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\title{Math 334 Sample Problems}
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\markboth{Sample Problems}{Sample Problems}


One notebook-sized page (one side) of notes will be allowed on the test.
You may work 
together on the sample problems -- I encourage you to do that.  Neither
Jerry nor I will tell you how to work these problems.  You may ask for
help if you bring your work in and show it to us, and tell us where you
are stuck.  The test (on Monday, October 22) will cover up to and
including section 2.5 in the text.  

There will be homework problems or example problems from the text
on the midterm. There will be four problems on the midterm, at least one
will be from this list or a slight modification of a problem on this list. 
This is a long list of problems that I have accumulated. Do your best
to work as many of them as you can. You may quote any theorems from the text
without proof but you may not cite example problems or homework problems. 

\bigskip

\begin{enumerate}

\item Suppose $\ds f'$ exists on $(a,b)$ and $\ds f'(x) \ne 0$ on $(a,b)$. 
Prove that either $\ds f'(x)>0$ for all $x \in (a,b)$ or $f'(x)<0$ for all
$x \in (a,b)$. 

\bigskip

\item Let $\ds f: \mathbf{R}^2 \to \mathbf{R}$ be defined by
\[
f(x,y)=\begin{cases}
(1-\cos \frac{x^2}{y})\sqrt{x^2+y^2} \text{ if $y\ne 0$}\\
0 \text{ if $y=0$.}
\end{cases}
\]
\begin{enumerate}
\item Prove that $f$  is continuous at $(0,0)$.
\item Compute all directional derivatives at $(0,0)$.
\item Prove that $f$ is not differentiable at $(0,0)$.
\end{enumerate}

\bigskip

\item

\begin{enumerate}

\item  Let $f$ be a continuous function from $[0,1]$to $\ds \mathbf{R}^n$.
 Prove that the graph of $f$, $\ds \Gamma_f = \{ (x,y) : y=f(x)\}$ is closed.
($\Gamma_f$ is a subset of $\ds [0,1]\times \mathbf{R}^n$.)

\item  Let $f$ be a function from $[0,1]$ to  $\ds \mathbf{R}^n$.  Suppose
$\Gamma_f$ is closed.  Is $f$ continuous?  Give a proof or counterexample. 

\end{enumerate}

\bigskip

\item Let $A \subset \mathbf{R}^n$ and $B \subset \mathbf{R}^m$ be
  compact.  Prove that $A\times B \subset \mathbf{R}^{n+m}$ is compact.

\bigskip

\item Let $\ds \mathbb{Q}$ be the set of rational numbers.  Prove that 
there does NOT exist a pair of non-empty sets $A,B$ such that 
$\ds \mathbb{Q}=A\cup B$ with $\ds \overline{A}\cap\overline{B}=\emptyset$. 

\bigskip

\item Prove that the temperature of a tetrahedron must have at least
  three distinct points on the edges or vertices of the tetrahedron with
  the same value.  Assume the temperature is a continuous function. (Hint:
the intermediate value theorem.)

\bigskip

\item Let $x \in \mathbf{R}, x\ne 0$ and $x> -1$. Let $a\in \mathbf{R}, a<0,
\text{or } a>1 $.  Prove that
\[
(1+x)^a > 1+ax.
\]



\item  Let $f$ be defined as follows

\begin{equation*}
f(x,y)=
\begin{cases}
x^{4/3} \sin (y/x) \text{ if $x\ne 0$ }\\
0 \text{ if $x=0$}.
\end{cases}
\end{equation*}
Where is $f$ differentiable?

\bigskip

\item Let $S$ be an open set and let $p\in S, q\notin S$.  Prove that there 
is a boundary point of $S$ on the line segment joining $p$ and $q$.

\bigskip

\item   Let a sequence be
  defined recursively by the rules: $\ds x_0=1,
  x_{n+1}= x_n +\frac{1}{x_n}$.  Prove that the sequence does not converge.

\bigskip

\item Suppose $\{a_n\}$ is a sequence with $\ds a_n > 0$ 
and $\ds b_n$ is defined by 
$\ds b_n=a_n+\frac{1}{a_n}$. 
\begin{enumerate}
\item Assume that  $\ds a_n \ge 1$ and that $\ds \{b_n\}$ converges.  
Prove that 
$\ds \{a_n\}$ converges.
\item If it is assumed that $\ds \{b_n\}$ converges but only that $\ds a_n>0$,
it does not necessarily follow that $\ds \{a_n\}$ converges.  Find such
an example.
\end{enumerate}

\bigskip

\item Prove that $\displaystyle \lim _{n\to \infty} \sin n$ does not exist.

\bigskip

\item Prove that the set $\ds \{(x,y,z): \frac{x^2}{2}+\frac{y^2}{3}+\frac{z^2}{4}=1\}$ is connected.

\bigskip

\item Suppose $A$ is a connected set and that 
$\ds A \subset B \subset \overline{A}$. Prove that $B$ is connected.

\bigskip

\item Let $f$ be continuous on $(0,1)$ and suppose $0<f(x)<x$ 
for all $x\in (0,1)$. Define $\ds f_n(x)$ inductively by 
$\ds f_1(x)=f(x), f_{n+1}(x) = f(f_n(x))$. Prove that 
$\ds \lim_{n \to \infty} f_n(x)$ exists and compute it.

\bigskip

\item Let $\displaystyle |a|<1$, where $a$ is a real number.  Prove that
  $ \displaystyle \lim _{n\to \infty} na^n=0$. Notice that $a$ is
  allowed to be negative.

\bigskip


\item Let $f$ be a continuous real valued function defined on $\ds
  [0,1]$ such that $f(0)=f(1)$.  Show that there is a pair of points
  $\ds a,b \in [0,1]$ such that $\ds b-a=1/2$ and  $\ds f(b)=f(a)$.

\bigskip

\item Let $\ds \{x_n\}_1^{\infty}$ be a sequence of real numbers.  Prove 
that there exists a sequence of intervals $I_n=[a_n,b_n], I_{n+1}\subset I_n$
with $\ds x_n \notin I_n$.  Use this to prove that there is a real number 
that is not in the sequence $\ds \{x_n\}_1^{\infty}$.  Thus prove that the
reals are not countable.

\item You will need to know the the following items

\begin{enumerate}
\item Cauchy's inequality
\item Triangle inequality
\item Open set
\item Closed set
\item Boundary of a set
\item Compact set
\item Bolzano-Weierstrass theorem
\item Connected set
\item Convergent sequence
\item Completeness
\item Cauchy's criterion
\item Continuity at a point
\item Continuity on a set
\item Uniform continuity
\item Partial derivatives
\item Differentiability 
\item Mean value theorem 
\item Chain rule
\end{enumerate}

\end{enumerate}

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