$$\newcommand{\norm}[1]{\left\|#1\right\|} \newcommand{\paren}[1]{\left(#1\right)} \newcommand{\abs}[1]{\left\lvert#1\right\rvert} \newcommand{\set}[1]{\left\{#1\right\}} \newcommand{\ang}[1]{\left\langle#1\right\rangle} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\N}{\mathbb{N}}$$

# Math 142B: Introduction to Analysis II

## Lecture B (Charlesworth)

### Announcements

• Final exams are now visible on Gradescope. I will be holding special office hours for issues related to the grading of the final exam or the course during the first two weeks of next quarter, on Friday April 6th from 1:30 PM - 2:30 PM, and Tuesday April 10th from 1:30 PM - 2:30 PM. You must email me in advance carefully explaining what the issue is.
• Below are posted some graphs of the first few partial sums of the Weierstrass function $f(x) = \displaystyle\sum_{k=0}^\infty \paren{\frac34}^k\sin\paren{9^k\pi x}$.
• The sixth homework assignment has been posted, and is due on March 14th. The last half of Problem 5 has been updated.
• Results from the second midterm are now posted on Gradescope. If there has been an error in grading your midterm, please email your TA no later than Monday March 5th at 2:00PM, with a clear description of the error that was made. Out of a possible 81, the median score was 33, the mean 33.02, and the standard deviation was 10.33. Grades on the midterm will be curved; while I haven't finalized the method of curving yet, it is likely to be something along the lines of adding 16 points to all scores.
• Below are posted some graphs of the Bernstein polynomials for $n = 10$ and their approximation to $\sin(2\pi x)$.

### Course Information

• Textbook: The textbook for this course is

Advanced Calculus, 2nd edition, by Patrick M. Fitzpatrick.

This is the textbook available at the UC San Diego bookstore. Assignments will be posted fully on this website rather than referencing problems in the book, and so it should in theory be possible to complete this course using an entirely different text or no text at all; however, attempting to do so is not necessarily a wise choice. Errata for the textbook are available here.

• Exams: There will be two midterm exams and a final exam: midterms will be held during lectures on the dates given below, while the final has been scheduled and this information is available through the usual channels. Note that there will be no make-up dates for the midterms or the final exam.

• Piazza: This course will use Piazza, an online discussion board. It will allow you to post messages (openly or anonymously) and answer posts made by your fellow students, about course content, homework, exams, etc. The instructors will also monitor and post to Piazza regularly. This is the preferred method of contact for questions related to the course, especially if your question is likely to be relevant to other students as well. Note: the same Piazza group is shared with Professor Dewey's section of Math 142B; the courses will have very similar content, but not necessarily identical. Therefore, when asking a question about an assignment, it is particularly important to include the statement of the problem you are asking about.

### Homework

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.

1. Show that if $\mathcal{P_1}, \ldots, \mathcal{P_n}$ are partitions of the interval $[a, b]$, then there is a partition $\mathcal{P}$ of the interval $[a,b]$ which is a refinitement of every $\mathcal{P}_i$; that is, show any finite set of partitions has a common refinement.
2. Suppose $\paren{\mathcal{P_n}}_n$ is an infinite sequence of partitions of $[a, b]$. Must there be a partition which is a refinement of every partition in the sequence? (Make sure to check the definition of "partition" carefully.)
1. Let $f : [0, 2] \to \R$ be defined by $f(t) = t$. Find a partition $\mathcal{P}$ of $[0, 2]$ such that $$U(f, \mathcal{P}) - L(f, \mathcal{P}) \lt \frac{1}{256}.$$
1. Suppose that $f : [a,b] \to \R$ is integrable, and $g : [a,b]\to\R$ is such that $g(x) = f(x)$ for all $x \in [a,b] \setminus\set{x_0}$. Show that $g$ is integrable, and $$\int_a^b f(t)\,dt = \int_a^b g(t)\,dt.$$
2. Using the previous part, show that if $f : [a,b] \to \R$ is integrable and $g : [a,b]\to\R$ is such that $g(x) = f(x)$ for all but finitely many points in $[a, b]$ then $g$ is integrable, and $$\int_a^b f(t)\,dt = \int_a^b g(t)\,dt.$$ (Hint: use induction.)
3. Suppose that $f : [a, b] \to \R$ is integrable, and $g : [a,b] \to \R$ is such that $g(x) = f(x)$ except for at a sequence of points $(x_n)_n$. Must $g$ be integrable? Prove it, or provide a counterexample.
2. Suppose $f : [a,b] \to \R$ is bounded, $c \in (a, b)$, and $f$ is integrable on both $[a, c]$ and $[c, b]$. Show that $f$ is integrable on $[a, b]$.
1. Find bounded functions $f, g : [0, 1] \to \R$ so that $f + g$ is integrable, but $f$ is not.
2. Find bounded functions $f, g : [0, 1] \to (0, \infty)$ so that $fg$ is integrable, but $f$ is not.
3. #### Bonus:

