Math 107 Fall 2003 ALTERNATE FINAL EXAM Name Dec. 16, 2003 Student ID Number:

Circle your instructor’s name.

Instructor: Deng Haataja Rogge Thomas Circle your TA’s name, if applicable.

TA: Eubanks Evans Everson Harrington Higgins Miller

INSTRUCTIONS:

• This exam should have 6 pages of questions plus this cover sheet; please verify that it does.

• Work each problem completely and clearly in the space provided. Show all your work for full credit. If you use the back of exam pages for scratch work, make note of that in the space provided for the problem.

• Simplify as much as possible, except as noted. For example, write \sqrt{2}/2 instead of cos(π/4).

• Give exact answers only, except as noted. For example, use π instead of 3.1415 if π is the answer.

• Calculators are allowed (except TI-89’s, TI-92’s and similar models), but an answer will only be counted if it is supported by all the work necessary to obtain that answer.

• Do not spend too long on any one problem; note the point value of the problem when deciding how much time to spend! You do not need to work the problems in the order they appear.

Question Points Score
1  14
2  14
3  14
4  14
5  12
6  36
7  20
8  8
9  8
10  18
11  8
12  16
13  18
Total  200

Math 107 Alternate Final Exam Page 1

1. (14 points) Construct a definite integral to determine the area enclosed by the graphs of y − x^{2} = 1 and y = −x + 3. DO NOT EVALUATE THIS INTEGRAL.

2. (14 points) Set up integral(s) for computing the volume of the solid formed by rotating the region bounded by y = x^{2} − 1 and y = 1 − x^{2} about the line x = 2.

(a) Sketch the region, carefully labeling all relevant quantities.

(b) Set up integral(s) to commpute the volume of this solid. DO NOT EVALUATE THE INTEGRAL(S).

3. (14 points) A water tank in the shape of a right circular cylinder has a base radius of 6 feet and a height of 11 feet. Suppose that the tank contains 9 feet of water. Set up an integral to determine the work done in pumping all of the water out of the top of the tank. DO NOT EVALUATE THIS INTEGRAL.

4. (a) (7 points) Find the exact value of cos^{−1}(cos(4π/3))

(b) (7 points) Find \frac{d}{dx} tan^{−1}(sin x)

5. (12 points) Use a comparison test to determine whether the improper integral converges or diverges.

∫_{1}^{∞} \frac{cos^{2} x + \sqrt{x}}{x^{2} + 1} dx

Math 107 Alternate Final Exam Page 3

6. (12 points each) Evaluate the integrals. (a) ∫ tan^{3} x sec x dx

(b) ∫_{1}^{e}(ln x)^{2} dx

(c) ∫ \frac{3}{\sqrt{x^{2} + 2x + 10}} dx

Math 107 Alternate Final Exam Page 4

7. (10 points each) Given the Taylor series centered at 0 (the Maclaurin series) for

cos x = 1 − \frac{x^{2}}{2!} + \frac{x^{4}}{4!} − \frac{x^{6}}{6!} + ⋯, for all x.

(a) Use this series to estimate cos(0.5) accurate to three decimal places. How many terms are required? You must show your work.

(b) Use the first three nonzero terms of the Maclaurin series for cos x to estimate ∫_{0}^{1} cos x^{2} dx. You must show your work.

8. (8 points) Determine if ∑_{k=1}^{∞} 6(\frac{4}{5})^{k} converges or diverges. If it converges, what does it converge to?

Math 107 Alternate Final Exam Page 5

9. (8 points) Find the radius of convergence of this power series: ∑_{k=0}^{∞} \frac{k^{2}}{2^{k}} x^{k}

10. (9 points each) Determine if each of the following series converges absolutely, converges conditionally, or diverges. Show enough work to justify your answer.

(a) ∑_{k=0}^{∞} \frac{(−1)^{k+1}}{\sqrt{k^{2} + 3}}

(b) ∑_{k=1}^{∞} \frac{5k}{(k + 2)3^{k}}

11. (8 points) Find an x-y equation for the curve given by x = cos t, y = 3 sin t.

12. Consider the curve given by the parametric equations x = t^{2} − 1, y = cos 4t, where 0 ≤ t ≤ 2.

(a) (6 points) Compute the slope of the tangent line of this curve at t = π/2.

(b) (10 points) Set up an integral to compute the surface area of the surface obtained by revolving this curve about y = −3. DO NOT EVALUATE THIS INTEGRAL.

13. (a) (6 points) Sketch the graph of r = 1 − 2 cos(2θ).

(b) (12 points) Set up an integral to find the area of \underline{one small loop} of this graph. DO NOT EVALUATE THIS INTEGRAL.