Sample graduate preliminary exam (3 hours, 9 problems counted equally)

1. Define what is meant by an inverse function, and give necessary and sufficient conditions for an inverse function to exist. Find the inverse function of $f ( x ) =$ $( x + 2 ) / ( 3 x - 1 )$ for x real, $x \neq 1 / 3$

2. Let a be a real number and let I be an open interval containing a. For a function $f : I  \mathbb { R }$ , give the definition of continuity at a. Prove that $f ( x ) = x ^ { 2 }$ is continuous at $a = 2$

3. Give the definition for convergence of a sequence of real numbers $\{ a _ { n } \} _ { n = 1 , 2 , . . }$ to a real number a. Prove that the sequence $a _ { n } = ( n ^ { 2 } + 2 ) / ( n ^ { 2 } + 3 n )$ converges to $a = 1$

4. Work out the Taylor series for f (x) = ln x around x = 1 and use it to approximate ln(1.1) accurate to 2 decimal places.

5. Evaluate the integral $\int _ { 0 } ^ { 1 0 \pi } | \sin x | d x .$

6. Define the Jacobian matrix of a mapping $f : \mathbb { R } ^ { n }  \mathbb { R } ^ { m }$ at a point $a = ( a _ { 1 } , . . . , a _ { n } )$ of $\mathbb { R } ^ { n }$ . Compute the Jacobian matrix of $f ( r , \theta ) = ( r \cos \theta , r \sin \theta )$ at $a = ( 0 , 0 )$

7. Assuming that $u , v ,$ and w are linearly independent ( in $\mathbb { R } ^ { n } )$ , prove that $2 u + v , u +$ $v + w _ { : }$ ,and $v - w$ are linearly independent.

8. Diagonalize the matrix $\left[ { \begin{array} { l l } { 2 } & { 1 } \\ { 1 } & { 2 } \end{array} } \right]$

9. Find the 3 cube roots of i and compute $( 1 + i ) ^ { 5 }$

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