Let the function $f$ be defined as follows: \begin{align*} f : [0,1] &\longrightarrow \R\\ x &\longmapsto \begin{cases}\frac1q & \text{ if } x = \frac{p}{q} \in \Q \text{ with } p, q \in \N, \gcd(p,q) = 1{\color{red},} \text{ and } x \neq 0\\ 0 & \text{ otherwise.} \end{cases} \end{align*} Show that $f$ is integrable, and find its integral.

Notice that $\frac{d}{dt}\int_0^t f(x)\,dx \neq f(t)$.

Assignment 1, due January 19th, 2018.
1. Suppose that $f : [a,b]\to\R$ is integrable. Show that $|f|$ is integrable. (Hint: notice that for any $x, y \in [a,b]$, $|f(x)| - |f(y)| \leq |f(x)-f(y)|$.)
2. Suppose that $f, g : [a,b]\to\R$ are integrable. Show that $\min(f, g)$ and $\max(f,g)$ are integrable. (Hint: there is a reason that this is Part B.)
1. Suppose $f : [a,b]\to[0,\infty)$ is continuous. Show that $\int_a^b f = 0$ if and only if $f(t) = 0$ for all $t \in [a,b]$.
2. 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.)

3. Suppose that $f : \R \to \R$ has a continuous second derivative.
1. Show that $f(t) = f(0) + f'(0)t + \int_0^t (t-x)f''(x)\,dx$ for all $t$. (Hint: compare the second derivatives of both sides.)
2. Show that if $G : \R \to \R$ is defined by $G(t) = \int_0^t(t-x)f(x)\,dx$, then $G''(t) = f(t)$ for all $t \in \R$.
4. 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 $$\frac{d}{dt}\paren{F(t) - \int_a^t f(x)\,dx} = 0.$$

#### Clarification:

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 $$\int_a^b F'(x)\,dx = F(b) - F(a).$$

5. #### Bonus:

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 $\epsilon$: $$S\subseteq \bigcup_{n=1}^\infty (a_n, b_n) \qquad\qquad\text{ and }\qquad\qquad \sum_{n=1}^\infty b_n-a_n \lt \epsilon.$$

1. Suppose that $f : [a, b] \to \R$ is bounded, and that $\set{x \in [a,b] : f \text{ is discontinuous at } x}$ is null. Show that $f$ is integrable.
2. Show that if $S \subset \R$ is countable, then $S$ is null.

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.

Assignment 2, due January 29th, 2018.
1. Let's verify that $e^x$ can be defined with the usual properties that we expect. In this problem, be sure not to use any properties of the logarithm before establishing them. Define the function $L$ as follows: \begin{align*}L : (0, \infty) &\longrightarrow \R\\x&\longmapsto\int_1^x \frac1t\,dt.\end{align*}
1. Explain why $L$ is differentiable.
2. Show that $\displaystyle\lim_{t\to\infty}L(t) = \infty$. (Hint: compare $L(n)$ to the harmonic series $\sum \frac1k$, which we know diverges from 142A.)
3. Show that $L(ab) = L(a) + L(b)$. (A question from the midterm may be useful here.)
4. Conclude that $\displaystyle\lim_{t\to0^+}L(t) = -\infty$.
5. The fact that $L$ is continuous means we know know that its range is $(-\infty, \infty)$. Show that $L$ is invertible. (Hint: there is a very useful criterion from 142A about monotonic functions and invertiblilty; it may be useful to show that $L$ is monotonic.)
6. Define the function $E : \R \to (0, \infty)$ to be the inverse of $L$, so $E(L(x)) = x$ and $L(E(y)) = y$. Show that $E'(x) = E(x)$.
7. Show that for any rational number $q \in \Q$, $E(q) = E(1)^q$. (Hint: use Part C; recall that in 142A we were able to prove that for any $\alpha \gt 0$ and any $x, y \in \R$, $\alpha^{x+y} = \alpha^x\alpha^y$.)
8. Use the fact that for any $\alpha \gt 0$ the function $x \mapsto \alpha^x$ is continuous to explain why $E(x) = E(1)^x$ for all $x \in \R$.
9. Show that the Taylor series for $E$ centred at $0$ converges to $E$ at every point.
10. Conclude that $E(1) = \sum_{j=0}^\infty \frac1{j!}$. This number, $E(1)$, is the familiar constant $e$.
2. Show that if $p$ is a polynomial such that $p(x_0) = 0$, then there is a polynomial $q$ so that $p(x) = (x-x_0)q(x)$. Conclude that a polynomial of degree $n$ has at most $n$ roots.
3. Suppose that $f, g : I \to \R$ have $n+1$ continuous derivatives. Show that they have contact of order $n$ at $x_0 \in I$ if and only if $$\lim_{x\to x_0}\frac{f(x)-g(x)}{(x\color{red}{-x_0})^n} = 0.$$
4. #### Bonus:

