Mathematics 134 – Intensive Calculus for Science 2 Final Examination – May 10, 2006

Directions: Do all work in the blue exam booklet. There are 200 regular points and 20 Extra Credit points, distributed as marked. Take a deep breath before plunging in and read each question carefully.

I. A) (10) Give a precise statement of the Fundamental Theorem of Calculus (both parts). B) (15) The following graph shows y = f(x). Let F be the antiderivative of f with

F (0) = 0 and F continuous. Sketch the graph y = F (x).

II. Compute each of the following integrals. You may use the table of integrals anywhere on these. If you do, say which table entry you are using.

A) (10) ∫ 3x^{6} − 4\sqrt{x} + sin(x) dx

B) (10) ∫ \frac{x^{3}}{x^{4}+1} dx

C) (15) ∫ x^{2}e^{−9x} dx

D) (10) ∫ \frac{1}{\sqrt{4x^{2}+9}} dx

E) (15) ∫ \frac{x+1}{x^{2}+5x+6} dx

III. Let R be the region bounded by y = x, the x-axis, and x = 1, x = 4.

A) (15) Find the volume of the solid obtained by rotating R about the line y = −2.

B) (10) A thin plate has the shape of the region R (x, y in cm) and density δ(x) = x^{−1/2} grams/cm^{2}. Find its total mass.

IV. At a particular location in Natick on the Mass Pike, a sensor was set up to measure the passage of traffic. The measurements made were used to derive a probability density function for the quantity x = time gap between successive cars (in minutes). The results gave the following formula as a good fit for the pdf: p(x) = 11(1 − x)^{10} if 0 < x < 1, and zero otherwise.

A) (10) Show that p satisfies the usual property for a probability density function: ∫_{0}^{1} p(x) dx = 1.

B) (15) What is the probability that the time gap between successive cars is between x = .1 minute and x = .2 minute?

V. A) (10) Using the definition of Taylor polynomials, compute the Taylor polynomial of

degree n = 3 for f(x) = \sqrt{1 + 2x} at a = 0. B) (10) Use our shortcut methods to check your work in part A. C) (5) Use your polynomial from part A to compute an approximation to \sqrt{1.2}. What is

the error in your approximation?

VI. (15) Solve for y by separation of variables: \frac{dy}{dx} = cos(x)(1 + y^{2}) with y(0) = 1.

VII. An avian flu epidemic has broken out in Birdsburgh, a large city with total population 10 million. Write N for the number of people who have been infected, as a function of time. The Birdsburgh Public Health department determines that:

(1)  \frac{dN}{dt} = kN(10 − N),

or in words: the rate of change of N is proportional to the product of N and the number of people not yet infected, where N is in millions of people, t in weeks, k a positive constant. A) (10) Which of the following slope field plots matches (1)? Explain how you can tell.

B) (10) For what value of k is N(t) = 10/(1 + 1000e^{−t}) a solution of (1)? C) (5) If the epidemic proceeds according to the function N(t) in part B, how many weeks

will pass before the number of infected people reaches 1 million?

Extra Credit (20) The hull of a boat is 20 feet long. At a distance s feet from the bow (the front), the cross section of the part of the hull below y = 0 (the waterline) has the shape of the region in the xy-plane below y = 0 and above the parabola y = ax^{2} − b, where a, b are given in the following table:

\begin{matrix}s & 5 & 10 & 15 & 20 \\ a & 2 & 3 & 4 & 5 \\ b & 2 & 3 & 4 & 4 \end{matrix}

Estimate the volume enclosed by the hull below the water line.

Have an enjoyable, safe, and productive summer!