Final3.pdf

from 2011 tests CODE 111 Mathematics 1225B Page 1 Practice Final Exam PART A (35 marks) NOTE: YOUR ANSWERS TO THE PROBLEMS IN PART A MUST BE CODED ON THE SCANTRON SHEET. ALSO CIRCLE YOUR ANSWERS IN THIS BOOKLET. Only the (scantron) answer sheet will be marked for Part A. Extra time will NOT be given for coding answers at the end of the exam. A1. If log3x − log3x = 2, find x. 1 1 A: 6 B: 3 C: D: 9 E: 3 9 √ A2. If f(x) = 4e x, find f (4). 1 2 2 2 2 2 A: 2e B: e C: 2e D: 4e E: 8e A3. If f(x) = 2−x , find f (0). A: 0 B: 1 C: −1 D: − 1 E: −ln2 ln2 A4. Find the slope of the tangent line to the graph of y = eat the point where x = ln3. A: 8 B: 6 C: 12 D: 18 E: 9 A5. If 8x − 11 = A + B , find B. (x − 2)(x − 1) x − 2 x − 1 A: 1 B: 2 C: 3 D: 4 E: 5 Z 3 A6. Find 12x lnxdx. 1 A: x4 3lnx − + C B: 3x (xlnx − 1) + C C: 12x (lnx − 1) + C 4 4 2 D: 3x4 lnx − 1 + C E: 3x (lnx) + C 4 2 Z 3 e 1 A7. Evaluate dx. e xlnx 3 1 2 A: ln3 B: e − e C: D: − E: 0 3 3 Mathematics 1225B CODE 111 from 2011 tests Practice Final Exam Page 2 Z ∞ −1 A8. Evaluate x5 dx. 1 A: 1 B: − 1 C: 1 D: −1 E: diverges 4 4 A9. Determine which one of the following integrals represents the area of the region bounded by y = x and y = 4. Z Z Z Z Z 2 2 4 2 2 √ 4 √ 4 √ A: (4 − x )dx B: (4 − x )dx C: y dy D: y dy E: 2 y dy 0 0 0 0 0 A10. Find the area of the region bounded by y = x − x and y = x. 8 8 4 4 16 A: B: − C: D: − E: 3 3 3 3 3 A11. Find √he volume of the solid of revolution obtained by rotating the region bounded by y = x, y = 2 and x = 0 about the x-axis. 8π 16π 32π A: 6π B: 8π C: D: E: 3 3 5 A12. Find √he volume of the solid of revolution obtained by rotating the region bounded by y = x, y = 2 and x = 0 about the y-axis. A: 6π B: 8π C: 8π D: 16π E:32π 3 3 5 A13. If f(x,y) = x − 3xy + y , find f y3,1). A: −7 B: 3 C: 24 D: −6 E: 1 A14. If f(x,y) = y , find fyxx,y). A: y lny B: xyx−1 C: yx−1 D: yx−1(1 + xlny) E: y x A15. Let f(x,y) = ye − 3x − y. Find the only critical point of f(x,y). A: (0,0) B: (0,3) C: (1,1) D: (3,1) E: (−3,−1) from 2011 tests Mathematics 1225B Page 3 CODE 111 Practice Final Exam A16. Find all the critical points of the function f(x,y) = 3x − x − 3xy . 2 A: (0,0) B: (1,1),(−1,−1) C: (1,0),(−1,0),(0,1),(0,−1) D: (0,0),(−1,1),(1,1) E: (1,−1),(1,1),(−1,1),(−1,−1) Use the following information for questions 17, 18 and 19. 3 2 f(x,y) = x − 6xy − y 2 fx(x,y) = 3x − 6y f yx,y) = −6x − 2y fxx (x,y) = 6x f yy,y) = −2 fxyx,y) = −6 A17. Which one of the following is true for the point (1,−3)? A: (1,−3) is not a critical point of f(x,y). B: f(x,y) has a saddle point at (1,−3). C: f(x,y) has a local minimum at (1,−3). D: f(x,y) has a local maximum at (1,−3). E: The second partials test yields no information. A18. Which one of the following is true for the point (−6,18)? A: (−6,18) is not a critical point of f(x,y). B: f(x,y) has a saddle point at (−6,18). C: f(x,y) has a local minimum at (−6,18). D: f(x,y) has a local maximum at (−6,18). E: The second partials test yields no information. A19. Which one of the following is true for the point (0,0)? A: (0,0) is not a critical point of f(x,y). B: f(x,y) has a saddle point at (0,0). C: f(x,y) has a local minimum at (0,0). D: f(x,y) has a local maximum at (0,0). E: The second partials test yields no information. A20. If the method of Lagrange multipliers is used to maximize the function f(x,y) = xy + 10 subject to the constraint x + 9y = 18, what system of equations must be solved? 2 2 A: xy + 10 + λ(x + 9y − 18) = 0 B: 2x + λy = 0 C: y + 2xλ = 0 18y + λx = 0 x + 18yλ = 0 xy + 10 = 0 2 2 x + 9y − 18 = 0 D: xy + 2λx = 0 E: xy + 10 = 0 2 2 xy + 2λy = 0 x + 9y − 18 = 0 x + 9y − 18 = 0 Mathematics 1225B from 2011 tests Practice Final Exam CODE 111 Page 4 A21. In using Lagrange’s method to solve a particular constrained optimization problem, the following system of equations needs to be solved: 1 + 2λx = 0 3 + 2λy = 0 2x + y 2 = 11 Find all possible solutions to this system. √ √ √ √ A: ( 5,1) and (− 5,−1) B: ( 5,−1) and (− 5,1) C: (1,−3) and