Back

Finals Exam Prep

Jerich Lee

December 16, 2024

So I finally finished all Final Exam practice problems that my professor gave to me. 36 of them! (Actually 32—one of them was a repeat and a couple of them were prev hw assignments) It is currently Sunday, and my final is on Tuesday. Let's go. I'm going to walkthrough all of my solutions, then analyze them with the solutions provided by my prof. Let's run it.
Problem.
[1 (Cauchy Concentration Test)] Suppose $a(1) \geq{} a(2) \geq{} \dots{} > 0$. Prove that $$\begin{align} \sum_{n} a(n) \quad \text{and} \quad \sum_{n} 2^n a(2^n) \end{align}$$ converge or diverge simultaneously. Hint. Let $s(m) = \sum_{k=1}^m a(k)$. Prove first that $$\begin{align} \frac{1}{2} \sum_{j=1}^n 2^j a(2^j) \leq s(2^n) \leq a(2^n) + \sum_{j=0}^n 2^j a(2^j). \end{align}$$ This can be achieved by grouping $a_k$'s into blocks of length $2^0, 2^1, 2^2, \dots{}$. (b) Use the result of (a) to show that $\sum_n{} \frac{1}{n^p}$ converges if and only if $p > 1$.
Solution.
Problem.
[2]
  1. State the definition of a compact set
  2. Give an example of an open cover of $\mathbb{{R}}$ which has no finite subcover
  3. Consider the set $E \subset \mathbb{{R}}^{5}$, consisting of all vectors $\vec{x} =\left( x_1, \ldots, x_5 \right) $ for which $$\begin{align} \sum_{k=1}^{5} \left\vert x_k \right\vert \leq 1 \end{align}$$ Is $E$ compact?
Solution.
Problem.
[3]
  1. Use Arithmetic-Geometric means inequality to prove that, for every $n$,
    1. $\left( 1+\frac{1}{n} \right)^{n}\leq \left( 1+ \frac{1}{n+1} \right)^{n+1} $
    2. $\left( 1+\frac{1}{n} \right)^{n+1}\geq \left( 1+\frac{1}{n+1} \right)^{n+2} $
  2. Conclude that the sequence $\left( \left( 1+\frac{1}{n} \right)^{n} \right)_n $ converges.
Solution.
Problem.
[4] Prove that the space $C(S)$ (the space of bounded continuous functions on a set $S$, with $d(f,g)=\sup_{x\in S}\vert f(x)-f(y) \vert $) is complete.
Solution.
Problem.
[5] Prove that $\vert \sin{} x - \sin{} y \vert > \frac{\left\vert x-y \right\vert}{2} $ for distinct $-\frac{\pi}{3}\leq{} x,y \leq{} \frac{\pi}{3} $.
Solution.
Problem.
[6] Prove that the function $f: \mathbb{{R}}\to{} \mathbb{{R}}: x\mapsto{} \frac{x}{x^{2}+1}$ is Lipschitz.
Solution.
Problem.
[7] Suppose the function $f$ is continuous on the interval $[1,9]$, differentiable in its interior, and satisfies $f(1)=3, f(4)=0$, and $f(9)=10$. Prove that there exists $c\in{} (1,9)$, such that $f^\prime{} (c)=1$.
Solution.
Problem.
[8] Prove that the set of all real numbers of the form $a+b\sqrt{5} $ (where $a,b \in{} \mathbb{{Q}}$) is a field.
Solution.
Problem.
[9] Suppose $(a_{n})$ is a bounded sequence of real numbers. Denote by $A$ the set of its subsequential limits. Prove that $A$ is a closed subset of $\mathbb{{R}}$.
Solution.
Problem.
[10] Suppose $f:\mathbb{{R}}\to{} \mathbb{{R}}$ is differentiable everywhere, and $\lim_{t \to 0} f^\prime{} (t)$ exists. Prove that $f^\prime{}$ is continuous at $0$.
Solution. So for this problem, I appealed to the Sequential Criterion for continuity (Theorem 17.1 + 17.2): $$\begin{align} f:S\to S^{*} \text{ is continuous at } x\in S \iff f(x_{n})\to f(x) \text{ whenever } x_{n}\to x. \label{seqcon} \end{align}$$ The question was labeled Difficult Problem, which always kind of confuses bc sometimes these labeled problems can be impossible but else they are actually not bad. This problem (I think) is of the latter...
