# Mathematics Preliminary Exam, Fall 2016

1. A sequence of functions $\{ f _ { n } : \mathbb { R } \to \mathbb { R } \} _ { n = 1 } ^ { \infty }$ converges uniformly to a function $f : \mathbb { R } $ R if for all $\epsilon > 0 .$ , there is a positive integer N such that for all $n > N$ and all $x \in \mathbb { R }$ , we have $| f _ { n } ( x ) - f ( x ) | < \epsilon .$

a) What do you need to check to show that a sequence does not converge uniformly?

b) Give an example such that for every $x \in \mathbb { R } , f _ { n } ( x )$ converges to $f ( x )$ , but the sequence does not converge uniformly.

2. Find an invertible matrix A and a diagonal matrix B such that $\left( \begin{array} { c c } { { 4 } } & { { - 6 } } \\ { { 3 } } & { { - 5 } } \end{array} \right) = A B A ^ { - 1 }$

3. Give an example (with proof) of a power series that converges exactly on the interval [3, 5).

4. Give an $\epsilon , \delta$ proof that $\mathrm { l i m } _ { x  2 } { \frac { 1 } { 3 + x } } = { \frac { 1 } { 5 } }$

5. a) Choose a path C from (2, 1) to (3, 5) and compute $\textstyle \int _ { C } ( 2 x + y ) d x + y d y$

b) Give (and apply) a criterion that shows the above integral either is or is not path independent.

6. Use induction to show that for all positive integers $1 ^ { 3 } + 2 ^ { 3 } + \cdot \cdot \cdot + n ^ { 3 } = ( 1 + 2 + \cdot \cdot \cdot + n ) ^ { 2 }$

7. Let $f : X \to Y$ be a map of sets.
   Give, with proof, necessary and sufficient conditions so that for all subsets $S \subset X , f ^ { - 1 } ( f ( S ) ) = S$

8. Draw representative level sets (i.e. $f ^ { - 1 } ( c )$ for $c \in \mathbb { R } )$ for $f ( x , y ) = x y$ . Next, compute and draw the gradient, $\nabla f$ , at a representative collection of points in $\mathbb { R } ^ { 2 }$ . Explain how these two objects must be related in general.

9. Give an example of a linear map defined on $\mathbb { R } ^ { 3 }$ with kernel generated by (1, 3, 5).