Mondays from 10:00am  11:00am, Wednesdays from 10:30am  11:30am, and Fridays from 12:30pm  1:30pm.
We will be communicating with you and making announcements through an online question and answer platform called Piazza. We ask that when you have a question about the class that might be relevant to other students, you post your question on Piazza instead of emailing us. That way, everyone can benefit from the response. Posts about homework or exams on Piazza should be content based. While you are encouraged to crowdsource and discuss coursework through Piazza, please do not post complete solutions to homework problems there. Questions about grades should be brought to the instructors, in office hours. You can also post private messages to instructors on Piazza, which we prefer over email.
If emailing us is necessary, please do the following:
Date  Time  Location  
Lecture B00 (Charlesworth)  Mondays, Wednesdays, Fridays  2:00pm  2:50pm  PCYNH 122 
Discussion B01 (Woonam Lim)  Tuesdays  6:00pm  6:50pm  AP&M 2402 
Discussion B02 (Yucheng Tu)  Tuesdays  7:00pm  7:50pm  AP&M 2402 
Discussion B03 (Yucheng Tu)  Tuesdays  8:00pm  8:50pm  AP&M 2402 
Final Exam  Monday, March 19  3:00pm  6:00pm  TBA 
Course: Math 142B
Title: Math 142B: Introduction to Analysis II
Credit Hours: 4 (Students may not receive credit for both Math 140B and 142B.)
Prerequisites: Math 142A or Math 140A, or consent of instructor.
Catalog Description: Second course in an introductory twoquarter sequence on analysis. Topics include: differentiation, the Rieman integral, sequences and series of functions, uniform convergence, Taylor and Fourier series, special functions. See the UC San Diego Course Catalog.
Textbook: Advanced Calculus, 2^{nd} edition, by Patrick M. Fitzpatrick.
Subject Material: We will mainly cover chapters 6, 8, and 9 of the text.
Lecture: Attending the lecture is a fundamental part of the course; you are responsible for material presented in the lecture whether or not it is discussed in the textbook. You should expect questions on the exams that will test your understanding of concepts discussed in the lecture.
Reading: 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.
Calendar of Lecture Topics: The following calendar is subject to revision during the term. The section references are only a guide; our pace may vary from it somewhat.
Week  Monday  Tuesday  Wednesday  Thursday  Friday 

1 
Jan 8
6.1

Jan 9
Discussion

Jan 10
6.2

Jan 11 
Jan 12
6.3

2 
Jan 15
Martin Luther King, Jr. Holiday

Jan 16
Discussion

Jan 17
6.3

Jan 18 
Jan 19
6.4

3 
Jan 22
6.5

Jan 23
Discussion

Jan 24
6.6

Jan 25 
Jan 26
6.6

4 
Jan 29
Review/catchup

Jan 30
Discussion

Jan 31
Midterm exam

Feb 1 
Feb 2
8.1

5 
Feb 5
8.2

Feb 6
Discussion

Feb 7
8.3

Feb 8 
Feb 9
8.4

6 
Feb 12
8.5

Feb 13
Discussion

Feb 14
8.6

Feb 15 
Feb 16
8.7

7 
Feb 19
Presidents' Day Holiday

Feb 20
Discussion

Feb 21
Review/catchup

Feb 22 
Feb 23
Midterm exam

8 
Feb 26
9.1

Feb 27
Discussion

Feb 28
9.2

Mar 1 
Mar 2
9.3

9 
Mar 5
9.4

Mar 6
Discussion

Mar 7
9.4

Mar 8 
Mar 9
9.5

10 
Mar 12
9.6

Mar 13
Discussion

Mar 14
Review/catchup

Mar 15 
Mar 16
Review/catchup

11  Mar 19  Mar 20  Mar 21  Mar 22  Mar 23 
Homework: Homework is a very important part of the course and in order to fully master the topics it is essential that you work carefully on every assignment and try your best to complete every problem. Homework will be assigned on the course webpage. Your homework can be submitted to the dropbox with your TA's name on it in the basement of the AP&M building. Homework is officially due at 4:00 PM on the due date.
Midterm Exams: There will be two midterm exams given during the quarter. You may bring one 8.5 by 11 inch sheet of paper with handwritten notes (on both sides) with you to each midterm exam; no other notes (or books) will be allowed. No calculators, phones, or other electronic devices will be allowed during the midterm exams. You may bring at most three fourleaf clovers, horseshoes, manekineko, or other such talismans for good luck. There will be no makeup exams.
Final Examination: The final examination will be held at the date and time stated above.
Administrative Links: Here are two links regarding UC San Diego policies on exams:
Regrade Policy:
Administrative Deadline: Your scores for all graded work will be posted to TritonEd.
Grading: Your course grade will be determined by your cumulative average at the end of the term and will be based on the following scale:
A+  A  A  B+  B  B  C+  C  C 
97  93  90  87  83  80  77  73  70 
In addition, you must pass the final examination in order to pass the course. Note: Since there are no makeup exams, if you miss a midterm exam for any reason, then your course grade will be computed with the second option. There are no exceptions; this grading scheme is intended to accommodate emergencies that require missing an exam.
Your single worst homework score will be ignored.
Academic Integrity: UC San Diego's code of academic integrity outlines the expected academic honesty of all studentd and faculty, and details the consequences for academic dishonesty. The main issues are cheating and plagiarism, of course, for which we have a zerotolerance policy. (Penalties for these offenses always include assignment of a failing grade in the course, and usually involve an administrative penalty, such as suspension or expulsion, as well.) However, academic integrity also includes things like giving credit where credit is due (listing your collaborators on homework assignments, noting books or papers containing information you used in solutions, etc.), and treating your peers respectfully in class. In addition, here are a few of our expectations for etiquette in and out of class.
Here are some additional resources for this course, and math courses in general.
Any remarks pertaining to particular lectures will be posted here throughout the term.
Below is a graph of the functions $f_k(x) = \binom{10}{k}x^k(1x)^{10k}$ for $0 \leq k \leq 10$.
Below is a graph of the functions $g_k(x) = \sin\paren{2\pi\frac{k}{10}}f_k(x)$, as well as their sum, and the function $\sin(2\pi x)$.