Mathematics 133 – Intensive Calculus for Science 1 Midterm Exam 3 – December 2, 2005

Directions: Do all work in the blue exam booklet. There are 100 possible points. You may use any of the following formulas without comment:

1) Cylinders: V = πr^{2}h

2) Circular cones: V = \frac{1}{3}πr^{2}h

3) Spheres: V = \frac{4}{3}πr^{3}, A = 4πr^{2}.

I. All parts of this question refer to the function f(x) = xe^{−x^{2}}.

A) (10) Compute f′(x).

B) (10) What are the (global) maximum and minimum values of f(x) on the interval −2 ≤ x ≤ 0?

C) (5) Does your minimum from part B occur at a local minimum of f(x)? Explain.

II. Compute the derivatives of the following:

A) (10)

f(x) = \frac{sin(x)}{cos(x) + sin(x)}

B) (10)

g(x) = ln (3x + \frac{1}{x})

C) (10)

h(x) = arcsin(4x)

D) (10) Find \frac{dy}{dx} if xy^{2} + x^{2}y = 3.

III. (20) The Holy Cross Buildings and Grounds department wants to build a new secure parking lot for fans of visiting football teams. To save costs they decide to make the lot by fencing in a rectangular area and using the existing fence at the non-enclosed end of Fitton Field (the end behind the scoreboard) as one of the four sides. The area of the lot must be 500 square yards. What should the dimensions of the lot be to minimize the length of new fencing needed?

IV. (15) The radius of a spherical ice ball is increasing at the rate of 3 mm/min. At what rate is the volume increasing when the radius is 6 mm?