Mathematics 136, section 2 – Advanced Placement Calculus Exam 2 – November 4, 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) = x^{3} + 6x^{2} − 15x.

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

[0, 3]?

II. (10) Evaluate the following limit:

lim_{θ→0} \frac{cos(5θ) − 1}{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 ; v(t) = velocity(ft/sec) 1 3.5 5.1 7.8 9.0 11.2

A) (5) Give an estimate for the total distance traveled by the car between t = 1 and t = 11

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 increasing between t = 1 and t = 11? Explain.

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

A) (5) ∫_{1}^{4} x^{2} − 3\sqrt{x} + 4 dx

B) (10) ∫ sec^{2}(2x)\sqrt[3]{tan(2x) + 4} dx

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

C) (10) Use parts: ∫ x^{2}e^{4x} dx (full credit only for showing all work; you can check your

result by the table)

D) (12.5) Use a trigonometric substitution: ∫ \frac{\sqrt{25 − x^{2}}}{x} dx.

E) (12.5) Use partial fractions:

∫ \frac{2x + 19}{(x + 1)(x^{2} + 16)} dx.

A) (15) A fish swimming against a current of u ft/sec (constant) for a distance L (constant) will

expend energy

E(v) = \frac{Lv^{3}}{v − u}

if it swims at v ft/sec (u < v). What v minimizes the total energy expended? B) (15) Using the definition (not the Evaluation Theorem) compute

∫_{2}^{3} x^{2} + 4x 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}.