College of the Holy Cross, Spring Semester, 2008

MATH 132, Section 01, Final Exam

Thursday, May 8, 2:30 PM Professor Little

Your Name:

Instructions: For full credit, you must show all work on the test pages and place your final answer in the box provided for the problem. Use the back of the preceding page if you need more space for scratch work. The numbers next to each part of the questions are their point values.

Please do not write in the space below

Problem Points/Poss
I  / 15
II  / 55
III  / 15
IV  / 15
V  / 30
VI  / 10
VII  / 20
VIII  / 15
IX  / 25
Total  /200

Have a safe, productive, and enjoyable summer!

(A) (5) Compute the derivative of the function g(x) = ∫_{0}^{3x} \frac{sin(t)}{t} dt.

g′(x) =

(B) (4) Let f(x) = \begin{cases}1 & if 0 ≤ x ≤ 2 \\ x − 1 & if 2 < x ≤ 4. \\ 11 − 2x & if 4 < x ≤ 6 \end{cases} Sketch the graph y = f(x) on the axes provided below.

(C) (6) Let F (x) = ∫_{0}^{x} f(t) dt, where f(t) is the function from part (B). Complete the following table of values for F (x):

x  0  1  2  3  4  5  6
F (x) 0

II. Compute the following integrals. Some of these may be forms covered by entries in the table of integrals. Half credit will be given for using a table entry; full credit only for showing all work leading to the final answer.

(A) (5) ∫ \frac{x^{4} − 3x^{3} + \sqrt{x}}{x^{2/3}} dx

Answer

(B) (5) ∫ x cos(x^{2}) dx

Answer

(C) (10) ∫ \frac{sec^{2}(5x) dx}{(tan(5x) + 7)^{5}}

Answer

(D) (10) ∫_{1}^{e} x^{7} ln(x) dx.

Answer

(E) (12.5) ∫ \frac{x^{2}}{\sqrt{25 − x^{2}}} dx Answer

5

(F) (12.5) ∫ \frac{x}{(x + 1)(x^{2} + 1)} dx Answer

6 (A) (5) Use a midpoint Riemann sum with n = 4 to approximate ∫_{0}^{2} e^{x^{2}} dx.

Answer

(B) (5) Use the Trapezoidal Rule with n = 4 to approximate ∫_{0}^{2} e^{x^{2}} dx.

Answer

(C) (5) Check the appropriate boxes: The midpoint approximation is a overestimate □/underestimate □, and the trapezoidal rule approximation is a overestimate □/underestimate □, because e^{x^{2}} is concave up □/concave down □ on [0, 2].

Note: From this point on in the exam, if an entry from the table of integrals applies, you may use it for full credit if you state which entry you are using and indicate what u and what constant values a, b, etc. are involved.

IV. For each of the following improper integrals, set up and evaluate the appropriate limits to determine whether the integral converges. If so, find its value; if not, say “does not converge.”

(A) (6) ∫_{0}^{2} \frac{1}{\sqrt[3]{x − 1}} dx.

Answer

(B) (9) ∫_{0}^{∞} \frac{1}{(x^{2} + 4x + 5)^{3/2}} dx.

Answer V. A region R in the plane is bounded by the graphs y = 9 − x^{2}, y = 2x, x = 0 and x = 1. (A) (5) Sketch the region on the following axes:

(B) (5) Compute the area of the region R.

Area =

(C) (10) Compute the volume of the solid obtained by rotating R about the x-axis.

Volume = (D) (10) Set up the integral(s) to compute the volume of the solid obtained by rotating R

about the y-axis. You do not need to compute the value.

Volume =

VI. (10) The daily solar radiation x per square meter (in hundreds of calories) in Florida in October has a probability density function f(x) = \frac{3}{32}(x − 2)(6 − x) if 2 ≤ x ≤ 6, and zero otherwise. Find the mean daily solar radiation.

Mean = VII. Does each of the following infinite series converge or diverge? For full credit, you must justify your answer completely by showing how the indicated test applies and leads to your stated conclusion.

(A) (10) ∑_{n=1}^{∞} \frac{n}{e^{2n}} – Integral Test

Answer

(B) (10) ∑_{n=1}^{∞} \frac{3^{n}}{n^{2}e^{n}} – any applicable method.

Answer (A) (7.5) Using the Ratio Test and testing the endpoints, determine the interval of conver-

gence for the power series ∑_{n=1}^{∞} \frac{2^{n}x^{n}}{n}.

Interval of convergence

(B) (7.5) Find the Taylor series for the function f(x) = sin(x/2). Express your answer in summation (“sigma”) notation.

Taylor series = (A) (5) A drug is administered to a patient intravenously at a constant rate of 10mg per

hour. The patient’s body breaks down the drug and removes it from the bloodstream at a rate proportional to the amount present. Write a differential equation for the function Q(t) = amount of the drug present (in mg) in the bloodstream at time t (in hours) that describes this situation. You do not need to solve the equation.

Answer

(B) (10) Find the general solution of the differential equation y′ = y\sqrt{e^{2x} + 1}.

General solution = (C) (10) A population P of tree frogs is undergoing logistic growth following the differential

equation P′ = (.1)P (1 − \frac{P}{100}), t in years. If the initial population is P (0) = 10, how long does it take for the population to reach 45?

Time to reach 45 =