Math 240, Fall 2003 Final Exam
Instructions: This is a closed book exam. No calculators are al-lowed. You are allowed two sides of a 5” by 7” index card of (hand-written) notes. Write your name and Penn ID # on the answer sheet on the last page of this exam, and also on the white booklet. Do all of your work in the white booklet. For all the questions 1 through 7 record your answers on the answer sheet. If you fill your white booklet, raise your hand with your booklet in it and we will give you another. We will not answer any questions during the exam about the problems. If you are sure a problem is wrong or has a typo indicate that on your answer sheet and move on to the next problem. At the end of the exam, detach the answer sheet, insert it into the white booklet and turn them in. Donotturn in the list of exam questions. Each problem is worth 10 points. A partial credit of 5 points is given if you make a minor mistake in an otherwise correct solution (this applies to the multiple choice questions 1 through 6, and also to the question 7). If you circle the correct answer but give a wrong solution in the answer booklet, or do not give any solution, you will not receive any credit for this problem.
1. Evaluate Z C
y dx+z dy+x dz,
whereC is the line segment starting at (0,0,0) and ending at (6,8,5). The answer is (a) 111 (b) 5 1/2−1 2 (c) 51/2 −1 4 (d) 53/2 4 (e) 5 3/2−1 2 (f) 53/2−1 4 (g) 53/2 2 (h) 51/2 2
2. Evaluate the work done by the force F= (2x+e−y)i+ (4y−xe−y)j along the curve y=x4, 0≤x≤1. The answer is
(a) 4π (b) 16π
3 (c)
4π
3. Find the surface area of that portion of the paraboloidz = 4−x2−y2
that is above the xy-plane. The anwer is (a) 1 (b) 3 2 (c) 2 3 (d) 7 6 (e) − 7 6 (f) 6 7 (g) − 2 3 (h) −1
4. Let F=y3i+x3j+z3k. Find
Z Z
S
F·ndS,
wheren is the outward normal andS is the surface bounded below by
z = 0, above by z = 4−x2−y2, and on the sides by x2+y2 = 3. The
answer is (a) 111 (b) 5 1/2−1 2 (c) 51/2 −1 4 (d) 53/2 4
5. (free response) Use the Laplace transform to solve the following system of diffferential equations.
2dx dt + dy dt −2x= 1 dx dt + dy dt −3x−3y= 2.
6. A large tank is partially filled with 100 gallons of fluid in which 10 pounds of salt is dissolved. Brine containing 1/2 pound of salt per gallon is pumped into the tank at a rate of 6 gal/min. The well-mixed solution is then pumped out at a slower rate of 4 gal/min. Find the number of pounds of salt in the tank after 30 minutes. The answer is
(a) 111 (b) 5 1/2−1 2 (c) 51/2 −1 4 (d) 53/2 4
7. Let y(t) be the solution to the initial value problem y′′+ 2y′ =δ(t−1), y(0) = 0, y′(0) = 1. Then y(1) equals (a) 111 (b) 5 1/2−1 2 (c) 51/2 −1 4 (d) 53/2 4 (e) 5 3/2−1 2 (f) 53/2−1 4 (g) 53/2 2 (h) 51/2 2
8. If f(t) is the “traingular wave” whose graph is below, and F(s) is its Laplace transform, then F(1) equals
(a) 111 (b) 5 1/2−1 2 (c) 51/2 −1 4 (d) 53/2 4
9. Let y(x) be the solution to the equation (6x+ 4y)dx+ (4x−8y)dy= 0. If y(0) = 0, what is y(1)? (a) 111 (b) 5 1/2−1 2 (c) 51/2 −1 4 (d) 53/2 4 (e) 5 3/2−1 2 (f) 53/2−1 4 (g) 53/2 2 (h) 51/2 2
10. Let y(x) be the solution to the equation exydy dx =e −y +e−2x−y. If y(0) = 0, what is y(1)? (a) 111 (b) 5 1/2−1 2 (c) 51/2 −1 4 (d) 53/2 4
11. Let y(x) be the solution to the equation dy dx −5xy = 2. If y(0) = 0, what is y(1)? (a) 111 (b) 5 1/2−1 2 (c) 51/2 −1 4 (d) 53/2 4 (e) 5 3/2−1 2 (f) 53/2−1 4 (g) 53/2 2 (h) 51/2 2
12. If F(s) =e−(s−2) s−2 s2+ 9 3 s2+ 9, f(0) equals (a) 111 (b) 5 1/2−1 2 (c) 51/2 −1 4 (d) 53/2 4
13. The Laplace transform of etsin(3t) equals (a) 111 (b) 5 1/2−1 2 (c) 51/2 −1 4 (d) 53/2 4 (e) 5 3/2−1 2 (f) 53/2−1 4 (g) 53/2 2 (h) 51/2 2
14. The Laplace transform of cos(t)U(t−π) equals (a) 111 (b) 5 1/2−1 2 (c) 51/2 −1 4 (d) 53/2 4
15. Let
g(s) = 2s−1
s2(s+ 1)3.
The inverse Laplace transform ofg(s) equals (a) 111 (b) 5 1/2−1 2 (c) 51/2 −1 4 (d) 53/2 4 (e) 5 3/2−1 2 (f) 53/2−1 4 (g) 53/2 2 (h) 51/2 2
16. A beam of length L is embedded at its left end and is free at its right end. If w(x) = 2 for 0< x < L, find y(x).
(a) 111 (b) 5 1/2−1 2 (c) 51/2 −1 4 (d) 53/2 4
17. Let A= 5 4 0 1 0 2 0 2 5 . Find the solution to
A X =X′. (a) 111 (b) 5 1/2−1 2 (c) 51/2 −1 4 (d) 53/2 4 (e) 5 3/2−1 2 (f) 53/2−1 4 (g) 53/2 2 (h) 51/2 2
18. Let A= 1 4 0 2 8 0 0 2 5 . What is the rank of A?
(a) 111 (b) 5 1/2−1 2 (c) 51/2 −1 4 (d) 53/2 4
Answer sheet
Name (Print Legibly; Last , First) ,
Penn ID #
Circle the correct answers to questions 1 through 6 and write down the answer to question 7. 1. (a) (b) (c) (d) (e) (f) (g) (h) 2. (a) (b) (c) (d) (e) (f) (g) (h) 3. (a) (b) (c) (d) (e) (f) (g) (h) 4. (a) (b) (c) (d) (e) (f) (g) (h) 5. (a) (b) (c) (d) (e) (f) (g) (h) 6. (a) (b) (c) (d) (e) (f) (g) (h) 7. (a) (b) (c) (d) (e) (f) (g) (h) 8. (a) (b) (c) (d) (e) (f) (g) (h) 9. (a) (b) (c) (d) (e) (f) (g) (h) 10. (a) (b) (c) (d) (e) (f) (g) (h) 11. (a) (b) (c) (d) (e) (f) (g) (h) 12. (a) (b) (c) (d) (e) (f) (g) (h)
13. (a) (b) (c) (d) (e) (f) (g) (h)
14. (a) (b) (c) (d) (e) (f) (g) (h)
15. (a) (b) (c) (d) (e) (f) (g) (h)
16. (a) (b) (c) (d) (e) (f) (g) (h)
