Name: Section: Recitation Instructor: INSTRUCTIONS Fill in your name, etc. on this first page. Without fully opening the exam, check that you have pages 1 through 11. Show all your work on the standard response questions. 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. You will be given exactly 90 minutes for this exam. Remove and utilize the formula sheet provided to you at the end of this exam. ACADEMIC HONESTY Do not open the exam booklet until you are instructed to do so. Do not seek or obtain any kind of help from anyone to answer questions on this exam. If you have questions, consult only the proctor(s). Books, notes, calculators, phones, or any other electronic devices are not allowed on the exam. Students should store them in their backpacks. No scratch paper is permitted. If you need more room use the back of a page. Anyone who violates these instructions will have committed an act of academic dishonesty. Penalties for academic dishonesty can be very severe. All cases of academic dishonesty will be reported immediately to the Dean of Undergraduate Studies and added to the student s academic record. I have read and understand the above instructions and statements regarding academic honesty:. SIGNATURE Page 1 of 11
Standard Response Questions. Show all work to receive credit. Please BOX your final answer. 1. Consider the function f(x, y) = e cos(x2 y). (a) (12 points) Find the linearization of f(x, y) at the point ( 1, π ). 2 (b) (6 points) Use the linearization to approximate f ( 19 20, 0.51π). (Even if you are unsure of part (a) you should attempt part (b) using your answer from (a)) Page 2 of 11
2. (a) (6 points) Find an equation of the line parallel to r(t) = 4t 3, t 1, 2t 1 and containing the point P (0, 1, 1). (b) (6 points) Find an equation of the plane containing r(t) and P (0, 1, 1). (c) (6 points) Find the area of the triangle with vertices P (0, 1, 1), Q(1, 0, 1), and R( 3, 1, 1). Page 3 of 11
3. Consider the function f(x, y) = 2 ln(25 9x 2 4y 2 ). (a) (6 points) Sketch the domain of f(x, y) in the plane below. y x (b) (6 points) What is the range of f(x, y)? (c) (6 points) Sketch the level curve of f(x, y) = 0 on the plane below. y x Page 4 of 11
4. Evaluate the following limits if possible or justify why it doesn t exist: (a) (4 points) lim (x,y) (1,1) x 2 x 2 + y 2 (b) (8 points) lim (x,y) (0,0) x 2 x 2 + y 2 5. (6 points) Find a vector function r(t) that satisfies r(1) = 1, 1, 0 and r (t) = 2ti + cos(πt)j + 3e t k. Page 5 of 11
6. (12 points) Let z = sin(x)e y2, where x = s 2 + t and y = s t 1. Find z s when s = 2 and t = 1. 7. (6 points) Compute x z, where x is a function of y and z given implicitly by xey yz = 3z 2 sin(x). Page 6 of 11
Multiple Choice. Circle the best answer. No work needed. No partial credit available. 8. (7 points) The center and radius of the sphere x 2 2x + y 2 + z 2 4z = 4 are A. ( 1, 0, 2) and 3 B. (1, 0, 4) and 2 C. ( 2, 0, 4) and 2 D. ( 2, 1, 4) and 2 E. (1, 0, 2) and 3 9. (7 points) If a = 0, 5, 0 and b = 1, 1, 0, then the angle between a and b is A. 0 B. π/4 C. π/3 D. π/2 E. π 10. (7 points) Find a nonzero vector orthogonal to the plane containing the points A = (2, 2, 2), B = (1, 1, 2), and C = ( 1, 2, 3): A. 6, 2, 0 B. 1, 0, 3 C. 2, 1, 3 D. 2, 2, 6 E. 1, 2, 3 Page 7 of 11
11. (7 points) 4y 2 3x 2 + 2z 2 = 7 is an equation of a A. Cone B. Hyperbolic paraboloid C. Elliptical paraboloid D. Hyperboloid of one sheet E. Hyperboloid of two sheets 12. (7 points) The curve r(t) = 3t, 1, 5t 2 + 8t intersect the paraboloid z = x 2 + 4y 2 at A. t = 0 B. t = 1 C. t = 1 D. t = 3 E. t = 3 13. (7 points) Suppose r(t) a differentiable vector function with r(0) = 0, 0, 0 r(1) = 1, 2, 2 r(2) = 1, 2, 1 What is the best approximation we can give for A. 6 B. 3 C. 6 D. 5 E. 8 2 0 r (t) dt? Page 8 of 11
14. (7 points) If f(x, y) = x y, then f x (1, 3) = A. 0 B. 1 C. 3 D. ln(3) E. 4 15. (7 points) Find the point at which r(t) = 2ti + t cos(πt 2 )j + 3e t k intersects the plane x + 3z = 9e 2 + 4. A. 0 B. 1 C. 2 D. 3 E. 3 16. (7 points) Which of these parametrizes the curve given by the intersection of the surfaces x 2 + 4y 2 = 1 and 2x + z = 2? A. t, 4 4t 2, 2 2t, where 0 t 1 B. t, 4 4t 2, 2 2t, where 1 t 1 C. cos(t), 2 sin(t), 2 2 cos(t), where 0 t 2π D. cos(t), sin(2t), 2 2 cos(t), where 0 t 2π E. cos(t), 1 sin(t), 2 2 cos(t), where 0 t 2π 2 Page 9 of 11
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 2 18 3 18 4 18 5 18 6 18 7 21 8 21 9 21 Total: 153 No more than 150 points may be earned on the exam. Page 10 of 11
FORMULA SHEET Vectors in Space Suppose u = u 1, u 2, u 3 and v = v 1, v 2, v 3 : Unit Vectors: i = 1, 0, 0 j = 0, 1, 0 k = 0, 0, 1 Length of vector u u = u 12 + u 22 + u 3 2 Dot Product: u v = u 1 v 1 + u 2 v 2 + u 3 v 3 = u v cos θ Cross Product: u v = Vector Projection: i j k u 1 u 2 u 3 v 1 v 2 v 3 Partial Derivatives proj u v = u v u 2 Chain Rule: Suppose z = f(x, y) and x = g(t) and y = h(t) are all differentiable then dz dt = f dx x dt + f dy y dt u Curves and Planes in Space Line parallel to v: r(t) = r 0 + tv Plane normal to n = a, b, c : a(x x 0 ) + b(y y 0 ) + c(z z 0 ) = 0 Arc Length of curve r(t) for t [a, b]. L = b a r (t) dt Unit Tangent Vector of curve r(t) sin 2 x + cos 2 x = 1 T(t) = r (t) r (t) Trigonometry sin 2 x = 1 (1 cos 2x) 2 cos 2 x = 1 (1 + cos 2x) 2 sin(2x) = 2 sin x cos x Page 11 of 11