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{\bf Math336 \hskip1.3in Exam 1 \hfill  February 20, 1998}

\bigskip
\underbar{Instructions}:

\medskip
(1) The exam is 50 minutes long.

(2)  SHOW ALL YOUR WORK.    Explain your answers.   You are more likely to get
partial credit if you explain what you are doing.


\medskip
\hrule

\bigskip

1. (20 points) Find an example of each of the following:
\medskip
\qquad
(a) A subset $A$ of $\Bbb R^3$ which is closed and unbounded.

\medskip
\qquad
(b) A compact set in $\Bbb R^2$ with no interior points.


 \bigskip 





2. (20 points) Answer true (T) or false (F) to the following statements (no proof is required):
\roster
\item The subset $(0, 1] = \{x | 0 < x \leq 1 \}$ is neither closed 
nor open in $\Bbb R^1$.

\item  The intersection of an {\it infinite} family $\Cal F$ of open sets in a metric space is {\it always}
 an open subset.


\item  The subset $S$ of all irrational numbers in $R^1$ is uncountable.

\item  The boundary of the set $A = \{ (x, y) | -1 < x < 1, -1 < y < 1 \}$
is \newline
$\partial A = \{ (x, y) | -1 \leq y \leq 1,  |x| = 1 \} \cup  \{ (x, y) |  -1 \leq x \leq 1, |y| = 1 \} $. 
\endroster

\bigskip

3. (20 points)
(a) Define what it means for a subset $A$ of a metric space $S$ to be closed.

\medskip
\qquad
(b) Define what it means for a subset $A$ of a metric space $S$ to be compact.






\bigskip

4. (20 points) Let $p_n = (\frac{(-1)^n n^2}{n^2 + 99 n + 7}, 1) \in \Bbb R^2$, for 
$n =1, 2, ...$. Does the sequence $\{ p_n \}^{\infty}_{n=1}$ in $\Bbb R^2$
have a convergent subsequence ? Explain your answer.

\bigskip

5. (20 points) Prove the following theorem: {\it ``Suppose that $A$ is a compact set in a
 metric space $S$. Then for each positive number  $\delta > 0$, there
is a finite number of points $p_1, p_2,..., p_n$ in $A$ such that 
$\cup_{i=1}^n B(p_i, \delta) \supset A$."}



\newpage 
{\bf Math336 \hskip1.3in Exam 2 \hfill  March 27, 1998}

\bigskip
\underbar{Instructions}:

\medskip
(1) The exam is 50 minutes long.

(2)  SHOW ALL YOUR WORK.    Explain your answers.   You are more likely to get
partial credit if you explain what you are doing.


\medskip
\hrule

\bigskip
\noindent
1. (20 points) Let $f: [0, 1] \times [0, 1] \rightarrow \Bbb R^1$ be the function  defined  by 
$f(x, y) = e^{x^2 + y^2}$. Is f uniformly continuous on the 
unit square $[0, 1] \times [0, 1] $ ? Carefully justify your answer.


\bigskip 


\noindent 
2. (20 points) (a) Let $A$ be a subset of a metric space $S$. State what it means 
for a function $ f: A \rightarrow \Bbb R^1$ to be continuous. 

\medskip
\qquad
(b) Let  $A = \{(x, y) | |x| \le  \pi, |y| \le 3  \}$ and let  $ f: A \rightarrow 
\Bbb R$  be the function defined  by $f(x, y) =  e^{y} \sin (x)  $. Is there  a point 
$(x_0, y_0) \in A$ with $f(x_0, y_0) = \frac 14$ ? Carefully justify your answer.

\bigskip

\noindent
3. (20 points) Let $A = \{(x, y) | x^2 + y^2 \le 4 \}$ and  $ f: A \rightarrow 
\Bbb R$ the function defined  by $f(x, y) = \sin (x-y) $. Does the range of $f$
contain its supremum? Carefully justify your answer.

\bigskip 

\noindent 
4. (20 points)  State and prove the following theorem: {\it ``Let $A$ be a compact subset 
of a metric space $S$. Suppose that $ f: A \rightarrow \Bbb R^1$ is continuous
on $A$. Then the range of $f$ is bounded."}


 

\bigskip 

\noindent 
5. (20 points) Compute the following partial derivatives:
\medskip
\qquad
(a) $\frac{\partial f}{\partial x_2}$, where $f(x_1, x_2, x_3) 
= x_1^{1000} + \sin (2x_2) + x_1x_2x_3^2$.

\medskip
\qquad
(b)  $\frac{\partial [f\circ g]}{\partial x_1}$, where $ f(u, v) = u^2  + 2 v^2$ and $g(x_1, x_2) 
= (2x_1 + 3x_2, x_1 + 2 x_2)$.