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 $x \in (-\color{red}{R}, \color{red}{R})$, $$\int_0^x f(t)\,dt = \sum_{k=1}^\infty f^{(k-1)}(0)\frac{x^k}{k!}.$$

(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.)

Assignment 3, due February 9th, 2018.
1. Use Taylor polynomials to show that for every $n \in \N$ there is a constant $C \gt 0$ so that for every $x \geq 0$, $e^x \gt Cx^n$.
2. Show that for every polynomial $p$, $$\lim_{x\to\infty}\frac{p(x)}{e^x} = 0.$$
1. Suppose that $I, J$ are open intervals, $g : J \to I$, and $f : I \to \R$, so that $f\circ g : J \to \R$.
1. How many derivatives of $f$ and how many of $g$ must exist for $f\circ g$ to be $n$ times differentiable?
2. Let $x_0 \in J$, and find an expression for the $n$-th Taylor polynomial of $f\circ g$ at $x_0$ in terms of Taylor polynomials for $f$ and $g$. Be sure to specify where these polynomials are centred!
2. Recall that (soon) we (will) have shown that the function $$f : x \longmapsto \begin{cases}e^{-\frac1{x^2}} & \text{ if } x \neq 0\\0&\text{ else}\end{cases}$$ is infinitely differentiable and has the property that $f^{(n)}(0) = 0$ for all $n$.
1. Show that if $I$ is an open interval, there is a function $f : \R \to \R$ which is infinitely differentiable so that $f(x) = 0$ when $x \notin I$, and $f(x) \gt 0$ when $x \in I$.
2. Use the above to show that if $J$ is a closed interval and $I$ is an open interval with $J \subset I$, there is a function $g : \R \to \R$ which is infinitely differentiable so that $g(x) = 0$ when $x \notin I$ and $g(x) = 1$ when $x \in J$.
3. We developed the following estimates when examining the Taylor series for logarithm: \begin{align*}\abs{\ln(1+y)-p_n(y)} &\leq \frac{1}{1+y}\frac{\abs{y}^{n+1}}{n+1}&\qquad\qquad&\text{if } -1 \lt y \leq 0,\\ \abs{\ln(1+x)-p_n(x)} &\leq \frac{x^{n+1}}{n+1} &&\text{if } 0 \leq x \leq 1.\end{align*}
1. Suppose $0 \lt x \lt 1$. Notice that we can approximate $\ln(1+x)$ by $p_n(x)$, or by choosing $y$ so that $\frac1{1+x} = 1+y$, we can approximate $\ln(1+x) = -ln(1+y)$ by $-p_n(y)$. Is either approximation better? Which one?
2. How large must $n$ be for the approximation of $\ln(1.01)$ by $p_n(0.01)$ to be accurate to at least 8 digits? Using the above method, how large must $n$ be for the approximation via $-p_n(y)$ to be accurate to at least 8 digits?
4. Show that the Weiestrass Approximation Theorem fails on sets which are not sequentially compact:
1. Suppose $S \subset \R$ is unbounded. Show that $f(x) := e^{x^2}$ cannot be uniformly approximated by polynomials on $S$. (Hint: Problem 1 may be helpful. Be sure not to make assumptions about $S$ which may not be true.)
2. Suppose $S' \subset \R$ is not closed. Find a function $g : S' \to \R$ which cannot be uniformly approximated by polynomials. (Hint: if $S'$ is not closed, there is a limit point of $S'$ which is not contained in $S'$.)
5. #### Bonus:

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.

