Mathematics 136 – AP Calculus Makeup Exam 2 – November 9, 2003

Directions. Do all work in the blue exam booklet. There are 100 points.

I. (20)

II. The velocity of a drag racing car is measured yielding the following data.

\begin{matrix}t (sec) & 0 & 1 & 2 & 3 & 4 & 5 \\ v(t) (ft/sec) & 0 & 10 & 25 & 40 & 45 & 50 \end{matrix}

A) (10) What is your best estimate of the total distance the car travels between t = 0

and t = 5 seconds? B) (10) How often would you need to measure the velocity in order to get an estimate of

the distance traveled accurate to within 1 foot?

III. A) (10) State the first and second parts of the Fundamental Theorem of Calculus. B) (10) The following graph shows y = f(x) for some function. Sketch the graph of the

antiderivative F (x) of g(x) with F (x) = 0. What is F (4)?

IV. Methods of integration. In B,C,D you may use the table of integrals. A) (10) Using integration by parts, show that

∫ x^{n} cos(ax) dx = \frac{x^{n}}{a} sin(ax) − \frac{n}{a} ∫ x^{n−1} sin(ax) dx.

B) (10)

∫ \frac{tan(\sqrt{x})}{\sqrt{x}}dx

C) (10)

∫ \frac{3x + 2}{x^{2} + 8x + 7} dx

D) (10)

∫ \frac{dx}{(4 − x^{2})^{3/2}}