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\title{Math 3110: Homework 10}
%\author{Zach Norwood}
\date{\textcolor{navy}{Due Wednesday, 24 April 2019}}

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\begin{document}
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\noindent
You will be graded on both the accuracy and the clarity of your solutions.
One purpose of the homework is to give you an opportunity to practice your proof-writing skills.

You are welcome\,---\,encouraged, even!\,---\,to collaborate on homework, but you should not 
copy solutions from any source, nor should you submit anything that you don't understand.

\addtocounter{problem}{-1}

\begin {problem} [don't submit any of this for a grade]
 Write out careful proofs for all the properties of integrable functions that we proved in class
 and make sure you understand how the proofs work.
 We didn't always give all the details, so make sure that you can fill in the blanks.
\end {problem}

\begin {problem}
 Suppose that $c\in [a,b]$ and that $f\colon[a,b]\to\R$ is a bounded function.
 Prove that, for any partition $P$ of $[a,b]$ such that $c\in P$, the lower sums and upper sums
 decompose as follows:
 \[u(f,P) = u(f\restr [a,c], P\cap[a,c]) + u(f\restr[c,b], P\cap [c,b])\]
 and
 \[l(f,P) = l(f\restr [a,c], P\cap[a,c]) + l(f\restr[c,b], P\cap [c,b]).\]
 (We made extensive use of this fact in our proof of ``$f$ integrable iff $f\restr[a,c]$ and $f\restr[c,b]$ each
 integrable.'')\footnote{To simplify notation, start by defining $f_1 = f\restr[a,c]$, $f_2 = f\restr[c,b]$,
 $P_1 = P\cap[a,c]$, and $P_2 = P\cap[c,b]$.}
\end {problem}

\begin {problem}
 Use the second part of the Fundamental Theorem of Calculus to prove the first part
 for continuous functions $f$.
 (Obviously you should not trivialize the problem by using the first part of the Fundamental Theorem.)
\end {problem}

\begin {problem}
 Prove that the set $\set{ \int_1^b \frac{1}{x^2}\,dx }{ b\in [1,\infty)}$ is bounded above
 and compute its supremum, which we might sensibly denote $\int_1^\infty \frac{1}{x^2}\,dx$.
\end {problem}

\begin {problem}
 Prove that if $f\colon[a,b]\to\R$ is increasing on $[a,b]$, then $f$ is integrable.
 Deduce that decreasing functions are integrable too.
\end {problem}

\begin {problem}
 Define $f\colon[0,1]\to\R$ by
 \[
  f(x) = \begin {cases}
  1	&\text{ if } x = 1/n \text{ for some } n\in\N \\
  0	&\text{ otherwise}
 	\end{cases}.
 \]
 Prove that $f$ is integrable and compute $\int_0^1 f$.
 (Notice that this gives us an example of an integrable function with infinitely many discontinuities.)
\end {problem}

\begin {problem}
 The \emph{average value} of an integrable function $f$ on an interval $[a,b]$ is defined to be
 \[
  \av(f) = \frac{1}{b-a} \int_a^b f.
 \]
 \begin{enumerate} [(a)]
  \item[($\star$)] Meditate on why this is a good definition of average value.
  \item Prove that if $f$ is the derivative of another function $F$, then $\av(f)$ is the 
  average rate of change of $F$ over $[a,b]$.
  \item
  Prove that
  \[
   \int_a^b \av(f) = \int_a^b f.
  \]
  \item 
  Suppose that $m\le f(x) \le M$ for all $x\in[a,b]$. Prove that $m\le \av(f) \le M$.
  \item
  Prove that if $f$ is continuous, then $f$ achieves its average value:
  precisely, there is some $c\in[a,b]$ such that $f(c) = \av(f)$.
  \item
  Give an example to show that the previous part cannot be extended to all discontinuous $f$.
 \end{enumerate}
\end {problem}

\begin {problem}[essentially 7.5.8 of the textbook]
 Define (!) a function $L\colon (0,\infty)\to\R$ by
 \[
  L(x) = \int_1^x \frac{1}{t} \,dt.
 \]
 \begin {enumerate} [(a)]
  \item What is $L(1)$? Explain why $L$ is differentiable and compute its derivative.
  \item Prove the identity $L(xy) = L(x) + L(y)$. (\emph{Hint}: For a fixed $y$, differentiate $x\mapsto L(xy)$.)
  \item Show that $L(x/y) = L(x) - L(y)$.
  \item Define
  \[
   \textstyle\gamma_n = \left( 1 + \frac12 + \frac13 + \cdots + \frac1n \right) - L(n).
  \]
  Prove that the sequence $(\gamma_n)_{n\ge1}$ converges. The constant $e = \lim \gamma_n$ is called
  \emph{Euler's constant}.
  \item
  By considering the sequence $\gamma_{2n} - \gamma_n$ or otherwise,
  show that $L(2)$ is the limit of the alternating harmonic series.
  \item[(bonus)]
  Prove\footnote{You should turn bonus problems in to me (Zach),
  not to Gradescope.}
  that $L(x^y) = yL(x)$ for all $x\in(0,\infty)$, $y\in\R$.
  Deduce that $L(e^x) = x$ for all $x\in\R$.
  Deduce also that $\frac{d}{dx} e^x = e^x$.
 \end {enumerate}
\end {problem}


\end {document}
