Mathematics 136, section 4 – AP Calculus Exam 3, December 5, 2003

Directions: Answer questions I - III in the blue exam booklet. There are 100 ordinary and 10 Extra Credit points. Please show all work you do to derive your solutions.

I. All parts of this question refer to the region R under the graph y = (16 − x^{2})^{\frac{1}{4}} , above the x-axis, between x = 0 and x = 4. A) (10) Set up an integral computing the volume of the solid obtained by rotating R

about the x-axis. B) (15) Using the table of integrals and the Fundamental Theorem of Calculus, compute

the volume. (Partial credit for identifying the correct table entry that applies.) C) (10) Set up the integral to compute the volume of the solid obtained if the region R

is rotated about the line y = 3. Note: Do not try to compute this integral.

II. The population of a prolific breed of hamsters increases at a rate proportional to the square of the population at all times. A) (10) Express the above statement as a differential equation for the hamster population

as a function of time. B) (10) There should be a constant of proportionality, say k, in your equation from A.

Set k = 1 for this part of the question only. Sketch the slope field for the equation, showing slopes at the points with integer coordinates, 0 ≤ t ≤ 3, 0 ≤ P ≤ 3.

C) (10) Find the general solution of your differential equation from part A by separation

of variables. D) (5) There were 2 hamsters at time t = 0 weeks and 4 hamsters at t = 3 months. What

happens between t = 3 and t = 6? Is this a realistic population model?

III. A) (15) Does the infinite geometric series

8 + \frac{16}{3} + \frac{32}{9} + \frac{64}{27} + \frac{128}{81} + ⋯

converge? If so, find the sum. If not, say why not. B) (15) Use the Ratio Test to find the interval of convergence of the power series:

∑_{k=1}^{∞} \frac{kx^{k}}{2^{k}} = \frac{x}{2} + \frac{x^{2}}{2} + \frac{3x^{3}}{8} + ⋯

Extra Credit (10) Find the arclength of the curve y = x^{3/2} from x = 1 to x = 4.