Mathematics 136, section 4 – AP Calculus Final Examination December 19, 2003

Directions

Answer questions I - VIII. There are 200 regular points and 10 Extra Credit points. Don’t panic and remember to have some fun!

I. Consider the functions:

y = f_{1}(x) = \frac{2 sin(3x) + 1}{1 + x^{2}},  y = f_{2}(x) = x + 3x^{5/3} + 1,  y = f_{3}(x) = xe^{2x} + e^{x^{2}}

A) (15) Find the derivatives of f_{1}, f_{2}, f_{3}. B) (5) What line would you see if you “zoomed in” on the point (0, 2) on the graph of

y = f_{1}(x) + f_{3}(x)?

II. The graph below is the graph y = f(x) for some function f(x):

A) (15) Using the graph above, sketch the graph y = f′(x), and explain the features of your graph, including its x-axis intercepts, its local maxima/minima, points where f′(x) is undefined, and so forth.

B) (15) Sketch the graph of the antiderivative F of f(x) with F (0) = 0.

III. A rectangular poster is to have an area no larger than 400 square inches. It will consist of a central printed area, surrounded by margins of 1 inch along the top and sides and a 2 inch margin along the bottom. A) (15) What overall dimensions for the poster will yield the maximum printed area? B) (5) How can you tell that your answer from A is really a maximum for the printed

area?

IV. A) (15) The hull of a boat is 15 feet long. At a distance s feet from the bow (the front),

the cross section of the hull perpendicular to its long axis has the shape of the region in the xy-plane below y = 0 (y = 0 is the water line) and above the parabola y = ax^{2} −b, where a, b are given in the following table:

\begin{matrix}s & 5 & 10 & 15 \\ a & 2 & 3 & 4 \\ b & 2 & 3 & 4 \end{matrix}

Estimate the volume enclosed by the hull below the water line using a suitable Riemann sum.

B) (10) The region R in the xy-plane is bounded by y = x^{3} + 1, x = 1, x = 2, and the x-axis. Find the volume of the solid obtained by rotating R about the x-axis.

V. Compute each of the following integrals. If you use an entry from the table, identify it by number.

A) (5) ∫ x sin(5x^{2}) cos(5x^{2}) dx

B) (10) ∫ x^{2} ln(4x) dx

C) (10) ∫_{0}^{1} \frac{x−2}{x^{2}−4x+5} dx

D) (10) ∫ \frac{1}{\sqrt{x^{2}+2x}} dx

VI. When an object (with no heat source or refrigeration of its own) is placed into a surrounding medium of constant temperature A, the rate of change of the temperature T of the object is proportional to the difference between T and A at all times.

A) (5) Restate the above as a differential equation for the temperature function T. B) (5) Find the general solution of your equation from A. C) (10) When a frozen chicken pot pie is removed from the freezer its temperature is

0^{∘} F. At that time, it is placed into a preheated 350^{∘} F oven (maintained at that temperature). After 10 minutes, the temperature of the pot pie has reached 40^{∘} F. How much longer will it take for the pot pie to reach 160^{∘} F and be ready to serve?

VII. A) (10) Sketch the slope field of the differential equation \frac{dy}{dx} = y^{2} − y, showing the slopes

at points on the lines y = −1/2, 0, 1/2, 1, 3/2. B) (10) Find the solution of this differential equation with y(0) = 1/2.

VIII. A) (10) Consider the function f(x) = \sqrt{1 + x}. Compute the Taylor polynomial of degree

4 for f at a = 0, and use it to approximate the value \sqrt{1.2}. B) (10) Find the Taylor series of g(x) = e^{−x/2} at a = 0, using any convenient method. C) (10) Use the Ratio Test to determine the radius of convergence of the power series

∑_{n=1}^{∞}(−1)^{n} \frac{x^{n}}{n^{2}} .

Extra Credit. For a water tank with cross-section area A(y) at height y above the bottom and an opening of area 1 at y = 0, Torricelli’s Law implies that as the water drains out through the opening, the height y of the water in the tank changes according to

A(y)\frac{dy}{dt} = −\sqrt{2gy}

(where g is the acceleration of gravity at the surface of the Earth). A) (5) Suppose that the tank has the shape of the solid of revolution formed by rotating

the area between a curve x = f(y) and the y-axis, 0 ≤ y ≤ h, about the y-axis. How should f(y) be chosen in order for the water level to decrease at a constant rate?

B) (5) Suppose you had such a tank. Explain how you could use it as a time measuring device. (Note: The ancient Greeks actually constructed tanks of this shape, and called them “water-clocks.” They were used to measure the time allotted to speakers in legal proceedings, for instance.)

Have a joyous and peaceful holiday season!