Name Sec MATH 152 ID Final Exam Sections 513 - 515 1-11 /66 Fall 2007 12 /12 P. Yasskin 13 /15 14 /10 Total /103 Multiple Choice: (6 points each) 1. A plate is bounded by the curves y = x 2 , y = −x 2 and x = 2 measured in meters. Find the total mass of the plate if the surface density is ρ = 3 kg/m 2 . a. 8 3 b. 16 3 32 c. 3 d. 8 e. 16 2. A plate is bounded by the curves y = x 2 , y = −x 2 and x = 2 measured in meters. Find the center of mass of the plate if the surface density is ρ = 3 kg/m 2 . a. b. c. d. e. 3 ,0 4 3, 1 4 10 3, 6 4 5 3 ,0 2 3, 6 2 5 1 3. Compute ∫ x 2 e 2x dx. 2 a. x e 2x − x e 2x − 1 e 2x + C b. c. d. e. 2 x2 2 x2 2 x2 2 x2 2 e 2x − e 2x − e 2x − e 2x − 2 x e 2x + 2 x e 2x − 2 x e 2x + 2 x e 2x − 4 4 1 e 2x + C 4 1 e 2x + C 2 1 e 2x + C 2 1 e 2x + C 4 x = 2t 2 y = t3 for 0 ≤ t ≤ 1 is rotated about the y-axis. Which integral gives the area of the surface swept out? 4. The parametric curve a. 2π ∫0 t b. 2π ∫0 t3 16 + 9t 2 dt c. 4π ∫0 t3 16 + 9t 2 dt d. 2π ∫0 t4 16 + 9t 2 dt e. 4π ∫0 t4 16 + 9t 2 dt 1 1 1 1 1 16 + 9t 2 dt 2 5. Which term appears in the partial fraction expansion of a. b. c. d. e. 3x 2 − 4x − 20 ? x − 2 2 x 2 + 4 −2 x − 2 2 1 x − 2 2 2 x − 2 2 −2 x − 2 1 x − 2 6. Find the solution of the differential equation x y1 = 2. dy = 2y + x 3 satisfying the initial condition dx a. y = x 3 − 7 x 2 4 b. y = x + x 2 3 c. y = 2x 3 − x 2 9 + 1 x3 5 5x 2 5 e. y = + 1 x3 2 3 3x d. y = 3 7. The region in the first quadrant bounded by the curves y = x 2 , y = 0 and x = 2 is rotated about the y-axis. Find the volume of the solid swept out. a. 32 π 5 b. 64 π 5 c. 8π d. 4π e. 2π 1, 3 by its 3 rd degree Taylor 2 2 polynomial centered at x = 1, namely T 3 x = x − 1 − 1 x − 1 2 + 1 x − 1 3 , then the 2 3 Taylor Remainder Theorem says the error |R 3 x| is less than 8. If you approximate fx = lnx on the interval Taylor Remainder Theorem: If T n x is the n th degree Taylor polynomial for fx centered at x = a then there is a number c between a and x such that the remainder is f n+1 c R n x = x − a n+1 n + 1! a. 1 b. 1 2 c. 1 4 d. 1 8 e. 0 4 9. Compute lim x0 sinx 2 − x 2 . 3 e x − 1 − x3 a. − 1 b. c. d. e. 6 1 6 2 3 −1 3 1 3 10. Find the volume of the parallelepiped with edge vectors ⃗ a = −2, 2, 1, ⃗ b = 3, 2, 4 and ⃗c = 1, −2, 3 a. −26 b. 26 c. 26 d. −46 e. 46 11. If ⃗ u points Down and ⃗v points North West, in which direction does ⃗ u × ⃗v point? a. North East b. South c. South West d. South East e. Up 5 Work Out: (Points indicated. Part credit possible.) 1 x2 12. (12 points) Compute ∫ dx 0 4 − x2 13. (15 points) A cone of radius 4 and height 6 and vertex down is filled with water up to height 2. Find the work done to pump the water out the top. Give your answer as a multiple of ρg. 6 14. (10 points) Start from the Maclaurin series: a. (6 pt) Find the Maclaurin series for 1 = 1+x ∞ ∑−1 n x n n=0 1 . 1 + x 2 HINT: Differentiate both sides of the given series. ∞ b. (4 pt) Compute ∑ n=1 −1 n−1 n . 2 n−1 7