Math Final Exam - 6/11/2015

Transcription

1 Math Final Exam - 6/11/2015 Name: Section: Section Class/Times Instructor Section Class/Times Instructor 1 9:00%AM ( 9:50%AM Papadopoulos,%Dimitrios 11 1:00%PM ( 1:50%PM Swartz,%Kenneth 2 11:00%AM ( 11:50%AM Papadopoulos,%Dimitrios 13 12:00%PM ( 12:50%PM Yang,%Dennis 3 11:00%AM ( 11:50%AM Lee,%Hwan%Yong 14 2:00%PM ( 2:50%PM Aran,%Jason 4 4:00%PM ( 4:50%PM Aran,%Jason 16 12:00%PM ( 12:50%PM Zhang,%Aijun 6 9:00%AM ( 9:50%AM Lee,%Hwan%Yong 17 4:00%PM ( 4:50%PM Yang,%Dennis 7 10:00%AM ( 10:50%AM Papadopoulos,%Dimitrios 18 1:00%PM ( 1:50%PM Yang,%Dennis 8 5:00%PM ( 5:50%PM Aran,%Jason 19 10:00%AM ( 10:50%AM Lee,%Hwan%Yong 10 2:00%PM ( 2:50%PM Swartz,%Kenneth 20 1:00%PM ( 1:50%PM Akin,%Myles The following rules apply: This is a closed-book exam. You may not use any books or notes on this exam. For free response questions, you must show all work. Answers without proper justification will not receive full credit. Partial credit will be awarded for significant progress towards the correct answer. Cross off any work that you do not want graded. For multiple choice questions, circle the letter of the best answer. Make sure your circles include just one letter. These problems will be marked as correct or incorrect; partial credit will not be awarded for problems in this section. You have 2 hours to complete this exam. When time is called, stop writing immediately and turn in your exam to the nearest proctor. You may not use any electronic devices including (but not limited to) calculators, cell phone, or ipods. Using such a device will be considered a violation of the university s academic integrity policy and, at the very least, will result in a grade of 0 for this exam. Page Points Score Total: 100

2 Math 200 Final Exam - Page 2 of 13 6/11/2015 Part I: Free Response 1. (10 points) Find an equation of the plane which contains the point P (2, 1, 2) and is perpendicular to the planes 3x + y z = 6 and 2x + 2y + z = 4

3 Math 200 Final Exam - Page 3 of 13 6/11/ (10 points) Let R be the closed region bounded by y = 9 x 2 and the x-axis, shown below. Calculate the absolute maximum and absolute minimum values of f(x, y) = xy + 3x on R and indicate where they occur.

4 Math 200 Final Exam - Page 4 of 13 6/11/2015 u = y + x 3. (15 points) Consider the transformation v = y x x = 1 (u v) 2 y = 1 (u + v) 2 (a) Suppose R is the region in the xy-plane enclosed by y = x, y = x + 2, y = 1 x and y = x, as shown below. Use the given transformation to find the corresponding region in the uv-plane. Sketch your results on the axes provided.

5 Math 200 Final Exam - Page 5 of 13 6/11/2015 (b) Find the absolute value of the determinant of the Jacobian (x, y) (u, v) (c) Use the transformation to evaluate R 2(y x)(y + x) da.

6 Math 200 Final Exam - Page 6 of 13 6/11/ (10 points) Evaluate the following integral by converting to Polar Coordinates. For full credit, you must sketch the region over which your are integrating. 3 3 x y 2 dy dx Hint: cos 2 θ = 1 2 (1 + cos 2θ) and sin2 θ = 1 (1 cos 2θ) 2

7 Math 200 Final Exam - Page 7 of 13 6/11/2015 Part II: Multiple Choice 5. (5 points) What is the area of the parallelogram which has u = 2, 0, 1 and v = 4, 1, 2 as adjacent sides? (a) 2 3 (b) 2 5 (c) 3 2 (d) 4 2 (e) 5 6. (5 points) If b and v are both nonzero vectors, which of the following vectors is always orthogonal to v proj b v? (a) b (b) v (c) b v (d) b + v (e) proj v b

