Mathematics 136, section 2 – Advanced Placement Calculus Final Examination – December 18, 2009

Directions

Do all work in the blue exam booklet. There are 200 regular points in problems I through IX, and 10 Extra Credit points. If a problem includes specific directions about how to solve it, full credit will be given only for a solution by that method. Take a deep breath before plunging in, don’t panic, and have some fun! I. Consider the functions:

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

A) (15) Find the derivatives of f_{1}, f_{2}, f_{3}.

B) (5) Find the equation of the tangent line to the graph y = f_{1}(x) + f_{3}(x) at (0, 2).

II. Figure 1 (at the top of the next page) shows the graph y = f(x) for some function f(x). A) (10) Sketch the graph y = −f(x + 1). B) (10) 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.

III.

A) (10) State the definition of the definite integral of a function f(x) from a to b.

B) (10) State both parts of the Fundamental Theorem of Calculus.

IV. The region R in the xy-plane is bounded by y = x^{3} − 1, x = 1, x = 2, and the x-axis. A) (10) Find the volume of the solid obtained by rotating R about the x-axis. B) (5) Set up the integral to compute the volume of the solid obtained by rotating R about the

line x = 1. You do not need to evaluate. C) (10) Find the coordinates of the centroid of R.

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

A) (5) ∫ x^{2} sin(3x^{3}) cos(3x^{3}) dx

Figure 1: Figure for Problem II

B) (5) ∫_{0}^{∞} \frac{dx}{x^{2} + 9}

C) (10) ∫ x^{2} ln(2x) dx (use parts)

D) (15) ∫ \frac{x^{2}}{\sqrt{25 − x^{2}}} dx (use a trigonometric substitution) VI. (15) Find the arclength of the catenary curve y = \frac{e^{x} + e^{−x}}{2} from x = 0 to x = 2. VII. All parts of this question refer to the differential equation

\frac{dy}{dx} = y^{2} − y.

A) (10) Sketch the slope field showing the slopes at points on the lines y = −1/2, 0, 1/2, 1, 3/2. B) (10) Find the solution with y(0) = 1/2.

VIII. A hive of honey bees is attacked by a fungal infection at t = 0, and the population declines at a rate proportional to the square root of the population.

A) (5) Express this statement about the rate of growth of the population P as a differential equation.

B) (10) Use separation of variables to find the general solution of your differential equation.

C) (5) At t = 0 there were 900 bees; 441 were left after 6 weeks. When did the bee population disappear entirely?

IX. Series.

A) (5) Find the sum of the infinite series

4 − \frac{8}{3} + \frac{16}{9} − \frac{32}{27} + · · · .

B) (10) Find the Taylor series of g(x) = e^{−x} at a = 0.

C) (10) Compute the Taylor polynomial of degree 2 for f(x) = \sqrt{x} at a = 1 and use it to approximate \sqrt{1.3}. How close is the approximation?

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 the equation

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 Egyptians actually constructed tanks of this shape and used them to measure time as early as the 16th century BC (i.e. long before calculus!). The Greeks took over the idea and their name for this sort of device was a clepsydra.)

Have a joyous and peaceful holiday season!