Holy Cross College, Spring Semester, 2005 MATH 132, Section 01, Final Exam Tuesday, May 10, 2:30 PM Professor Little

1. The following graph shows the function y = f(x).

(a) [10 points] Let F (x) be the antiderivative of f(x) with F (0) = 0. Compute the entries in the following table of values:

x  0  1  2  3  4
F (x) 0

(b) [10 points] Sketch the graph y = F (x) on the interval 0 ≤ x ≤ 4:

2. Integrate each of the following. You must show all work and cite any table entries you use for full credit.

(a) [5 points] ∫ \frac{x^{2} + 3x + 5}{x^{3/2}} dx

(b) [10 points] ∫ \frac{e^{t} + 1}{(e^{t} + t)^{2}} dt

(c) [10 points] ∫ \frac{dx}{(9 − x^{2})^{3/2}} (use a trigonometric substitution)

(d) [10 points] ∫_{0}^{1} x^{2}e^{2x} dx

(e) [10 points] ∫ \frac{(x − 1) dx}{x^{3} + 25x}

3. All parts of this problem refer to the region R in the plane bounded by y = 2 + cos(x), y = 0, x = 0, and x = π, shown below:

(a) [10 points] Find the volume of the solid with base R whose cross-sections by planes perpendicular to the x-axis are squares.

(b) [10 points] True or False: The solid obtained by rotating R about the x-axis has volume π times the volume from part a. (For full credit, give the integral that would compute the volume of this solid of revolution, and answer the question.)

(c) [15 points] A thin metal plate has the shape of the region R (units in meters) and constant mass density 1kg/m^{2}. Find the x-coordinate of its center of mass.

(d) [10 points] Set up, but do not evaluate the integral giving the arclength of the top edge of R.

4. The following graph shows a probability density function (pdf) for a quantity x:

(a) [5 points] Which of the following two plots shows the corresponding cumulative distribution for x:

,

(b) [5 points] Using the appropriate graph, estimate the median of x.

5. Does each of the following series converge? For full credit, you must justify your answer completely by showing how the indicated test leads to your stated conclusion.

(a) [5 points] ∑_{n=1}^{∞} \frac{(−1)^{n}}{n^{1/3}} – alternating series test.

(b) [5 points] ∑_{n=1}^{∞} \frac{1}{n^{9/10}} – integral test

(c) [5 points] ∑_{n=1}^{∞} \frac{3^{n}}{π^{n}} – any applicable method

6. Using the ratio test, determine the radius of convergence of the following power series

∑_{n=0}^{∞} \frac{(x + 2)^{n}}{n^{2}4^{n}}

([15 points]).

7. All parts of this problem refer to f(x) = cos (\frac{x}{2}).

(a) [15 points] Using the definition of Taylor polynomials, compute the Taylor polynomial of degree n = 6 for f(x) at a = 0.

(b) [10 points] Use your answer from part a to approximate cos(1.2).

8. When a course ends, it’s an unfortunate fact of life that students start to forget the material they have learned. One mathematical model for this process states that the rate at which a student forgets material is proportional to the difference between the material currently remembered and some positive constant a < 1. Let y be the fraction of the original material remembered t weeks after the course has ended. Then y(0) = 1, and the model assumes y(t) > a for all t.

(a) [5 points] Which of the following differential equations is the correct translation of the model described above:

I : \frac{dy}{dt} = y − a II : \frac{dy}{dt} = \frac{−k}{y − a}
III : \frac{dy}{dt} = k(y − a) IV : \frac{dy}{dt} = t(y − a).

(b) [10 points] Solve the equation you selected in part a by separation of variables.

(c) [10 points] Suppose a = .2, and after 3 weeks the student only remembers 80% of the material from the course (that is, y(3) = .8). How long will it be before the student only remembers one half of the material?