1. Show that the only functions from $\R$ to $\R$ which can be uniformly approximated by polynomials are the polynomials.
2. 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 $\bar{z} = a-bi$. 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 $|f(z) - \bar{z}| \gt \epsilon$.

Assignment 4, due February 21st, 2018.
1. Suppose that $(a_n)$ is a sequence of real numbers such that for some $N \in \N$ and some $r \lt 1$, $|a_n|^{\frac1n} \leq r$ for all $n \gt N$. Show that $\displaystyle\sum_{n=0}^\infty a_n$ converges.
2. Suppose that $(a_n)$ is a sequence of real numbers such that for some $r \gt 1$, $|a_n|^{\frac1n} \gt r$ for infinitely many $n$. Show that $\displaystyle\sum_{n=0}^\infty a_n$ diverges.
3. Suppose that $(a_n)$ is a sequence of real numbers. Let $R = \displaystyle\limsup_{n\to\infty} |a_n|^{\frac1n} = \lim_{n\to\infty}\paren{\sup\set{|a_k|^{\frac1k} : k \gt n}}$ if the sequence $|a_n|^{\frac1n}$ is bounded, and $R = \infty$ otherwise. Show that the series $\displaystyle\sum_{n=0}^\infty a_nx^n$ converges if $|x| \lt R^{\color{red}{-1}}$ and diverges if $|x| \gt R^{\color{red}{-1}}$. (Hint: recall that for a sequence $(b_n)$, its limit superior is the unique number with the property that for any $\epsilon \gt 0$, $b_n \gt \displaystyle\limsup_{k\to\infty}b_k + \epsilon$ for only finitely many $n$, while $b_n \gt \displaystyle\limsup_{k\to\infty} b_k - \epsilon$ for infinitely many $n$.)
1. Suppose that $(a_n)$ is a sequence of real numbers, that $R = \displaystyle\limsup_{n\to\infty} |a_n|^{\frac1n}$, and $0 \lt r \lt R^{\color{red}{-1}}$. Show that the sequence of functions \begin{align*}f_k : [-r, r] &\longrightarrow \R\\x&\longmapsto \sum_{n=0}^k a_nx^n\end{align*} converges uniformly. (Hint: Show it is uniformly Cauchy.)
2. Suppose that $(f_n)$ is a sequence of functions which converges pointwise to a function $f$. Show that for any finite set of points $\set{x_1, \ldots, x_m}$ and any $\epsilon \gt 0$ there is some $N$ so that $|f_n(x_j) - f(x_j)| \lt \epsilon$ for every $n \gt N$ and every $1 \leq j \leq m$.
3. Suppose $I \subset \R$ is a closed, bounded interval, and $f_n : I \to \R$ is a sequence of monotonic functions which converges pointwise to a function $f$. (Note: a sequence of monotone functions, not a monotone sequence of functions.)
1. Show that $f$ is monotonic.
2. Suppose that $f$ is continuous. Show that the convergence is uniform. (Hint: use uniform continuity to break up $I$ into pieces on which $f$ does not change too much, choose $N$ from Problem 3 on the endpoints of these pieces, and then use monotonicity of $f$ and $f_n$.)
4. Consider the sequence of functions defined as follows: \begin{align*}g_k : (0, 1) &\longrightarrow \R\\x&\longmapsto \frac{1}{\color{red}{k}x+1}.\end{align*} Show that $(g_k)$ converges pointwise (and find its limit), but $(g_k)$ does not converge uniformly. Show further that the convergence is uniform on $(1/4, 3/4)$.
5. #### Bonus:

Does the Bolzano-Weierstrass Theorem hold for pointwise converge of sequences of continuous functions? That is, if $f_n : [0,1]\to\R$ is bounded in the sense that for some $M$ and all $n \in \N$, $x \in [0,1]$ we have $|f_n(x)| \lt M$, does $(f_n)$ have a subsequence which converges pointwise?

