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\vspace*{-20pt}
{\bf Math 508, Fall 2014 \hfill Jerry L. Kazdan}
\medskip
\begin{center}
{\large\bf Problem Set 8}
{\sc Due:} Thurs. Nov. 6, 2014. \emph{Late papers will be accepted until
1:00 PM Friday}.
\end{center}
\medskip
{\bf This week}. Please read all of Chapter 6 in the Rudin text. Note
that we will only discuss the Riemann integral, not the
Riemann-Stieltjes integral.
\smallskip
\comments{
\makeboxbox{\parbox{5.0in}
{\sc Revision Note: } This revised version of Homework Set 3 has
clarifications and hints for problems \# 6, 10, and 11. Problem 7 is
solved in the Class Notes on Compactness and Problem 12 is now a Bonus
Problem.}
}
\vskip15pt
{\bf Note:} We say a function is \emph{smooth} if its derivatives of
all orders exist and are continuous.
\bigskip
\begin{problems}
\item Use the definition of the derivative as the limit of a difference
quotient to show that $\cos x$ is differentiable for all $x$. [You may
use without proof that $\lim_{\theta\to 0}\sin \theta/\theta = 1$ and
$\lim_{\theta\to 0}(1-\cos\theta )/\theta=0$.]
\item Let $A(t)$ be an $n \times n$ matrix whose elements depend
smoothly on $t\in\R$. Assume $A(t)$ is invertible at $t=t_0$.
\begin{parts}
\item Compute the derivative of $A^2(t)$ in terms of $A$ and $A'$.
\item Show that $A(t)$ is invertible for all $t$ near $t_0$. [Problem
Set 5 \#10].
\item Show that $A^{-1}(t)$ is differentiable at $t=t_0$ and find a
formula for it. Of course, from the special case of $1\times 1$
matrices you have a guess what it should (roughly) be.
\item Find a formula for the derivative of $A^{-2}(t)$ at $t=t_0$.
\end{parts}
\item In class we proved that the only solution of the differential equation
$u'(x)=u(x)$ with $u(0)=1$ is $u(x)=e^{x}$.
\begin{parts}
\item Use this to find the unique solution of $v'=v$ with $v(0)=c$, where $c$ is a constant.
\item Apply this to show that $e^{x+a}=e^{a}e^{x}$ for all real $a$ and $x$.
\item If for some constant $\gamma$ the differentiable function $v(x)$
satisfies \ $v'-\gamma v\le 0$, show that $v(x)\le v(0)e^{\gamma x}$
for all $x\ge 0$. [{\sc Hint: } Consider $g(x):= e^{-\gamma x}v(x)$.]
\end{parts}
\item A continuous function is called {\it piecewise linear} if it
consists only of straight line segments (see
\url{https://en.wikipedia.org/wiki/Piecewise_linear_function})
Let $f:[a,b]\to \R$ be a continuous function. Show that given any
$\epsilon>0$, there is a piecewise linear function $g:[a,b] \to \R$
such that $\abs{f(x)-g(x)} < \epsilon$ for all $x \in [a,b]$. In
other words, any continuous function on $[a,b]$ can be approximated
``uniformly'' by a piecewise linear function.
\newpage
\item Let $f:\R \to \R$ be a smooth function.
\begin{parts}
\item If $f'(1)=0$, $f''(1)=0$, $f'''(1)=0$ and $f''''(1)>0$, show
that $f$ has a local minimum at $x=1$.
\item If $f'(1)=0$, $f''(1)=0$, and $f'''(1)>0$, what can you say
about the behavior of $f$ near $x=1$?
\end{parts}
\item Say a smooth function $u(x)$ is a solution of the differential equation
$$
u'' + 3u'-(1+x^{2})u=0.
$$
\vspace{-20pt}
\begin{parts}
\item Show that $u$ cannot have a positive local maximum (that is, a local maximum where $u$ is positive).
\item Similarly, show that $u$ cannot have a negative local minimum.
\item If $u(x)$ satisfies the above equation on the interval $[0,2]$ with the boundary conditions $u(0)=0$ and $u(2)=0$, show that $u(x)=0$ in $[0,2]$.
\item Generalize all of the above to solutions of
$$
u'' +b(x)u' - c(x)u = 0 \quad\text{on } \quad \{\alpha \le x \le \beta\},
$$
where $b(x)$ and $c(x)$ are any continuous functions with $c(x)>0$.
\end{parts}
\item \begin{parts}
\item A strictly increasing, continuous, real-valued function $f$ on
an open interval $I \subset \R$ has an inverse function $f^{-1}$ which
is also strictly increasing, continuous, and defined on an open
interval $U$. Suppose $f \in C^1(I)$ and $f'(t_0) > 0$ at some point
$t_0 \in I$ [here $C^{1}(I)$ means the function is differentiable on
$I$ and this derivative is a continuous function).
Prove that there is an open sub-interval $I' \subset I$ on which
$f^{-1}$ exists, is strictly increasing, and continuous.
\item Using $f^{-1}$ from the previous part, prove that
$f^{-1} \in C^{1}(U')$ (where $U'$ is its domain) and that
\[ \frac{d}{dy} f^{-1}(y) = \frac{1}{f'(x)} \]
if $x$ is chosen to equal $f^{-1}(y)$. This is a special case of the
{\it Inverse Function Theorem}, which you will most likely study
further (in higher dimensions) in Math 509. [{\sc Hint: } Let $a :=
f^{-1}(y)$ and $b := f^{-1}(y+h)$. What does the Mean Value Theorem
say about $f(b) - f(a)$?]
\end{parts}
\item Use the definition of the integral as a Riemann sum to compute
$\int_0^b \sin x\,dx$. You will need the formula for
$\sin \theta + \sin 2\theta +\sin 3\theta+\cdots+\sin n\theta$; see
\url{http://www.math.upenn.edu/~kazdan/202F13/notes/sum-sin_kx.pdf}
\item Let $f(x) = \sin(1/x)$ for $0