# Department of Mathematics Department of Mathematics August 12, 2014

3 hours.
Work all of the following problems.
Justify your answers.

R denotes the field of real numbers;

N denotes the set $\{ 1 , 2 , 3 , \ldots \}$ of natural numbers.

1. Let f be a function mapping R of to R. Give the definitions of each of the following concepts in words, without using logical symbols or the word $\mathrm { ^ { 6 6 } n o t ^ { 9 } }$

a) f is surjective

b) f is not injective

c) f is uniformly continuous

d) f is not uniformly continuous

2. Let $\left( a _ { n } \right)$ be a sequence defined recursively by $a _ { 1 } = 1$ and $\textstyle a _ { n + 1 } = { \frac { a _ { n } } { 3 } } + 5$ for $n \geq 1$ . Prove inductively that $a _ { n } \leq a _ { n + 1 } \leq 1 0$ for each positive integer n. Then explain why the sequence $\left( a _ { n } \right)$ converges and find its limit.

3. Suppose f and g are functions mapping R into itself with lim $_ { 1 } { _ { x  0 } } f ( x ) = 0$ a) Prove from the $\epsilon - \delta$ definition that if g is bounded, then $\begin{array} { r } { \operatorname* { l i m } _ { x  0 } f ( x ) g ( x ) = 0 } \end{array}$ as well.

b) Give an example to show that the boundedness hypothesis cannot be omitted from Part a).

4. Let $v _ { 1 } , v _ { 2 } , v _ { 3 }$ form a basis for a vector space V . Prove that $v _ { 1 } + v _ { 2 } , \ v _ { 2 } \ -$ $v _ { 3 } , \ v _ { 2 } + 2 v _ { 3 }$ form a basis for V .

5. Let R be the region in the first quadrant bounded by the curves $x ^ { 2 } + y ^ { 2 } = 2 x$ and $y = 0$ . Let C be the boundary of the region $R ,$ oriented counterclockwise.
   Evaluate the line integral

$$
\int _ { \cal C } \ x e ^ { x } d x \ + \ ( y e ^ { y } + x ^ { 2 } ) d y .
$$

6. Take $A : = \left\lceil { \begin{array} { c c } { 2 } & { - 2 } \\ { - 2 } & { 5 } \end{array} } \right\rceil$ . Find an orthogonal matrix P for which $P ^ { - 1 } A P$ is diagonal.
   Then find the minimum value of the dot products $A x \cdot x$ as x ranges through the unit vectors in $\mathbb { R } ^ { 2 }$

7. Take $A = \{ 0 , 1 , 2 , 3 , 4 , 5 \}$ . Define a relation R on A by x $R y$ if and only if $x ^ { 2 } - 4 x = y ^ { 2 } - 4 y .$

a) Prove that R is an equivalence relation.

b) Exhibit the partition of A whose members are the equivalence classes of R.

8. Let $f : \mathbb { R } ^ { 2 } \to \mathbb { R }$ be continuous.

a) Define what it means for $f$ to be (totally) differentiable at the origin.

b) Suppose $\textstyle { \frac { \partial f } { \partial x } }$ is continuous at (0, 0), while $\frac { \partial f } { \partial y }$ exists at (0, 0). Prove that f is differentiable at (0, 0).

Note: You can “buy” the answer to Part a) for 4 points.
Also you can earn partial credit on Part b) by making the stronger assumption both partial derivatives are continuous at (0, 0).