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 13. 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. 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. 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. Score: /108 (no more than 100 points may be earned on the exam.) Page 1 of 13
Standard Response Questions. Show all work to receive credit. Please BOX your final answer. 1. Let P (1, 0, 1), Q(2, 1, 3) and R( 1, 2, 0) (a) (3 points) Find the projection of the vector P Q onto the vector P R. (b) (3 points) Compute P Q P R. (c) (3 points) Find the area of P QR. (d) (3 points) Find the distance from (0, 0, 2) to the plane determined by P, Q, and R. Page 2 of 13
2. (6 points) Evaluate the double integral : 1 1 e x2 0 y dx dy 3. (6 points) Find the area of the part of the surface z xy = π that lies within the cylinder x 2 + y 2 = 16. Page 3 of 13
4. (6 points) Find the volume of the solid enclosed by z = x 2 + y 2 and z = 8 x 2 y 2. 5. (6 points) Let F(x, y) = x 2 yi + (xy 2 + 2 3 x3 ) j. Use Green s theorem to find positively oriented circle x 2 + y 2 = 1. C F dr, where C is the Page 4 of 13
6. (12 points) Given the vector field (a) Show F is a conservative vector field. F(x, y, z) = yzi + xzj + (xy + 2z)k (b) Find a function f such that f = F. (c) Evaluate C F dr, where C is any smooth curve from P (0, 0, 1) to Q(1, 4, 1) Page 5 of 13
7. (12 points) For the vector field F(x, y, z) = 2i + (x 2 z 2 )k, use Stokes s theorem to evaluate curl F ds where S is the portion of the paraboloid z = 4 x 2 y 2, z 0, oriented upward. S Page 6 of 13
8. (8 points) Use the divergence theorem to evaluate x 3, y 3, z 3 ds, where S is the sphere x 2 + y 2 + z 2 = 1, oriented outward. S 9. (4 points) Find the length of the curve given by r(t) = ti + 2 cos(t)j + 2 sin(t)k between t = 1 and t = 3. Page 7 of 13
Multiple Choice Questions. Select the best answer. No partial credit. 10. (3 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, 2, 1 is tangent to S at (1, 1, 1). C. 1, 1, 1 is normal to S at (1, 1, 1). D. 1, 2, 1 is normal to S at (1, 1, 1). E. None of the above 11. (3 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). E. None of the above 12. (3 points) The graph of x 2 + 2y 2 2z = 0 is a(n): A. Ellipsoid B. Elliptical Paraboloid C. Elliptical Cone D. Hyperbolic Paraboloid E. None of the above Page 8 of 13
13. (3 points) Which of the following vector fields would match the one given in the figure: A. F = x, x y F B. F = x, y C. F = x 2, x 2 y D. F = y, x 14. (3 points) If u and v are vectors, and θ the angle between them, then which of the following is equal to u v? A. u + v B. u v C. u v cos(θ) D. u v sin(θ) u v E. u v 15. (3 points) If u and v are vectors, and θ the angle between them, then which of the following is equal to (u v) u? A. cos(θ) B. sin(θ) C. 1 2 sin(θ) D. 1 E. 0 Page 9 of 13
16. (3 points) Which of the following is a normal vector to the plane (x 2) + 3(z 1) = 0? A. 2, 0, 1 B. 2, 1, 0 C. 1, 0, 3 D. 1, 3, 0 E. 2, 3, 1 17. (3 points) A vector equation of the line through the point (0, 1, 2) in the direction v = 3, 1, 0 can be written as A. r(t) = 3, 1 + t, 2t B. r(t) = 3t, 1 + t, 2 C. r(t) = 3t, 1, 2 2t D. r(t) = 3 3t, 1, 2t E. r(t) = 3t, t, 2t 18. (3 points) The domain of the function f(x, y) = ln(x 2 y + 2) is A. D = {(x, y) y = x 2 + 2} B. D = {(x, y) y x 2 + 2} C. D = {(x, y) y < x 2 + 2} D. D = {(x, y) y x 2 + 2} E. D = {(x, y) y > x 2 + 2} Page 10 of 13
19. (3 points) The range of the function f(x, y) = ln(x 2 y + 2) is A. (, ) B. (, 2] C. (, 2) D. [2, ) E. (2, ) 20. (3 points) Let f(x, y) = x 2 + 2 cos(y), then the gradient vector field f is A. 2x, 2 cos(y) B. x 2, sin(y) C. 2x, 2 sin(y) D. 2x, 2 sin(y) E. x 2, 2 cos(y) 21. (3 points) Let f(x, y) = x 2 + 2 cos(y). The rate of change of f at (1, 0) in the direction of 1, 1 is A. 1 B. 2 C. D. π E. 0 3 2 Page 11 of 13
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. 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 12 of 13
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 13 of 13