Proof. We know that $f^\prime{} (0)$ is defined on $f^\prime{} $, as $f$is diff'able everywhere. WTF: $\lim_{n \to 0}f^\prime{} (x)=f^\prime{} (0) $ We know that $\lim_{x \to 0}f^\prime{} (x) $ exists. So, we want to show that it equals $f^\prime{} (0)$ by \autoref{seqcon}. $$\begin{align} \lim_{x \to 0} f^\prime (x)=L \end{align}$$ We know that the above implies: $$\begin{align} \lim_{n \to \infty} f^\prime (a_{n})=L \ s.t. \ \lim_{n \to \infty} a_{n}\to 0 \\[10pt] \left\vert f^\prime (a_{n})-L \right\vert <\varepsilon \\[10pt] \frac{f^\prime (a_{n})-f^\prime (0)}{a_{n}}=L \\[10pt] f^\prime (0) = -La_{n}+f^\prime (a_{n}) \\[10pt] f^\prime (a_{n}) = La_{n}+f^\prime (0) \\[10pt] \left\vert La_{n}+f^\prime (0)-L \right\vert <\varepsilon \\[10pt] \left\vert L(a_{n}-1)+f^\prime (0) \right\vert < \varepsilon \\[10pt] \left\vert f^\prime (0) -L \right\vert < \varepsilon \implies f^\prime (0) = L \label{y} \end{align}$$ By \autoref{y}, $$\begin{align} \lim_{x \to 0} f^\prime (x)=f^\prime (0) \end{align}$$.
∎

Problem.
[11] Suppose $f: \mathbb{{R}}\to{} \mathbb{{R}}$ is continuous and strictly decreasing, with $f(2)=3$ and $f^\prime{} (2)=-5$. Find $g^\prime{} (3)$, where $g=f^{-1}$ (the inverse function of $f$).
Solution. $f:\mathbb{{R}}\to{} \mathbb{{R}}$ is cont., strictly decreasing, $f(2)=3, f^\prime{} (2)=-5$. By the Derivative of an Inverse Function, $$\begin{align} g^\prime (d)&=\frac{1}{f^\prime (c)}= \frac{1}{f^\prime (g(d))} \label{in} \\[10pt] g^\prime (3) &= \boxed{\frac{1}{-5}} \end{align}$$
Problem.
[12] Compute $\lim_{n \to \infty} \sum_{j=1}^{n} \frac{1}{n+5j}$.
Solution.
Problem.
[13] Suppose the functions $f$ and $g$ are integrable on $[a,b]$.
  1. Prove that the inequalities $\int_{a}^{b} (tf+g)^{2} \geq 0$ and $\int_{a}^{b} (tf-g)^{2} \geq 0$ holds for any $t$.
  2. Prove that, for any $t>0$, we have $2\left\vert \int_{a}^{b} fg \right\vert \leq t\int_{a}^{b} f^{2} +\frac{1}{t}\int_{a}^{b} g^{2}$. \label{bun}
  3. Show that, if $\int_{a}^{b} f^{2} = 0$, then $\int_{a}^{b} fg=0 $.
  4. Prove Bunyakovsky-Cauchy-Schwarz Inequality for Integrals: $$\begin{align} \left( \int_{a}^{b} fg \right)^{2} \leq \left( \int_{a}^{b} \left\vert fg \right\vert \right)^{2} \leq \left( \int_{a}^{b} f^{2} \right) \cdot \left( \int_{a}^{b} g^{2} \right) \label{bunya} \end{align}$$
Solution. This one was a monster. Prob the hardest problem in the entire pset. Lots of duh moments.
  1. By the Linearity and Comparison of Integrals, we know that $$\begin{align} g\leq f \implies \int_{a}^{b} g \leq \int_{a}^{b} f \label{linint} \end{align}$$ Let $g=0$ and $f = (tf+g)^{2}$. We know that $f\geq g$, as the square of a integrand is always positive. By \autoref{linint}, $\int_{a}^{b} (tf+g)^{2} \geq 0$ and $\int_{a}^{b} (tf-g)^{2}\geq 0 $.