8 Math 200 Final Exam - Page 8 of 13 6/11/ (5 points) The position of a particle moving through space is given by r (t) = cos(2t), sin(2t), e t. Find the velocity of the particle at time t = 0. (a) 2, 0, 1 (b) 0, 2, 1 (c) 1, 2, 0 (d) 2, 2, 1 (e) 2, 2, 1 8. (5 points) Suppose f(x, y) is an integrable function. Which of the following integrals results from reversing the order of integration of x f(x, y) dy dx x f(x, y) dy dx (a) (b) (c) (d) (e) 3 y y 0 y 3 y 3 y 3 y x 3 f(x, y) dx dy f(x, y) dx dy f(x, y) dx dy f(x, y) dx dy f(x, y) dx dy x 0 f(x, y) dx dy

9 Math 200 Final Exam - Page 9 of 13 6/11/ (5 points) Consider the region in the first octant bounded by y = 1, x + z = 2, and the coordinate planes, shown below. Which of the following integrals represents the volume of this region? (a) (b) (c) (d) (e) x z 2 1 x z dzdydx dzdxdy dzdydx dxdydz dzdydx

10 Math 200 Final Exam - Page 10 of 13 6/11/ (5 points) A differentiable function f(x, y) has two critical points: (0, 2) and (0, 2). And, it satisfies the following: Which of the following is correct? f xx (x, y) = 2 f xy (x, y) = 0 f yx (x, y) = 0 f yy (x, y) = 6y (a) f has a local minimum at (0, 2). (b) f has a local maximum at (0, 2). (c) f has a local minimum at (0, 2). (d) f has a local maximum at (0, 2). (e) f has saddle points at (0, 2) and (0, 2). 11. (5 points) In which direction is f(x, y) = x 2 y x increasing most rapidly at P ( 2, 1)? y2 (a) 5, 0 (b) 0, 0 (c) 2, 2 (d) 3, 1 (e) 3, 2

11 Math 200 Final Exam - Page 11 of 13 6/11/ (5 points) What is the distance from the point P (1, 4, 4) to the plane x + 2y + z = 7? (a) 1 (b) 2 (c) 6 (d) 2 3 (e) (5 points) Consider the following definition. Definition: The average value of a function z = f(x, y) over a region R in the xy-plane is: f(x, y) da R f avg (x, y) = 1 da R Determine the average value of f(x, y) = 2xy on R = {(x, y) 0 x 3 and 0 y 2}. (a) 3 (b) 6 (c) 9 (d) 12 (e) 18

12 Math 200 Final Exam - Page 12 of 13 6/11/ (5 points) Consider the solid enclosed by z = 3x 2 + 3y 2 and z = 4 x 2 y 2, shown below. Which of the following represents the volume of this solid using cylindrical coordinates? (a) (b) (c) (d) (e) 2π 1 4 r r 2 2π π 4 r 2 2π 1 4 r r 2 2π 3 4 r r 2 2π 4 3 r dz dr dθ 3r 2 dz dr dθ r 2 sin θ dr dz dθ r dz dr dθ r sin θ dr dz dθ

13 Math 200 Final Exam - Page 13 of 13 6/11/ (5 points) Consider the solid which is bounded above by x 2 + y 2 + z 2 = 4 and below by z = 1, shown below. Which of the following represents the volume of this solid using spherical coordinates? (a) (b) (c) (d) (e) 2π π/ π π/ π π/ sec φ 2π π/ sec φ 2π π/ sec φ 1 dρ dφ dθ ρ 2 sin φ dρ dφ dθ 1 dρ dφ dθ ρ 2 sin φ dρ dφ dθ ρ 2 sin φ dρ dφ dθ