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 90 minutes for this exam. Remove and utilize the formula sheet provided to you at the end of this exam. This is a practice exam. The actual exam may differ significantly from this practice exam because there are many varieties of problems that can test each concept. 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) Find the volume of the sphere x 2 + y 2 + z 2 64 that lies between the cones z = x 2 + y 2 x 2 + y 2 and z =. Hint: Use spherical coordinates. 3 Page 2 of 12
2. Evaluate the following integrals (a) (9 points) 27 3 0 3 x e y4 dy dx. (b) (9 points) 2 8 y 2 (x 2 + y 2 ) dx dy 0 y Page 3 of 12
3. (18 points) Find the absolute maximum and minimum values of f(x, y) = x 3 + 2y 2 26 on the set D where D is the closed region bounded by y = 0 and y = 16 x 2 Page 4 of 12
4. Let F(x, y) = 2xy + 5 3y, x 2 10x be a vector field. y 2 y 3 (a) (6 points) Show that F is conservative.. (b) (6 points) Find a potential function for F, that is, a function f such that f = F. (c) (6 points) Evaluate the line integral F dr, where C is the curve parametrized by r(t) = t sin(t), πe t cos(t) for 0 t π. C Page 5 of 12
5. (18 points) Find the area of the part of the paraboloid x = y 2 + z 2 that lies inside the cylinder y 2 + z 2 = 49. 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) Find the curl of the vector field F(x, y, z) = x + yz, y + xz, z + xy. A. 0, 1, 0 B. 0, 0, 0 C. 1, 1, 1 D. 1, 1, 1 E. 0, 0, 1 7. (7 points) Find the divergence of F(x, y, z) = (3x + yz) i + ( y xz) j + (4z + 6xy) k A. 3 B. 6 C. 12 D. 0 E. None of the above 8. (7 points) In cylindrical coordinates, z = r means A. Cone B. Elliptical Paraboloid C. Sphere D. Hyperbolic Paraboloid E. Ellipsoid Page 7 of 12
9. (7 points) In spherical coordinates, what does r = 3 represent? A. Sphere B. Cone C. Line D. Plane E. Hyperbolic paraboloid 2π 2 0 0 4 4 x 2 10. (7 points) Convert A. B. C. D. 4 2 2 x 2 4 x 2 dy dx 2 2 4 x 2 2 x 2 dy dx 2 2 4 x 4 x 2 dy dx 2 4 x dy dx E. None of the above r dr dθ from polar to Cartesian coordinates: 11. (7 points) Find the gradient of f(x, y, z) = 6xy 2 z 3 3xyz A. 6y 2 z 3 6yz, 12xyz 3 6xz, 18xy 2 z 2 6xy B. 3y 2 z 3 3yz, 6xyz 3 3xz, 9xy 2 z 2 3xy C. 6y 2 z 3 3yz, 12xyz 3 3xz, 18xy 2 z 2 3xy D. 12xyz 3 3yz, 6y 2 z 3 3xz, 18xy 2 z 2 3xy E. None of the above Page 8 of 12
12. (7 points) Which of the following functions has a constant gradient vector field? A. f(x, y) = 2x 2 + y B. f(x, y) = y 2 C. f(x, y) = x y D. f(x, y) = sin(xy) E. None of the above 13. (7 points) Evaluate A. π(1) 2 B. 1 C. xyz D. 0 1 1 1 0 0 0 E. None of the above dy dz dx. 14. (7 points) Evaluate the line integral ( 1, 1), ( 1, 1), (1, 1) and (1, 1). Hint: Use Green s theorem. A. B. C. 0 cos(2)+1 2 (6) cos(2)+1 2 (12) D. None of the above C 3 cos(2y) dx+3x 2 sin(2y) dy where C is the square with vertices 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 150 points may be earned on the exam. Page 10 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 0 + tv Unit Vectors: Length of vector u Dot Product: Cross Product: i = 1, 0, 0 j = 0, 1, 0 k = 0, 0, 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 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) 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) = 0 and f y (a, b) = 0. Let D = f xx (a, b)f yy (a, b) [f xy (a, b)] 2 (a) If D > 0 and f xx (a, b) > 0, then f(a, b) is a local minimum. (b) If D > 0 and f xx (a, b) < 0, then f(a, b) is a local maximum. (c) If D < 0 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) = 0 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