  2. Using the result of part 1: $$\begin{align} \int_{a}^{b} (tf+g)^{2} \geq 0\\[10pt] &= \int_{a}^{b} t^{2}f^{2} + 2tfg + g^{2} \\[10pt] &= \int_{a}^{b} t^{2}f^{2} + 2\int_{a}^{b} tfg + \int_{a}^{b} g \geq 0 \\[10pt] &= t^{2}\int_{a}^{b} f^{2} + 2t\int_{a}^{b} fg + \int_{a}^{b} g^{2} \\[10pt] &= -2\int_{a}^{b} fg \leq t\int_{a}^{b} f^{2} + \frac{1}{t}\int_{a}^{b} g^{2} \end{align}$$ The other case, $\int_{a}^{b} (tf-g)^{2}\geq $ is handled similarly to obtain the absolute value of $2\left\vert \int_{a}^{b} fg \right\vert $.
  3. Proof. Suppose by contradiction, $fg=y>0$ for some $x\in{}[a,b]$. By continuity of the interval of integrable functions, $\exists{} $ interval $[c,d] \subset{} [a,b]\ s.t. \ f\geq{} \frac{y}{2}$ on $[c,d]$. Then, $$\begin{align} \int_{a}^{b} f^{2} = \int_{a}^{b} f^{2}+\int_{c}^{d} f^{2}+\int_{d}^{b} f^{2}\geq \frac{y}{2}(d-c)>0 \end{align}$$ Which is a contradiction.
    ∎

  4. Proof. $$\begin{align} 2\left\vert \int_{a}^{b} fg \,\mathrm{d}x \right\vert^{2}&< \left( t\int_{a}^{b} f^{2} +\frac{1}{t}\int_{a}^{b} g^{2} \right)\left( t\int_{a}^{b} f^{2} +\frac{1}{t}\int_{a}^{b} g^{2} \right) \\[10pt] &\leq t^{2}\underbrace{\left( \int_{a}^{b} f^{2} \right)^{2} }_{\alpha}+2\left( \int_{a}^{b} f^{2} \right)\left( \int_{a}^{b} g^{2} \right) + \frac{1}{t^{2}}\underbrace{\left( \int_{a}^{b} g^{2} \right)^{2} }_{\beta} \\[10pt] \end{align}$$ We wish to minimize the function $$\begin{align} h(t) &= \alpha t + \frac{\beta}{t} \\[10pt] h^\prime (t) &= \alpha - \frac{\beta}{t^{2}} = 0 \implies \alpha=\frac{\beta}{t^{2}} \\[10pt] t^{2}=\frac{\beta}{2}\\[10pt] t = \left( \frac{\beta}{\alpha} \right)^{2} \label{t} \end{align}$$ Subbing \autoref{t} into \autoref{bun}, we achieve \autoref{bunya}.
    ∎

Problem.
[14] Recall that the metric $d$ on $\mathbb{{R}}^{n}$ is defined as follows: for $\vec{a}, \vec{b}\in{} \mathbb{{R}}^{n}, d(\vec{a},\vec{b})=\lVert \vec{a} -\vec{b} \rVert $, where, for $\vec{c}=(c_1, \ldots{} , c_{n}),\lVert \vec{c} \rVert =( \sum_{i} \vert c_i \vert )^{\frac{1}{2}} $. Suppose $\vec{x}, \vec{y} \in{} \mathbb{{R}}^{n}$. Prove that the function $\phi{}:\mathbb{{R}}\to{} \mathbb{{R}}:t \mapsto{} \lVert \vec{x} +\vec{ty} \rVert $ is convex.
Solution.
Proof. $$\begin{align} \phi\left( \frac{t+s}{2} \right) &\leq \frac{\phi(t)+\phi(2)}{2}\\[10pt] \left\lVert \vec{x_i}+t \vec{y_i} \right\rVert &\leq \left\lVert \vec{x_{i}}\right\rVert + \left\lVert \vec{ty_{i}}\right\rVert \\[10pt] &= \left\lVert \vec{x_{i}}\right\rVert + \left\lVert \vec{t}\right\rVert \cdot\left\lVert \vec{y_{i}}\right\rVert \\[10pt] c&= \frac{t+s}{2}\\[10pt] \phi\left( c \right) &= \left\lVert \vec{v} +c \vec{y} \right\rVert \\[10pt] &= \left\lVert \vec{x} + \left( \frac{t+s}{2} \right) \vec{y} \right\rVert \\[10pt] &= \left\lVert \vec{x} +\frac{t}{2}\vec{y} +\frac{s}{2}\vec{y} \right\rVert \\[10pt] &\leq \left\lVert \vec{x} \right\rVert + \left\lVert \frac{t}{2}\vec{y} \right\rVert + \left\lVert \frac{s}{2}\vec{y} \right\rVert \\[10pt] &= \left\lVert \vec{x} \right\rVert + \frac{t}{2}\left\lVert \vec{y} \right\rVert + \frac{s}{2}\left\lVert \vec{y} \right\rVert \label{yuh} \end{align}$$ WTF: $$\begin{align} \frac{\phi(t)+\phi(s)}{2}&=\frac{\left\lVert \vec{x} +t\vec{y} \right\rVert + \left\lVert \vec{x} +s \vec{y} \right\rVert}{2} \\[10pt] &\leq \frac{\left\lVert \vec{x} \right\rVert}{2}+\frac{\left\lVert \vec{x} \right\rVert}{2}+\frac{\left\lVert t \vec{y} \right\rVert}{2} + \frac{\left\lVert s \vec{y} \right\rVert}{2} \\[10pt] &= \left\lVert \vec{x} \right\rVert + \frac{t+s}{2}\left\lVert \vec{y} \right\rVert \label{huh} \end{align}$$ \autoref{yuh}= \autoref{huh}.
