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) In an alternate universe, the “super-attractive” force exerted by two objects of mass m_{1}, m_{2} is directly proportional to the square of the distance between them: F = Sm_{1}m_{2}r^{2}, where S is a constant and r is the distance between the masses. Four 1kg masses are fixed at points x = −12, −4, 2, 7 respectively along a straight line in this alternate universe. At what location x along that line should another 2kg mass be placed to minimize the sum of the super-attractive forces exerted on it by the four unit masses?

II. In a car moving at 90 ft/sec, the driver suddenly saw an obstacle 400 feet ahead and braked to a stop in 10 seconds. The car’s velocity was recorded by a sensor in its onboard computer every two seconds:

\begin{matrix}t (sec) & 0 & 2 & 4 & 6 & 8 & 10 \\ v(t) (ft/sec) & 90 & 75 & 50 & 25 & 5 & 0 \end{matrix}

A) (10) Can you say for sure from the information here whether the car hit the obstacle

or not? Explain, using the left- and right-hand Riemann sums for v(t). B) (10) From the given information, what is your best estimate about whether the car

hit the obstacle?

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) ∫ tan(\sqrt{x})/\sqrt{x} dx

C) (10) ∫ (3x + 2)/(x^{2} + 8x + 7) dx

D) (10) ∫ (4 − x^{2})^{−3/2} dx