Name: PID: Section: Recitation Instructor: DO NOT WRITE BELOW THIS LINE. GO ON TO THE NEXT PAGE. Page Points Score 3 18 4 18 5 18 6 18 7 18 8 18 9 18 10 21 11 21 12 21 13 21 Total: 210 No more than 200 points may be earned on the exam. Page 1 of 15
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 15. 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 120 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 2 of 15
Standard Response Questions. Show all work to receive credit. Please BOX your final answer. 1. (a) (9 points) Find the length of the curve given by r(t) = 9t, 4t 3/2, t 2 between t = 0 and t = 5. (b) (9 points) Let z = sin(2x) ln(y), x = t 2 s 2, y = 2st. Use the chain rule to find z t at the point (s, t) = ( 1 2, 1). Page 3 of 15
2. (18 points) Evaluate the double integral : 1 1 e x2 0 y dx dy Page 4 of 15
3. (18 points) Find the volume of the solid enclosed by z = x 2 + y 2 and z = 50 x 2 y 2. Page 5 of 15
4. (a) (9 points) Find the area of the part of the surface z xy = π that lies within the cylinder x 2 +y 2 = 1. (b) (9 points) Use Green s Theorem to evaluate the line integral (7y + 3e 3 x )dx + (8x + cos(y 4 ))dy where C is the boundary of the region enclosed by the parabolas y = x 2 and x = y 2. C Page 6 of 15
5. Given the vector field F(x, y, z) = yz, xz, xy + 2z (a) (5 points) Show that F is a conservative vector field. (b) (8 points) Find a potential function f such that f = F. (c) (5 points) Evaluate F dr, where C is any smooth curve from P = (0, 0, 1) to Q = (1, 4, 1) C Page 7 of 15
6. (18 points) Use Stokes Theorem to evaluate (6xyi + zj + 3yk) dr where C is the intersection of the C plane x + z = 5 and the cylinder x 2 + y 2 = 1 oriented counterclockwise as viewed from above. Page 8 of 15
7. (18 points) Use the Divergence Theorem to calculate the outward flux of F = r r where r = x, y, z across S: the hemisphere z = 9 x 2 y 2 and the disk x 2 + y 2 9 in the xy-plane. Page 9 of 15
Multiple Choice. Circle the best answer. No work needed. No partial credit available. 8. (7 points) Let S be the surface given by x 2 + y 2 + z 2 = 1, then A. 1, 1, 1 is tangent to S at (1, 1, 1). B. 1, 1, 2 is tangent to S at (1, 1, 1). C. 1, 1, 1 is normal to S at (1, 1, 1). D. 1, 1, 2 is normal to S at (1, 1, 1). E. None of the above 9. (7 points) Let f(x, y) = xy(3 x y), then A. (3, 0) is not a critical point of f(x, y). B. (3, 0) is a local maximum of f(x, y). C. (3, 0) is a local minimum of f(x, y). D. (3, 0) is a saddle point of f(x, y). 10. (7 points) The graph of x 2 + y 2 2z 2 = 0 is a(n): A. Ellipsoid B. Elliptical Paraboloid C. Cone D. Hyperbolic Paraboloid E. None of the above Page 10 of 15
11. (7 points) For any vectors u and v, what is ( u v) u? A. 1 B. 1 C. 0 D. It depends on what u and v are. 12. (7 points) The plane (x 7) + 4(z 1) = 0 is perpendicular to which vector? A. 2, 0, 1 B. 7, 0, 1 C. 1, 0, 4 D. 1, 1, 3 13. (7 points) Which of the following vector fields is conservative? A. y, x, 0 B. x 2, xz, xy C. sin(z), cos(y), cos(z) D. e x, e y, e z Page 11 of 15
14. (7 points) The line through the point (1, 1, 1) in the direction v = 3, 4, 5 can be written as A. r(t) = 1, 1, 1 + t 3, 4, 5 B. r(t) = 3, 4, 5 + t 1, 1, 1 C. r(t) = 1, 1, t D. r(t) = 3, 4, 5t 15. (7 points) The domain of the function f(x, y) = ln(x 2 + y 2 + 2) is the region described by: A. y 2 > x 2 + 2 B. y 2 x 2 + 2 C. R 2 D. y x 2 + 2 16. (7 points) If f(x, y) = y 2 + 2 cos(x), then the gradient vector field f is A. 2 sin(x), 2y B. 2y, 2 sin(x) C. y 2, 2 cos(x) D. 0, 0 Page 12 of 15
17. (7 points) The vector projection of 1, 3 onto the vector 1, 1 is A. 1, 1 B. 2, 2 C. 2, 2 1 D. 1 2, 2 18. (7 points) What is the area of the triangle with vertices (1, 0, 1), (2, 0, 2), and (3, 0, 3)? A. 1 B. π C. 2 D. 0 19. (7 points) What is the distance from the point (1, 3, 2) to the plane z = 0? A. 14 B. 2 C. 1 D. 3 Page 13 of 15
Vectors in Space FORMULA SHEET PAGE 1 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 14 of 15
Multiple Integrals Area: A(D) = 1 da D Volume: V (E) = Transformations D E 1 dv Polar/Cylindrical f(x, y) da = f(x, y, z) dv = E Transformations E E f(x, y, z) dv = r 2 = x 2 + y 2 x = r cos θ y = r sin θ y/x = tan θ FORMULA SHEET PAGE 2 f(r cos θ, r sin θ) r dr dθ D f(r cos θ, r sin θ, z) r dz dr dθ E Spherical x = ρ sin φ cos θ y = ρ sin φ sin θ z = ρ cos φ ρ 2 = x 2 + y 2 + z 2 f(ρ sin φ cos θ, ρ sin φ sin θ, ρ cos φ)(ρ 2 sin φ) dρ dφ dθ curl(f) = F div(f) = F Additional Definitions 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 = C D Integrals over Surfaces Stokes Theorem curl F ds = Divergence Theorem F ds = S S E (Q x P y ) da C F dr div F dv Page 15 of 15