Assignment 5, due March 7th, 2018.
1. Suppose that $f_n : S \to \R$ is a sequence of functions which converges uniformly to a function $f$.
1. Show that $f$ is bounded if and only if there are some $N$ and $M$ so that for all $n \gt N$ and $x \in S$, $|f_n(x)| \lt M$.
2. Show that $f$ is unbounded if and only if there is some $N$ so that for all $n \gt N$, $f_n$ is unbounded.
(Notice that we have shown that if $(f_n)$ converges uniformly, there cannot be infinitely many terms which are bounded and infinitely many terms which are unbounded.)
2. Suppose that $f_n : S \to \R$ is a sequence of functions which converge uniformly to a function $f : S \to \R$. Suppose also that $g : \R \to \R$ is continuous.
1. Show that if $g$ is uniformly continuous, then $(g\circ f_n)$ converges to $g\circ f$ uniformly.
2. Show that if $f$ is bounded, then $(g\circ f_n)$ converges to $g\circ f$ uniformly.
3. Show that this may fail if $f$ is unbounded and $g$ is not uniformly continuous.
3. Suppose $f : [0, 1] \to \R$ is continuous. For $r \geq 1$, we define $$\norm{f}_{r} := \paren{\int_0^1 \abs{f(t)}^r\,dt}^{\frac1r}.$$ (Note: it happens to be true that $\norm{\cdot}_r$ obeys the triangle inequality ($\norm{f+g}_r \leq \norm{f}_r + \norm{g}_r$) and a few other nice properties (e.g., Hölder's inequality which states that $\norm{fg}_1 \leq \norm{f}_p\norm{g}_q$ if $1 = \frac1p + \frac1q$). However, you should not use any properties like this that you happen to know unless you prove them; they are not necessary to complete the questions below.)

1. Explain why $\norm{f}_r$ exists when $f$ is continuous.
2. Suppose that $f_n : [0, 1] \to \R$ are continuous functions which converge uniformly to a function $f$. Show that $\displaystyle\lim_{n\to\infty}\norm{f_n}_r = \norm{f}_r$.
3. Find a sequence of continuous functions so that $\displaystyle\lim_{n\to\infty} \norm{f_n}_1 = 0$ but $(f_n)$ does not converge uniformly. (Note: you're only asked to do this for $r = 1$, although a similar approach works for any $r$.)
4. Find a power series expansion of $f(x) = \frac1{(1-x)^3}$. (Hint: start with a power series you know and take derivatives...)
5. Suppose that $f(x) = \displaystyle\sum_{j=0}^\infty \paren{\frac34}^j\sin(9^j \pi x)$. On Monday we will see that this function has the peculiar property that it is continuous everywhere, but nowhere differentiable. You should be able to manipulate this to solve the following problems.
1. Find a function $g: \R \to \R$ which is once differentiable everywhere, and twice differentiable nowhere.
2. Show that if $x^2f(x)$ is differentiable at $0$, but nowhere else.
3. Show that if $h^{(j)}(0) = 0$ for $0 \leq j \leq n$ and $h(x) \neq 0$ when $x \neq 0$, then $\displaystyle\lim_{x\to0}\frac{h(x)f(x)}{x^j} = 0$ for $0 \leq j \leq n$, while $h(x)f(x)$ is not differentiable anywhere except $0$. (This is similar to saying that $h(x)f(x)$ has contact of order $n$ with the constant $0$ function; if $h(x)f(x)$ had $n-1$ derivatives in some neighbourhood of $0$, then this statement would be the same. Since the derivative is only defined at $0$, one can't talk about the second (or higher) derivatives there.)
4. Conclude that there is a function which satisfies the above for all $n \in \N$ and is not differentiable anywhere but $0$. (Hint: recall pathological functions from earlier in the course...)
6. #### Bonus:

Suppose $f : [0, 1] \to \R$ is continuous, and let $\norm\cdot_r$ be defined as in Question 3. Show that $\displaystyle\lim_{r\to\infty}\norm{f}_r = \norm{f}_\infty$.

Assignment 6, due March 14th, 2018.

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 "lah-tech" or "lay-tech"; 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.

### Instructional Staff

 Name Role Office E-mail 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:

• begin the subject line with "Math 142B:"; this ensures that your message will not be caught in a spam filter;
• include your full name and student number; this ensures that we know who is contacting us; and
• state your issue clearly; this ensures that we know what you need help with.

