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2 Standard Response Questions. Show all work to receive credit. Please BOX your final answer. 1. (14 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 =. 3 Page 2 of 12

3 2. Evaluate the following integrals (a) (7 points) x e y4 dy dx. (b) (7 points) 2 8 y 2 (x 2 + y 2 ) dx dy 0 y Page 3 of 12

4 3. (14 points) Find the absolute maximum and minimum values of f(x, y) = 8 3 x3 + 2y 26 on the set D where D is the closed region bounded by y = 0 and y = 1 x 2 Page 4 of 12

5 4. Let F(x, y) = 2xy + 5 3y, x 2 10x be a vector field. y 2 y 3 (a) (4 points) Show that F is conservative. (b) (5 points) Find a potential function for F, that is, a function f such that f = F. (c) (5 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

6 Multiple Choice. Circle the best answer. No work needed. No partial credit available. No credit will be given for choices not clearly marked. 5. (4 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 6. (4 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 7. (4 points) In cylindrical coordinates, what does z = r represent? A. Cone B. Elliptical Paraboloid C. Sphere D. Hyperbolic Paraboloid E. Ellipsoid Page 6 of 12

7 8. (4 points) In spherical coordinates, what does r = 3 represent? A. Sphere B. Cone C. Line D. Plane E. Hyperbolic paraboloid 9. (4 points) Convert A. B. C. D. 4 4 x x 2 2π x 2 dy dx x 2 2 x 2 dy dx x 4 x 2 dy dx 2 4 x dy dx E. None of the above r dr dθ from polar to Cartesian coordinates: 10. (4 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 7 of 12

8 11. (4 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 12. (4 points) Evaluate dy dz dx. A. π(1) 2 B. 1 C. xyz D. 0 E. None of the above 13. (4 points) Evaluate the line integral ( 1, 1), ( 1, 1), (1, 1) and (1, 1). A. B. C. D. 0 cos(2)+1 2 (3) cos(2)+1 2 (6) cos(2)+1 2 (12) E. None of the above C 3 cos(2y) dx+3x 2 sin(2y) dy where C is the square with vertices Page 8 of 12

9 Challenge Question(s). Show all work to receive credit. 14. (14 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 9 of 12

10 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 Total: 106 No more than 100 points may be earned on the exam. Page 10 of 12

11 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. Geometry / Trigonometry Area of an ellipse x2 a + y2 = 1 is A = πab 2 b2 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

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

### Without fully opening the exam, check that you have pages 1 through 12.

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MTH 34 Solutions to Exam November 9, 8 Name: Section: Recitation Instructor: INSTRUCTIONS Fill in your name, etc. on this first page. Without fully opening the exam, check that you have pages through.

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