∎

Problem.
[15] Suppose $E$ is a compact subset of a metric space $(S,d)$.
  1. Prove that any $x\in S$ has a \emph{nearest point} in $E$—i.e.,the point $y$ such that $d(x,y)=\inf \left\{ d(x,s):s\in E \right\} $.
  2. Give an example of $x$ and $E$ where the nearest point is not unique—i.e.,there exists distinct $y,z\in E$ such that $d(x,y)=d(x,z)=\inf \left\{ d(x,s) : s\in E \right\} $.
  3. Prove that, if $S\in \mathbb{{R}}^{n}$ with its usual Euclidean metric, and $E \subset S$ is compact and convex, then, for any $x\in S$, the point nearest to it in $E$ is unique.
Solution. Why is $E\subset{} \mathbb{{R}}^{2}$ $E=\{ (u,v):u^{2}+v^{2}=1 \} $
Problem.
[16] The function $g: \mathbb{{R}}\to{} \mathbb{{R}}$ is defined by setting $$\begin{align} g(x) = \begin{cases} x^{3}, &\text{ if } x\in \mathbb{{Q}};\\ -x^{3}, &\text{ if } x \neq \mathbb{{Q}}. \end{cases} \end{align}$$ Compute the derivative $g^\prime{} $ at all points where it exists.
Solution.
Problem.
[17] Compute $\lim_{n} \int_{0}^{1} e^{-nt^{2}} \,\mathrm{d}t $
Solution.
Problem.
[18] Suppose $f$ and $g$ are continuous functions on $[0,1]$, which coincide at all rational points. Prove that $f=g$ everywhere.
Solution.
Problem.
[19] Compute the following integrals:
  1. $\int_{0}^{1} t\sqrt{1+t^{2}} \,\mathrm{d}t $
  2. $\int_{0}^{1} t \sin t \,\mathrm{d}t $
Solution.
Problem.
[20]
  1. Give an example fo a continuous map $f: S\to S^{*}$ such that there exists an open set $U \subset S$ with the property that $f(U)$ is not open in $S^{*}$.
  2. Suppose $(S,d)$ is a compact metric space, and $f:S\to S^{*}$ is continuous and bijective. Prove that, for every open $U \subset S$, $f(U)$ is open.
Solution.
Problem.
[21] Determine whether the following series converge:
Solution.
  1. $\sum_{k=1}^{\infty} \frac{k+\sqrt{k} \sin k}{k^{3}+1}$
  2. $\sum_{k=1}^{\infty} \frac{(-4)^{k}}{k^{4}}$
  3. $\sum_{k=1}^{\infty} \frac{2k^{2}-1}{k^{3}+1}$
Problem.
[22] The sequence $(a_{n})$ is defined by $a_1=5, a_{n+1}=\sqrt{2a_{n}+3} $ for $n\geq{} 1$. Determine whether this sequence converges. If it does, find its limit.
Solution.
Problem.
[23] Define the function $$\begin{align} f(x) = \begin{cases} x\sin(\frac{1}{x}), &\text{ if } x\neq 0;\\ 0, &\text{ if } x=0. \end{cases} \end{align}$$
  1. Prove that $f$ is not differentiable at $0$
  2. Prove that there is not continuous function $g:[-1,1]\to \mathbb{{R}}$ satisfying $\int_{0}^{x} g(t) \,\mathrm{d}t =f(x)$ for any $x\in [-1,1]$.
