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 12. 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 9 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 12
Standard Response Questions. Show all work to receive credit. Please BOX your final answer. 1. (18 points) Consider the function f(x, y) = x3 3 x + y2 y + 3 (a) Find the critical points of f. (b) Find the absolute maximum and minimum values of f on D, the closed rectangle shown below. y 2 1 D 1 2 3 x Page 2 of 12
2. (18 points) (a) Express the triple integral below in spherical coordinates. 6xe (x2 +y 2 +z 2 ) 2 dv where E is the portion of the ball x 2 + y 2 + z 2 1 that lies in the first octant. E (b) Evaluate the integral from (a) Page 3 of 12
3. (18 points) Evaluate C 1 dx + 4x dy where C is the simple closed curve composed of two semi circles and two line segments shown to the right, transversed clockwise starting and ending at (2, ). y = x 1 2 Page 4 of 12
4. (18 points) Consider the vector field F(x, y, z) = ( y 2 z 3 + 1 ) i + ( 2xyz 3) j + ( 3xy 2 z 2) k (a) Compute the curl(f) (b) Find a function f such that F = f. (c) Evaluate a line integral F T ds where C is the curve shown below. C z (2, 2, 2)> y x (1,, 1) Page 5 of 12
5. (18 points) Find the surface area of that part of the paraboloid z = x 2 + y 2 that is below the plane z = 4 and above the plane z = 1. Page 6 of 12
Multiple Choice. Circle the best answer. No work needed. No partial credit available. No credit will be given for choices not clearly marked. 6. (7 points) Let f(x, y) = x 3 y. Find the value of the directional derivative at (1, 2) in the direction in which f decreases most rapidly. A. 52 B. 1 C. 37 D. E. 7 7. (7 points) An object occupies the region in the first octant bounded by the coordinate planes and by the paraboloid z = 4 x 2 y 2. The mass density at a given point in the object is equal to the distance from the xy plane. The total mass m of the object is given by the triple integral: A. B. C. D. E. 2 4 x 2 4 x 2 y 2 2 4 x 2 4 x 2 y 2 2 4 x 2 4 x 2 y 2 4 4 x 2 4 x 2 y 2 2 4 y 2 4 x 2 y 2 z 2 dz dy dx z dz dy dx z dz dy dx z dz dy dx z 2 dz dx dy 8. (7 points) Find the volume of the solid region bounded below by the surface z = r (in cylindrical coordinates) and above by the plane z = 1. A. π/6 B. π/5 C. π/4 D. 2π/7 E. π/3 Page 7 of 12
9. (7 points) Reversing the order of integration A. B. C. D. E. 2 ln x 1 ln y 2 1 ln 2 2 ln 2 e y 1 2 ln 2 e y 1 ln 2 e y 1 xy 2 dx dy e y x 2 y dx dy x 2 y dx dy x 2 y dx dy x 2 y dx dy x 2 y dy dx = 1. (7 points) In spherical coordinates, the equation of the upper half of the cone z 2 = 3(x 2 + y 2 ) is given by A. φ = π/4 B. φ = π/3 C. φ = π/2 D. φ = π/6 E. φ = 11. (7 points) If F(x, y, z) = (y + z)i + 2(z + x)j + (x + y)k, then curl(f) = A. B. i C. i + k D. i k E. i + k Page 8 of 12
12. (7 points) Evaluate F dr, where F(x, y) = y 2, x, and C is the line segment from (, ) to (1, 2). A. 5/2 B. 7/3 C. 2 D. 3/2 E. 4/3 C 13. (7 points) If C is the complete boundary of the triangle with vertices (, ), (1, ), (1, 1) and C is oriented counterclockwise, then (y cos x + 2y)dx + sin xdy = C A. 1 B. 2 C. 4 D. 2 E. 1 14. (7 points) Which of the following is a correct parametrization and bounds for the surface z = x 2 + y 2 under the plane z = 9? A. r(u, v) = v sin u, v cos u, v 2 with u [, 2π] and v [, 3] B. r(u, v) = v sin u, v cos u, v with u [, 2π] and v [, 3] C. r(u, v) = v cos u, v sin u, v 2 with u [, 2π] and v [, 9] D. r(u, v) = v cos u, v sin u, 9 with u [, 2π] and v [, 3] E. r(u, v) = u cos v, u sin v, v with u [, 2π] and v [, 3] Page 9 of 12
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 15 points may be earned on the exam. Page 1 of 12
FORMULA SHEET PAGE 1 Vectors in Space Curves and Planes in Space Suppose u = u 1, u 2, u 3 and v = v 1, v 2, v 3 : Line parallel to v: r(t) = r + tv Unit Vectors: Length of vector u Dot Product: Cross Product: i = 1,, j =, 1, k =,, 1 u = u 12 + u 22 + u 3 2 u v = u 1 v 1 + u 2 v 2 + u 3 v 3 = u v cos θ 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 Plane normal to n = a, b, c : a(x x ) + b(y y ) + c(z z ) = Arc Length of curve r(t) for t [a, b]. L = b a r (t) dt Unit Tangent Vector of curve r(t) T(t) = r (t) r (t) More on Surfaces Directional Derivative: D u f(x, y) = f u Second Derivative Test Suppose f x (a, b) = and f y (a, b) =. Let D = f xx (a, b)f yy (a, b) [f xy (a, b)] 2 (a) If D > and f xx (a, b) >, then f(a, b) is a local minimum. (b) If D > and f xx (a, b) <, then f(a, b) is a local maximum. (c) If D < then f(a, b) is a saddle point. 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 12
FORMULA SHEET PAGE 2 Multiple Integrals Area: A(D) = 1 da D Volume: V (E) = Transformations D E 1 dv Polar/Cylindrical r 2 = x 2 + y 2 x = r cos θ y = r sin θ y/x = tan θ f(x, y) da = f(r cos θ, r sin θ) r dr dθ f(x, y, z) dv = E D f(r cos θ, r sin θ, z) r dz dr dθ E Spherical Additional Definitions curl(f) = F div(f) = F F is conservative if curl(f) = Line Integrals Fundamental Theorem of Line Integrals f dr = f(r(b)) f(r(a)) C Green s Theorem P dx + Q dy = (Q x P y ) da C D Transformations x = ρ sin φ cos θ y = ρ sin φ sin θ z = ρ cos φ ρ 2 = x 2 + y 2 + z 2 E E f(x, y, z) dv = f(ρ sin φ cos θ, ρ sin φ sin θ, ρ cos φ)(ρ 2 sin φ) dρ dφ dθ Page 12 of 12