18.01 Practice Exam 2

1. Recall that the volume of a (right circular) cone is V = \frac{1}{3}πr^{2}h, where r is the radius of the base, and h is the height. Suppose that the distance from the tip to the base’s rim is 1 unit.

(a) (12 points) Show that V = \frac{1}{3}π(h − h^{3}), where h is between 0 and 1.

(b) (18 points) Determine the maximum volume of such a cone.

2. (24 points) Sketch the graph of f(x) = x^{2} + x^{−1}. Indicate the intervals where the function is increasing, decreasing, concave up or concave down. Label all relative maxima and minima and inflection points.

3. (12 points) An elastic band of length x and (circular cross-sections) of radius r is being stretched at the rate of 1 cm/sec. Assuming the volume remains constant, at what rate is the radius shrinking when x = 10 cm and r = 0.1 cm.

4. (6 points) State the Mean Value Theorem.

5. Approximate \sqrt{4.04} by

(a) (8 points) applying one iteration of Newton’s method to the function f(x) = x^{2} − 4.04;

(b) (8 points) using a linear approximation of \sqrt{x} at x = 4.

6. Evaluate the following limits (or explain why the limit does not exist):

(a) (6 points) lim_{x→0} \frac{e^{2x} − 1}{sin x};

(b) (6 points) lim_{x→∞} x^{1/x}.