\newpage 


{\bf Math336 \hskip0.8in Exam 3 (Takehome) \hfill Due April 24, 1998 in class}

\bigskip
\underbar{Instructions}:

\medskip
(1) This take home-exam is due April 24, 1998  in
class. No late take-home exam will be accepted.

(2)  This test will be conducted under the Honor Code. You should work out
all problems by yourself. You are NOT allowed to give or receive 
any unauthorized aid on this exam. 

(3) Please read and sign the following statement: 


\bigskip


   {\it  On my honor, I have abided by the code of honor and have 
committed no act of academic dishonesty on this examination. I have
neither given nor receive 
any unauthorized aid on this exam.}

\bigskip



    Student's signature: \hrule

\bigskip
\bigskip


    Date: \hrule


\bigskip
\bigskip

\bigskip



\noindent
1. (10 points) Let  $f(x,y) = 0$ and 
$$
   f(x, y) = \frac{xy}{2x^2 + y^2}, \text{  for   }  (x, y) \neq (0, 0).
$$

\medskip
\qquad
(a) 
Show that $(D_1f)(x, y) $ and $(D_2f)(x, y) $ both exist.

\medskip
\qquad
(b)  Compute $f(x, x)  $ and show that $f$ is not continuous at $(0, 0)$.


\bigskip 


\noindent 
2. (15 points) Let $f(0,0) = 0$ and  
$$
   f(x, y) = x^2 + y^2 - 2 x^2y - \frac{4x^6y^2}{(x^2 +y^2)^2},
$$ 
for  $(x, y) \neq (0, 0).$
\medskip
\qquad
(a) Show  that 
$$
    4x^4y^2 \le (x^2 +y^2)^2
$$
\qquad and $|f(x, y)| \leq |x^2 + y^2 - 2 x^2y| + x^2$. 
Conclude that $f $ is continuous. 

\medskip
\qquad
(b) Is $(0, 0)$  a critical point of $f$ ?

\medskip
\qquad
(c) Compute  $f(x, x^2)$. Does $f(x, y)$  have a local minimum at the point $(0, 0)$ ?


\newpage 
\bigskip

\noindent
3. (10 points) Let $ f(x, y) = e^{x^2 + y^2}$. Find the Taylor formula
of $f$ with the remainder around the point $(0, 0)$  with  
$n = 3$. (Hint: see
Theorem 7.7 on page 185).

\bigskip 

\noindent 
4. (10 points)  Let $f(x, y, z) = x^2 - y^2 + z^2$, $a = (1, 1, 1)$ and
$ h = (3, 4, 5)$. Evaluate $ \bigtriangledown f (a) $ and the directional 
derivative of $f$ at $(1, 1, 1)$ in direction $h$, $<\bigtriangledown f(a), 
 h>$. 
 

\bigskip 

\noindent 
5. (20 points) Investigate the absolute convergence or divergence of 
$\sum_{n=1}^{\infty} a_n$ if :
\medskip
\qquad
(a) $ a_n = \frac{\sqrt{n+1} - \sqrt{n}}{n}. $


\medskip
\qquad
(b)  $a_n = \frac{(-1)^n}{n^3}$.


\bigskip 

\noindent 
6.  (10 points) Find the radius of convergence of each of the following power series:


\medskip
\qquad
(a) $\sum_{n=1}^{\infty} n^3 x^n$


\medskip
\qquad
(b) $\sum_{n=1}^{\infty} \frac{2^n}{n!} (x-1)^n $



\bigskip

\noindent 
7.  (15 points) (a) Starting with the geometric series for $\frac{1}{1- x}$,
use Theorem 9.14 to show $ [- \log (1 - x)] = x + \frac{x^2}{2} +
\frac{x^3}{3} + \cdots  = \sum_{n=1}^{\infty}  \frac{x^n}{n} $
for $|x| < 1/2$. 

\medskip
\qquad 
(b) Evaluate $\int^x_0 \log (1-t) dt$.


\medskip
\qquad 
(c) Using the results obtained in  (a), (b) and Theorem 9.14,  show that   
$$\sum_{n=1}^{\infty}  \frac{x^{n+1}}{n(n+1)} = -x \log(1- x) + x + \log(1-x)$$
for $|x| < 1/2$.

\bigskip

\noindent 
8.  (10 points) Let $f(x, y) = 3x^2 - 4xy + 3 y^2 + 8x -17y + 5$. Find the 
point $(a, b)$ at which $f(x,y)$ attains its local minimum. 


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