MTH 34 Solutions to Exam 1 Feb. nd 016 Name: Section: Recitation Instructor: READ THE FOLLOWING INSTRUCTIONS. Do not open your exam until told to do so. No calculators, cell phones or any other electronic devices can be used on this exam. Clear your desk of everything excepts pens, pencils and erasers. If you need scratch paper, use the back of the previous page. Without fully opening the exam, check that you have pages 1 through 10. Fill in your name, etc. on this first page. Show all your work. Write your answers clearly! Include enough steps for the grader to be able to follow your work. Don t skip limits or equal signs, etc. Include words to clarify your reasoning. Do first all of the problems you know how to do immediately. Do not spend too much time on any particular problem. Return to difficult problems later. If you have any questions please raise your hand and a proctor will come to you. There is no talking allowed during the exam. You will be given exactly 90 minutes for this exam. I have read and understand the above instructions:. SIGNATURE Page 1 of 10
MTH 34 Solutions to Exam 1 Feb. nd 016 Standard Response Questions. Show all work to receive credit. Please BOX your final answer. 1. (10 points) Consider the position function r(t) = sin (6t), t, 5 cos (6t). Calculate the velocity, and speed when t = π 3. v(t) = 6 cos(6t), 1, 30 sin(6t) v(π/3) = 6 cos(4π), 1, 30 sin(4π) = 6, 1, 0 Velocity at t = π/3 v(π/3) = 36 + 1 + 0 = 37 Speed at t = π/3. (a) (8 points) Find the linearization of the function f(x, y) = 4x ln (xy 1) 1 at the point (, 1). Now we have f x = 4 ln(xy 1) + L(x, y) = f x (, 1)(x ) + f y (, 1)(y 1) + f(, 1) = f x (, 1)(x ) + f y (, 1)(y 1) 1 4xy xy 1 and f y = 4x xy 1 = 8(x ) + 16(y 1) 1 = 8x + 16y 33 so therefore: (b) (4 points) Use the linearization to approximate f(.0, 1.). L(x, y) = 8x + 16y 33 L(.0, 1.) = 8(.0) + 16(1.) 33 = 16.16 + 19. 33 =.36 Page of 10
MTH 34 Solutions to Exam 1 Feb. nd 016 3. Evaluate the following limits if they exist or prove that they do not exist. 3xy (a) (8 points) lim (x,y) (0,0) x + 3y Consider the two paths Path 1: x = 0 and y 0. Then lim x=0 y 0 3xy x + 3y = lim y 0 3xy 0 3y = 0 3x 4x = 3 4 Path : y = x and x 0. Then lim y=x x + 3y = lim x 0 x 0 Since these two paths yield different results the limit does not exist. (b) (8 points) lim (x,y) (0,0) 5xy x + y Because y x + y we know that so by the Squeeze Theorem since we know lim (x,y) (0,0) 5xy x + y = 0 5 x y 1 when (x, y) (0, 0). Therefore x + y 5xy x + y 5 x lim 5 x = 0 and lim 5 x = 0 (x,y) (0,0) (x,y) (0,0) Page 3 of 10
MTH 34 Solutions to Exam 1 Feb. nd 016 4. (1 points) Find an equation of a plane containing the points: A(1, 1, ), B(3,, 1), and C( 1, 1, ). AB =, 1, 1 and AC =, 0, 0. So therefore n = AB AC = i j k 1 1 0 0 = (0 0)i (0 )j + (0 ( ))k = 0,, An equation of the plane would be 0 = (y 1) + (z ) or equivalently y + z = 3. 5. (10 points) Parametrize the curve given by the intersection of the surfaces shown below. Hint: make sure you get the entire curve and don t forget to write down the domain of your parameter! z x + y = 1 x + z = x y x + y = 1 can be parametrized by x = cos t and y = sin t with t [0, π]. Then since z = x we can use parametrize z by z = cos t giving us the final solution: r(t) = cos t, sin t, cos t t [0, π] Page 4 of 10
