Mathematics 136 – AP Calculus Exam 2 – October 31, 2003

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

I. (20) On a dark and scary night you find yourself hiking in a forest, 10 km from the cabin where you will spend the night. Parallel to the path to the cabin there is a paved road. You can walk 8 km/hr on the road, but only 3 km/hr through the forest. If your path is as in the following diagram, how should you choose x to minimize the total time it takes to get to the cabin?

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 = g(t) for some function. Sketch the graph of the

antiderivative G(t) of g(t) with G(0) = −1. What is G(5)?

IV. Methods of integration. In B,C,D you may use the table of integrals. A) (10) Using integration by parts, derive the formula for #13 in the table of integrals. B) (10) ∫ e^{sin(2x)} cos(2x) dx

C) (10)

∫ \frac{5x + 6}{x^{2} + 12x + 40} dx

D) (10)

∫ \frac{x^{2}}{\sqrt{9 − x^{2}}} dx