Problems are weighted equally.
Justify your answers.
R stands for the real numbers.

1. Suppose  and  are injective maps of a set  into itself.
   Show that the fcomposite function $f \circ g$ is also injective.

2. Use the $\epsilon , \delta$ definition to prove that $\operatorname* { l i m } _ { x \to 2 } x + { \frac { 1 } { x } } = { \frac { 5 } { 2 } } .$

3. Let $f : \mathbb { R } \to \mathbb { R }$ by $f ( x ) = x \mathrm { e } ^ { 2 x }$ . Find a general formula for the ’th derivative $f ^ { ( n ) } ( x )$ f f x x and prove it by induction.

4. Find a basis for the subspace of $\mathbb { R } ^ { 4 }$ spanned by the vectors $( 1 , 1 , 1 , 1 ) , ( 3 , 4 , 6 , 7 )$ and (5 6 8 9).

5. Suppose  is a three by three matrix with real entries having eigenvalues $- 1 , 0 .$ Aand 1. Prove that $A ^ { 3 } = A$

6. Find the Maclaurin series of the function $f ( x ) = \arctan x$ and use an appropriate f x xMaclaurin polynomial to approximate arctan 01 with an error of less than $1 0 ^ { - 5 }$

7. Suppose $\left( f _ { n } \right)$ is a sequence of bounded functions taking R into R which converge funiformly to a function . Prove that there is a number $M > 0$ satisfying $| f _ { n } ( x ) | \leq M$ g M >for all natural numbers  and all real numbers .

8. Suppose $f : \mathbb { R } ^ { 2 } \to \mathbb { R }$ has continuous partial derivatives at (0 0). Prove that fhas directional derivatives in every direction at the origin.

9. Describe the construction of a field having eight elements.