Mathematics 136, section 2 – Advanced Placement Calculus Make-up Exam 2 – November 10, 2009

Directions

Do questions I - V and either part A or part B of question V on the back. If you submit work for both parts of V, I will count it for Extra Credit. There are 100 regular points and 15 possible Extra Credit points. Do all work in the blue exam booklet.

I. Consider the function f(x) = xe^{−x^{2}}.

A) (10) Find all the critical numbers of f and classify each as a local maximum, local minimum, or neither.

B) (5) What are the absolute (global) maximum and minimum values of f(x) on the interval [−2, 0]?

II. (10) Evaluate the following limit:

lim_{θ→0} \frac{sin^{2}(θ)}{θ^{2}}.

III. The velocity of a accelerating car is measured at each of the following times, yielding a table

of values:

t = time(sec) 1 3 5 7 9 11 13 ; v(t) = velocity(ft/sec) 15 13.5 9.1 7.8 4.0 3.0 0

A) (5) Give an estimate for the total distance traveled by the car between t = 1 and t = 13 using a left-hand Riemann sum for the velocity function and Δt = 2.

B) (5) Is your result from A) less than or greater than the actual total distance traveled, assuming v is always decreasing between t = 1 and t = 13? Explain.

IV. Compute each of the following integrals. Show all work for A and B.

A) (5) ∫_{1}^{8} x^{1/3} − \frac{3}{x^{2}} + 2 dx

B) (10) ∫ \frac{cos(4x) dx}{(sin(4x) + 8)^{3}}

Integrate using the indicated method and the table (say which entry you are using if you use one).

C) (10) Use parts: ∫ x^{2} cos(3x) dx (full credit only for showing all work; you can check

your result by the table)

D) (12.5) Use a trigonometric substitution: ∫ \frac{x^{3}}{\sqrt{1 + x^{2}}} dx.

E) (12.5) Use partial fractions:

∫ \frac{x + 8}{(x + 2)^{2}(x − 1)} dx.

A) (15) Using the definition (not the Evaluation Theorem) compute

∫_{1}^{4} 2x^{2} + 3 dx.

The following formulas may be useful:

∑_{i=1}^{n} i = \frac{n(n + 1)}{2}  ∑_{i=1}^{n} i^{2} = \frac{n(n + 1)(2n + 1)}{6}.

B) (15) A rectangular box with a square base and top is to be made from cardboard costing 5 cents per square foot. The volume of the box must be 10 cubic feet. What dimensions will minimize the cost of the cardboard used for the box?