Name: Second Midterm

Write your name on this paper. Explain all your answers. Use common sense. Do not use calculators/book/notes. There are five parts all equally weighted. Here is the first one:

1. Find the derivative of the following functions.

(a) f(x) = \frac{1}{500}(5x + 6)^{100} + sin(32x) (b) g(x) = \frac{1}{x} + 2^{x} (c) h(x) = tan(x − 1)

2. Find the following limits.

(a) lim_{x→5π} \frac{x − 5π}{sin(x)}  (b) lim_{x→∞} \frac{x + e^{x}}{1 + e^{x}}

3. Consider the function

f(x) = \frac{1}{x − 10} + \frac{1}{x − 20}.

(a) Find the horizontal and the vertical asymptotes of the graph of f.

(b) There is a unique inflection point on the graph of f. What is it?

(c) Roughly sketch the graph of f using the information you obtained in answering (a) and (b).

4. Examples of extreme values. (a) State the extreme value theorem.

(b) Give an example of a continuous function on (0, 3) with neither an absolute maximum nor and absolute minimum. (Give a formula for f – no explanation necessary for this part.)

(c) Give an example of a nonconstant, differentiable function f on (0, 2) with an absolute maximum at x = 1. (Give a formula for f – no explanation necessary for this part.)

5. Optimization: Consider the following continuous function

f(x) = \{ \begin{matrix}−2x & if & x <= 0 \\ x & if & x > 0 \end{matrix}

and consider the point P = (1, 13). Which point on the graph of f is closest to the point P?