MTH 34 Solutions to Exam 1 Feb. nd 016 6. (1 points) Given that u = i j + k and v = 3i k find the proj u (v). proj u (v) = u v u u = 3 1 + 1 + 4 1, 1, = 1 1, 1, 6 7. (1 points) Compute w r at (r, s) = (, 0), given that w = x y 3 with x = 1 r + rs 3 and y = r + se s. Note that x(, 0) = y(, 0) = x r = r + s 3 y r = 1 x r (, 0) = y r (, 0) = 1 w x = x w y = 3y w x (, ) = 4 So therefore by the chain rule w r = 4() 1(1) = 4 (r,s)=(,0) w y (, ) = 1 Page 5 of 10
MTH 34 Solutions to Exam 1 Feb. nd 016 8. (16 points) Consider the function f(x, y) = (a) Sketch the domain of f 1 y x y 4 3 1 4 3 1 1 1 3 4 x 3 4 (b) Sketch the level curve f(x, y) = 1 y 4 3 1 4 3 1 1 1 3 4 x 3 4 (c) What is the range of f? The range of f is (, 0). Page 6 of 10
MTH 34 Solutions to Exam 1 Feb. nd 016 Multiple Choice. Circle the best answer. No work needed. No partial credit available. 9. (6 points) If f(x, y, z) = xy z 3 + sin 1( x z ) then f xyz (3, 1, 1) = A. 6 B. 6 C. 0 D. 18 E. 18 x y 10. (5 points) Evaluate the limit: lim (x,y) (1,1) y x A. 1 B. 1/ C. 0 D. 1/ E. 1 F. The limit does not exist 11. (5 points) Consider the planes: x + y z = 1 and x + y + z = 3. A line of intersection of the planes is: A. 1, 1, 0 t +, 1, B. 1, 1, 0 t + 1, 1, 0 C. 1, 1, 0 t + 1, 1, 0 D. 1, 1, 0 t + 1, 0, 1 E., 1, t + 0, 1, 1 Page 7 of 10
MTH 34 Solutions to Exam 1 Feb. nd 016 1. (6 points) If P = (6, 4, z) is on the plane that goes through (1, 0, 1), (0, 1, 0), and (1, 1, 1), what must z equal? A. 6 B. 7 C. 8 D. 8 E. 10 13. (5 points) If particle A moves on a path defined by t, t and particle B moves on a path defined by t, t 3, at what positive time t do they have the same speed? A. B. 4 3 C. 1 D. E. 3 14. (6 points) What is the length of the curve defined by t + 1, sin t, + cos t with t [1, 3]? A. B. 3 C. 5 D. 3 5 E. 10 Page 8 of 10
MTH 34 Solutions to Exam 1 Feb. nd 016 15. (6 points) The tangent plane of the paraboloid z = 5 x y at (1,, 0) is given by: A. z = (x 1) 4(y ) B. z = (x 1) + 4(y ) C. z = (x 1) 4(y ) D. z = (x 1) + 4(y ) E. z = 3(x 1) + (y ) 16. (5 points) The intersection of the quadric surfaces defined by x + y + z = 1 and x y + z = 1 is: A. one point B. two points C. a straight line D. a parabola E. a circle 17. (6 points) What is the y-coordinate of the point where the curve defined by t 1, 3t, 1 intersects the cone z = x + y for t > 0? A. 3/5 B. 5/ C. 3/10 D. 8/5 E. 15 Page 9 of 10
MTH 34 Solutions to Exam 1 Feb. nd 016 Congratulations you are now done with the exam! Go back and check your solutions for accuracy and clarity. Make sure your final answers are BOXED. When you are completely happy with your work please bring your exam to the front to be handed in. Please have your MSU student ID ready so that is can be checked. DO NOT WRITE BELOW THIS LINE. Page Points Score 3 16 4 5 4 6 16 7 16 8 17 9 17 Total: 150 Page 10 of 10