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\begin{center}
{\LARGE Math 10860: Honors Calculus II, Spring 2021 \\ Homework 3}
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\begin{enumerate}
\item Some questions on uniform continuity.
\begin{enumerate}
\item Recall that we argued in class that the function $f\colon (0,1] \rightarrow \R$ given by $f(x)=1/x$ is continuous but not uniformly continuous, and we further argued that the issue was what was happening near $0$ (the function is ``blowing up'', with unboundedly increasing slope). Find a function $f\colon (0,1] \rightarrow \R$ that is continuous but not uniformly continuous, {\em and is bounded on $(0,1]$}.
\item Show that if $f, g\colon A\rightarrow \R$ are both uniformly continuous on $A$ (some interval in $\R$), {\em and both bounded}, then $fg$ is uniformly continuous on $A$.
\item Give an example of an interval $A$, and functions $f, g\colon A\rightarrow \R$ that are both uniformly continuous on $A$, with $f$ {\em not} bounded on $A$, $g$ bounded on $A$, such that $fg$ is not uniformly continuous on $A$.
\end{enumerate}
\item Consider the function $f\colon [0,2] \rightarrow \R$ defined by
\[f(x) = \begin{cases}
0 & \text{if $x \neq 1$},\\
1 & \text{if $x = 1$}.
\end{cases}\]
Prove that there does not exist a function $g\colon [0,2] \rightarrow \R$ with the property that $g'=f$.
\item Find the derivatives of the following functions.
\begin{enumerate}
\item $F(x) = \int_a^{x^3} \sin^3 t~dt$
\item $F(x) = \int_x^{15} \left(\int_8^y \frac{dt}{1+t^2+\sin t}\right)~dy$
\item $F(x) = \int_a^b \frac{x~dt}{1+t^2 + \sin^2t}$
\end{enumerate}
\item For each of the following functions $f$, consider $F(x) = \int_0^x f$, and determine at which points $x$ is $F'(x)=f(x)$. Caution: there may be some $x$ for which $F'(x)=f(x)$ even though the hypotheses of the obvious theorem do not apply.
\begin{enumerate}
\item $f(x) = \begin{cases} 0 & \text{if $x \leq 1$}, \\ 1 & \text{if $x > 1$}. \end{cases}$
\item $f(x) = \begin{cases} 0 & \text{if $x \neq 1$}, \\ 1 & \text{if $x = 1$}. \end{cases}$
\item $f(x) = \begin{cases} 0 & \text{if $x \leq 0$}, \\ x & \text{if $x \geq 0$}. \end{cases}$
\end{enumerate}
\item Let $f$ be integrable on $[a,b]$, let $c$ be in $(a,b)$ and let
\[F(x) = \int_a^x f \quad \quad (a \leq x \leq b).\]
For each of the following statements, either give a proof or a counter-example.
\begin{enumerate}
\item If $f$ is differentiable at $c$ then $F$ is differentiable at $c$.
\item If $f$ is differentiable at $c$ then $F'$ is continuous at $c$.
\item If $f'$ is continuous at $c$, then $F'$ is continuous at $c$.
\end{enumerate}
\item Two unrelated, but hopefully quick, parts.
\begin{enumerate}
\item
Show that, as $x$ ranges over the interval $(0,\infty)$, the value of the following expression does not depend on $x$:
\[\int_0^x \frac{dt}{1+t^2} + \int_0^{1/x} \frac{dt}{1+t^2},\]
and then (using this fact, or otherwise) deduce that
\[\int_0^1 \frac{dt}{1+t^2} = \int_1^\infty \frac{dt}{1+t^2}.\]
\item
Find $F'(x)$ if $F(x) = \int_0^x xf(t)~dt$. {\bf Hint}: the answer is {\em not} $xf(x)$.
\end{enumerate}
\item Define $F(x) = \int_1^x \frac{dt}{t}$ and $G(x) = \int_b^{bx} \frac{dt}{t}$ (for $b \geq 1$).
\begin{enumerate}
\item Find $F'(x)$ and $G'(x)$.
\item Use the result of the last part to prove that for $a,b \geq 1$,
\[\int_1^a \frac{dt}{t} + \int_1^b \frac{dt}{t} = \int_1^{ab} \frac{dt}{t}.\]
\end{enumerate}
\item Prove that if $h$ is continuous, $f$ and $g$ are differentiable, and
\[F(x) = \int_{f(x)}^{g(x)} h(t)~dt\]
then
\[F'(x) = h(g(x))g'(x) - h(f(x))f'(x).\]
\end{enumerate}
\begin{itemize}
\item {\bf An extra credit problem}: Let $I$, $J$ and $K$ be intervals. Suppose that $g\colon I\rightarrow J$ and $f\colon J\rightarrow K$ are both integrable ($f$ on $J$ and $g$ on $I$). What can you say about the composition function $f \circ g\colon I \rightarrow K$?. Note that it will be one of three things: exactly one of
\begin{description}
\item[A] $f\circ g$ is integrable (on $I$)
\item[B] $f \circ g$ is not integrable
\item[C] $f \circ g$ is sometimes integrable, sometimes not, depending on the specific choices of $f$ and $g$
\end{description}
is true. Which one? If {\bf A} or {\bf B}, give a proof; if {\bf C}, give examples to show that both behaviors are possible.
\end{itemize}
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