MTH 234 Exam 2 April 10th, Without fully opening the exam, check that you have pages 1 through 12.

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1 Name: Section: Recitation Instructor: INSTRUCTIONS Fill in our name, etc. on this first page. Without full opening the eam, check that ou have pages 1 through 12. Show all our work on the standard response questions. Write our answers clearl! Include enough steps for the grader to be able to follow our work. Don t skip limits or equal signs, etc. Include words to clarif our reasoning. Do first all of the problems ou know how to do immediatel. Do not spend too much time on an particular problem. Return to difficult problems later. If ou have an questions please raise our hand and a proctor will come to ou. You will be given eactl 90 minutes for this eam. Remove and utilize the formula sheet provided to ou at the end of this eam. ACADEMIC HONESTY Do not open the eam booklet until ou are instructed to do so. Do not seek or obtain an kind of help from anone to answer questions on this eam. If ou have questions, consult onl the proctor(s). Books, notes, calculators, phones, or an other electronic devices are not allowed on the eam. Students should store them in their backpacks. No scratch paper is permitted. If ou need more room use the back of a page. Anone who violates these instructions will have committed an act of academic dishonest. Penalties for academic dishonest can be ver severe. All cases of academic dishonest will be reported immediatel 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 honest:. SIGNATURE Page 1 of 12

2 Standard Response Questions. Show all work to receive credit. Please BOX our final answer. 1. (18 points) Consider the function f(, ) = (a) Find the critical points of f and classif them as local minima, local maima, or saddle points. (b) Find the absolute maimum and minimum values of f on the BOUNDARY of D, the closed rectangle shown below D Page 2 of 12

3 2. (18 points) Evaluate 3z( z 2 ) dv where E is the portion of the ball z 2 2 that lies in the first octant. E Page 3 of 12

4 3. (18 points) Evaluate C 1 d + 4 d 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, 0). = 1 2 Page 4 of 12

5 4. (18 points) Consider the vector field F(,, z) = ( 2 z 3 3 ) i + ( 2z ) j + ( 3 2 z 2) k (a) Compute the curl(f) (show our calculations) (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)> (1, 0, 1) Page 5 of 12

6 5. (18 points) Find the surface area of that part of the paraboloid z = that is below the plane z = 4 and above the plane z = 1. Page 6 of 12

7 Multiple Choice. Circle the best answer. No work needed. No partial credit available. No credit will be given for choices not clearl marked. 6. (7 points) Consider the function f(, ) = 2 2 at the point ( 1, 2). In what direction does f increase most rapidl? A , 2 B , 2 C , 2 D , 1 E , 1 7. (7 points) An object occupies the region in the first octant bounded b the coordinate planes and b the paraboloid z = The mass densit at a given point in the object is equal to the distance from the z plane. The total mass m of the object is given b the triple integral: A. B. C. D. E z dz d d dz d d z dz d d dz d d z 2 dz d d 8. (7 points) Find the volume of the solid region above the surface z = r 2 (in clindrical coordinates) and below the plane z = 1. A. π/4 B. π/3 C. π/2 D. π E. 4π/3 Page 7 of 12

8 9. (7 points) Reversing the order of integration A. B. C. D. E d d 3 2 d d 3 2 d d 3 2 d d 3 2 d d d d = 10. (7 points) Which of the following could be the vector field F =, A. B. C. D. E. 11. (7 points) If F(,, z) = zi + (2 + z)j + 3k, then curl(f) = A. 0 B. i + j + 3k C. i j + 2k D. i j + 3k E. i + j + 2k Page 8 of 12

9 12. (7 points) Evaluate 2, 3 2 dr, where C is the line segment from (0, 0) to (1, 2). A. 3 B. 4 C. 5 D. 7 E. 9 C 13. (7 points) If C is the complete boundar of the triangle with vertices (0, 0), (1, 0), (0, 4) and C is oriented counterclockwise, then ( )d + 4 d = C A. 6 B. 3 C. 1 D. 5 E (7 points) Which of the following is a correct parametrization and bounds for the surface z = above the plane? A. r(u, v) = v sin u, v cos u, 9 v with u [0, 2π] and v [0, 3] B. r(u, v) = v cos u, v sin u, 9 v 2 with u [0, 2π] and v [0, 9] C. r(u, v) = v cos u, v sin u, 0 with u [0, 2π] and v [0, 3] D. r(u, v) = v sin u, v cos u, 9 v 2 with u [0, 2π] and v [0, 3] E. r(u, v) = u cos v, u sin v, 9 v with u [0, 2π] and v [0, 3] Page 9 of 12

10 Congratulations ou are now done with the eam! Go back and check our solutions for accurac and clarit. Make sure our final answers are BOXED. When ou are completel happ with our work please bring our eam to the front to be handed in. Please have our MSU student ID read so that is can be checked. DO NOT WRITE BELOW THIS LINE. Page Points Score Total: 153 No more than 150 points ma be earned on the eam. 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(, ) and = g(t) and = h(t) are all differentiable then dz dt = f d dt + f d dt u Plane normal to n = a, b, c : a( 0 ) + b( 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(, ) = f u Second Derivative Test Suppose f (a, b) = 0 and f (a, b) = 0. Let D = f (a, b)f (a, b) [f (a, b)] 2 (a) If D > 0 and f (a, b) > 0, then f(a, b) is a local minimum. (b) If D > 0 and f (a, b) < 0, then f(a, b) is a local maimum. (c) If D < 0 then f(a, b) is a saddle point. Trigonometr sin 2 = 1 (1 cos 2) 2 cos 2 = 1 (1 + cos 2) 2 sin(2) = 2 sin cos 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/Clindrical r 2 = = r cos θ = r sin θ / = tan θ f(, ) da = f(r cos θ, r sin θ) r dr dθ f(,, 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 d + Q d = (Q P ) da C D Transformations = ρ sin φ cos θ = ρ sin φ sin θ z = ρ cos φ ρ 2 = z 2 E E f(,, 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