### Class Meetings

 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

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### Syllabus

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 two-quarter 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, 2nd 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/catch-up
Jan 30
Discussion
Jan 31
Mid-term 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/catch-up
Feb 22 Feb 23
Mid-term 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/catch-up
Mar 15 Mar 16
Review/catch-up
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 four-leaf clovers, horseshoes, maneki-neko, 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.

• It is your responsibility to ensure that you do not have a schedule conflict involving the final examination; you should not enroll in this class if you cannot take the final examination at its scheduled time.
• No calculators, phones, or other electronic devices will be allowed during the final examination. You may bring a single piece of paper of size no more than 8.5 inches by 11 inches containing hand-written notes, and at most three four-leaf clovers, horseshoes, maneki-neko, or other such talismans for good luck.
• There will be no makeup final exam.

• Your exams will be graded using Gradescope.  You will be able to request a regrade directly from your TA for a specified window of time.  Be sure to make your request within the specified window of time; no regrade requests will be accepted after the deadline.

• It is your responsibility to check your scores and contact your TA before the end of the 10th week of the quarter to resolve recording errors.
• Questions regarding missing or incorrectly recorded scores will not be considered after the last day of instruction.

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
• Note: The above scale is a lower bound: for example, if your cumulative average is 80, your final grade will be at least B-. However, your instructor may adjust the above scale to be more generous.
Your cumulative average will be the best of the following two weighted averages:
• 20% Homework,  20% Midterm Exam I,  20% Midterm Exam II,  40% Final Exam
• 20% Homework,  20% Best Midterm Exam,  60% Final Exam

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 zero-tolerance 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.

• Entering/exiting class: Please arrive on time and stay for the entire class/section period. If, despite your best efforts, you arrive late, please enter quietly through the rear door and take a seat near where you entered. Similarly, in the rare event that you must leave early (e.g. for a medical appointment), please sit close to the rear exit and leave as unobtrusively as possible.
• Noise and common courtesy: When class/section begins, please stop your conversations. Wait until class/section is over before putting your materials away in your backpack, standing up, or talking to friends. Do not disturb others by engaging in disruptive behavior. Disruption interferes with the learning environment and impairs the ability of others to focus, participate, and engage.
• Electronic devices: Please do not use devices (such as cell phones, laptops, tablets, iPods) for non-class-related matters while in class/section. No visual or audio recording is allowed in class/section without prior permission of the instructor (whether by camera, cell phone, or other means).
• E-mail etiquette: You are expected to write as you would in any professional correspondence. E-mail communication should be courteous and respectful in manner and tone. Please do not send e-mails that are curt or demanding.

#### Accommodations:

Students requesting accommodations for this course due to a disability must provide a current Authorization for Accommodation (AFA) letter issued by the Office for Students with Disabilities (OSD) which is located in University Center 202 behind Center Hall. Students are required to present their AFA letters to Faculty (please make arrangements to contact me privately) and to the OSD Liaison in the department in advance (by the end of Week 2, if possible) so that accommodations may be arranged. For more information, see here.

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### Resources

Here are some additional resources for this course, and math courses in general.

• UCSD Podcasts
• The lectures for this course will be podcast; you will be able to find the audio podcasts at the UCSD podcast site.
• Sites with resources for assistance and tutoring
• OASIS. The Office of Academic Support & Instructional Services (OASIS) offers weekly workshops to support many undergraduate classes. For more information, see the OASIS webpage.
• Calculus Tutoring  Links to information regarding the Calculus Tutoring Lab, OASIS, and private tutors
• Wolfram|Alpha  A powerful computational knowledge engine
• Interesting sites relating to Mathematics (If you see a Math site you like that's not here, email me the URL.)

### Lecture Notes

Any remarks pertaining to particular lectures will be posted here throughout the term.

Graphs of Bernstein polynomials.

Below is a graph of the functions $f_k(x) = \binom{10}{k}x^k(1-x)^{10-k}$ 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 $\sin(2\pi x)$.

Graphs of partial sums of the Weierstrass function. Below are graphs of the functions $f_n(x) = \displaystyle\sum_{k=0}^n \paren{\frac34}^k\sin\paren{9^k\pi x}$, with $n = 0, 1, 2, 3$, and $10$.
CSS and page template greatfully taken from Todd Kemp's earlier offering of a different course.