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\author{Math 25}

\title{Homework 6 --- due Oct 31st}

\begin{document}

\maketitle
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There are two sections: A and B.
Please return each part
to the proper CA.
Remember that you are \emph{not} allowed to use a calculator nor a
computer (in problem \ref{lod} for example!)

\renewcommand{\thesection}{\Alph{section}}

\section{Problems graded by Benjamin}

\subsection{Filling holes} Let $f$, $g$ be two continuous functions on the interval
$[a,b]$ and fix $c \in (a,b)$. Suppose that $f$ is $C^1$ on $[a,b]
\setminus \{c\}$ and that $f'=g$ on $[a,b] \setminus \{c\}$. 

\begin{enumerate}
\item Prove
that $f$ is differentiable at $c$ and that $f'=g$ on $[a,b]$. Hint:
consider the function $x \mapsto f(x)-\int_c^x g(t)dt$.
\item Prove that the function $f$ defined by $f(0)=1$ and
  $f(x)=x/\sin(x)$ is $C^1$ on $[-\pi/2,\pi/2]$.
\end{enumerate}

\subsection{Some Riemann sums I}
Let $I=[0,1]$ and let $f: I \ra \RR$ be a continuous function. For $n
\geq 1$, define
a step function $f_n$ by:
\[ f_n(x) =  
\begin{cases}
f(1/n) & \text{if $0 \leq x \leq 1/n$} \\
f(k/n) & \text{if $1 \leq k \leq n-1$ and $k/n < x \leq (k+1)/n$.}
\end{cases} \]

\begin{enumerate}
\item Prove
that $|f_n-f|_I \ra 0$ as $n \ra +\infty$;
\item use this to show that \[ \lim_{n \ra +\infty} \frac{1}{n}
  \sum_{k=1}^n f\left( \frac{k}{n} \right) = \int_0^1 f \]
\item prove that the above formula still holds if $f$ is a ruled
  function on $I$.
 \end{enumerate}

\subsection{Some Riemann sums II}
Using the results of the previous exercise, compute the following
limits:
\[  \lim_{n \ra +\infty} 
  \sum_{k=1}^n \frac{1}{k+n}, \quad
\lim_{n \ra +\infty} 
  \sum_{k=1}^n \frac{n}{k^2+n^2}, \quad
 \lim_{n \ra +\infty}
  \sum_{k=1}^n \frac{k^3}{n^4} \cos  \left( \frac{4k}{n} \right). \] 

\subsection{Some Riemann sums III}
Let $n$ be an integer $\geq 2$. If $1 \leq i \leq n$, we write
$n=q_{i,n} i + r_{i,n}$ with $0 \leq r_{i,n} \leq i-1$. Let $a_n$ be
the number of $i$'s with $1 \leq i \leq n$ such that $r_{i,n} \geq
i/2$. Let $\phi : I \ra \RR$ be defined by $\phi(0)=0$ and $\phi(x) =
\lfloor 2/x \rfloor - 2 \lfloor 1/x \rfloor$, where $\lfloor y
\rfloor$ denotes the integer part of $y$.  
\begin{enumerate}
\item prove that $q_{i,n}$ and $r_{i,n}$ above are uniquely determined;
\item prove that $\phi$ is a ruled function; 
\item compute $\int_{1/k}^1 \phi$ if $k
\geq 2$ and use this to compute $\int_0^1 \phi$;
\item prove that $a_n = \sum_{k=1}^n \phi(k/n)$;
\item prove that $\lim_{n \ra +\infty} a_n / n$ exists and compute it.
\end{enumerate}

\section{Problems graded by Inna}

\subsection{Euler's constant}
\begin{enumerate}
\item prove that there exists a constant $M \geq 0$ such that
  $0 \leq x-\log(1+x) \leq M x^2$ if $0 \leq x \leq 1$;
\item use this to show that 
\[ \lim_{n \ra +\infty} 
  \left( \sum_{k=1}^n \frac{1}{k} \right) - \log(n) \] 
exists. It is called Euler's constant.
\end{enumerate}

\subsection{Cauchy-Schwarz' inequality}
Let $f,g : [a,b] \ra \RR$ be two ruled functions. Prove that 
\[ \left( \int_a^b fg \right)^2 \leq  \int_a^b f^2 \int_a^b g^2. \]
Hint: obviously, $\int_a^b (f-gt)^2 \geq 0$. Now judiciously choose $t$.

\subsection{Four inequalities} 
\begin{enumerate}
\item Prove that if $f,g : [0,1] \ra \RR$ are
non-decreasing ruled functions, then \[ \int_0^1 fg \geq \left(\int_0^1 f \right)
\left( \int_0^1 g \right). \] 
\item prove that if $f: [0,1] \ra \RR \setminus \{0\}$ is a
continuous function, then \[ \left(\int_0^1 f \right)
\left( \int_0^1 1/f \right) \geq 1. \]
\item Let $f: [0,1] \ra \RR$ be a $C^1$ function such that
$f(0)=f(1)=0$. Prove that 
\[  \left| \int_0^1 f \right| \leq \frac{1}{4} |f'|_{[0,1]}. \]
Can one replace $1/4$ by a smaller number?
\item  Let  $f: [0,\pi] \ra \RR$ be a $C^1$ function such that
$f(0)=f(\pi)=0$. Prove that
\[ \int_0^{\pi} f(t)f'(t) \on{cotan}(t)dt = \frac{1}{2} \int_0^{\pi}
f^2(t) (1+\on{cotan}^2(t)) dt. \]
and use this to show that
\[ \int_0^{\pi} f^2 \leq \int_0^{\pi} (f')^2. \] 
\end{enumerate}

\subsection{Lots of derivatives} \label{lod}
Recall that $\sinh(x)=(e^x-e^{-x})/2$. Let
$f(x)=\sin(\sinh(x))-\sinh(\sin(x))$. Find the least integer $n \geq
0$ such that $f^{(n)}(0) \neq 0$ and compute $f^{(n)}(0)$. Hint: you
can use Taylor's expansion for $\sin$ and $\sinh$. 

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