Name: Second Midterm

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

1. Find the derivative of the following functions.

(a) f(x) = 100x^{2} + e^{x^{2}+x}.

(b) g(t) = \frac{1}{1+sin^{2}(t)}.

2. Consider the equation

2^{x} + 2^{y} + 2^{2y} = 8

(a) Find \frac{dy}{dx}.

(b) Find the tangent line to the curve at the point (1, 1).

3. Concavity. (a) What is the definition of an inflection point?

(b) Find the inflection points of the graph of f(x) = 2x^{6} − 5x^{4} + 20x − 10.

4. Consider the function f(x) = 2|x − 1| − 2x^{2} + 3x on the interval [−1, 2]. (a) Find the critical points of f.

(b) Find the local minima and maxima.

(c) Find the absolute minima and maxima.

(d) Sketch the graph of f using the data you found above.

5. Rolle’s theorem. (a) State Rolle’s theorem.

(b) Suppose that f is a differentiable function on (−1, 4) and suppose that f(0) = f(1) = f(2) = f(3) = 0. Is it possible that f′ has fewer than 3 zeros?

(c) Give an example of a function f as in (b) whose derivative has exactly 3 zeros. No explanation necessary for this part.

6. Compute the following limits. (a)

lim_{x→∞} \frac{x + ln(x)}{x − 1}

(b)

lim_{x→0} \frac{cos(x) − cos(2x)}{x}