Solution.
Problem.
[24 (Alternating Series Theorem)] Suppose $a_1\geq{} a_2\geq{} \ldots{} \geq{} 0 $. Prove that $\sum_{k=1}^{\infty{}} (-1)^{k-1}a_{k}=a_{1}-a_{2}+a_3 - \ldots{} $ converges iff $\lim_{k}a_{k}=0 $.
Solution.
Problem.
[25] Denote by $\mathbb{{T}}$ the unit circle in $\mathbb{{R}}^{2}$—i.e.,$\mathbb{{T}}=\{ (x,y)\in{} \mathbb{{R}}^{2}:x^{2}+y^{2}=1 \} $. Consider $f:[0,2\pi{}]\to{} \mathbb{{R}}^{2}: t\mapsto{} (\cos{} t, \sin{} t)$. Prove that
  1. $f$ is continuous
  2. $f([0,2\pi)]=\mathbb{{T}}$
  3. $f^{-1}:\mathbb{{T}}\to [0,2\pi)$ is not continuous
Solution.
Problem.
[26] Suppose $S$ is a non-empty metric space. Prove that $S$ is connected iff it has exactly two subsets which are both open and closed—$\varnothing{} $ and itself.
Solution. $\sum_{k=1}^{n} (k-1)$
Problem.
[22.4] Consider the following subset of $\mathbb{R}^2$: $$\begin{align} E = \left\{ \left( x, \sin\frac{1}{x} \right) : x \in (0, 1] \right\}; \end{align}$$
  1. Determine its closure $ E^- $. See Fig. 19.4.
  2. Show $ E^- $ is connected.
  3. Show $ E^- $ is not path-connected.
$E$ is simply the graph of $g(x) = \sin{}\frac{1}{x}$ along the interval $(0, 1]$.
Solution.
Problem.
[26.5] Let $f(x) = \sum_{n=0}^\infty{} \frac{1}{n!} x^n $ for $x \in{} \mathbb{R}$ .
  1. Show $ f'(x) = f(x) $.
  2. Do \textbf{not} use the fact that $ f(x) = e^x $; this is true but has not been established at this point in the text.
Solution.
Problem.
[26.6] Let $s(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots{}$ and $c(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots{}$ for $x \in{} \mathbb{R}$ .
  1. Prove $ s'(x) = c(x) $ and $ c'(x) = -s(x) $.
  2. Prove $ (s^2(x) + c^2(x))' = 0 $.
  3. Prove $ s^2(x) + c^2(x) = 1 $.
Solution.
Problem.
[29.5] Let $f$ be defined on $\mathbb{R}$, and suppose $|f(x) - f(y)| \leq{} (x - y)^2$ for all $x, y \in{} \mathbb{R}$. $$\begin{align} \end{align}$$
  1. Prove that $ f $ is a constant function.
Solution.
Problem.
[32.2] Let $$\begin{align} f(x)\begin{cases} x & \text{for rational } x, \\ 0 & \text{for irrational } x. \end{cases} \end{align}$$
  1. Calculate the upper and lower Darboux integrals for $ f $ on the interval $[0, b]$.
  2. Is $ f $ integrable on $[0, b]$?
Solution.
Problem.
[33.5] Show $$\begin{align} \left| \int_{-2\pi}^{2\pi} x^2 \sin^8(e^x) \, dx \right| \leq \frac{16\pi^3}{3}. \end{align}$$
Solution.
Problem.
[34.2]
  1. Calculate $$\begin{align} \lim_{x \to 0} \frac{1}{x} \int_{0}^{x} e^{t^2} \, dt. \end{align}$$
  2. Calculate $$\begin{align} \lim_{h \to 0} \frac{1}{h} \int_{3}^{3+h} e^{t^2} \, dt. \end{align}$$
Solution.
Problem.
[34.5] Let $f$ be a continuous function on $\mathbb{R}$ and define $$\begin{align} F(x) = \int_{x-1}^{x+1} f(t) \, dt \quad \text{for } x \in \mathbb{R}. \end{align}$$
  1. Show that $ F $ is differentiable on $ \mathbb{R} $.
  2. Compute $ F' $.
Solution.
Problem.
[38.3] Show that there is a differentiable function on $\mathbb{R}$ whose derivative is nowhere differentiable.
Solution.