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. 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 9 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. I have read and understand the above instructions and statements regarding academic honesty:. SIGNATURE Page of
MTH 34 Solutions to Exam November 9, 8 Standard Response Questions. Show all work to receive credit. Please BOX your final answer.. (9 points) Let g(x,y) = x y 4xy 6y. Find and classify each critical point of g as a local minimum, a local maximum, or a saddle point. Solution: g x = xy 4y g y = x 4x y So the critical points are P(, /3), O(,), and Q(4,). Now let D = g xx g yy gxy = (y)( ) (x 4) D(O) = 6 < = g has a saddle point at O D(Q) = 6 < = g has a saddle point at Q D(P) = 8 > and g xx (P) = /3 = g has a local maximum at P. (5 points) Let E be the hemisphere x +y +z 9 with z. Express that triple integral below in spherical coordinates. Do Not Evaluate. E e x +y +z dv Solution: e x +y +z dv = ˆ π ˆ π/ ˆ 3 E e ρ ρ sinφdρdφdθ Page of
MTH 34 Solutions to Exam November 9, 8 3. Let F = (3yz e x sinz)i+(3xz +y)j+(3xy e x cosz)k and answer the questions below. (a) (8 points) Find a function f so that f = F. Solution: f = 3xyz e x sin(z)+y (b) (3 points) Let C be the line segment from P(,,π/) to Q(/,,π). Evaluate the integral ˆ (3yz e x sinz)dx+(3xz +y)dy +(3xy e x cosz) dz C Solution: = f(q) f(p) = 3π +4 ( ) (c) (3 points) Let P and Q be as defined above and let R = R(,3,π). Now let C be the line segment from P to R and let C be the line segment from R to Q. Evaluate the integral ˆ C C (3yz e x sinz)dx+(3xz +y)dy +(3xy e x cosz) dz Solution: Same as part (b). Page 3 of
MTH 34 Solutions to Exam November 9, 8 4. (7 points) A solid D lies within the cylinder x +y = 4 and is bounded below by the plane z = and above by the plane z = 6. The temperature at each point in D is given by T(x,y,z) = x+y +z +. Find the average temperature over the solid D. Solution: The volume of the solid D is 4π. It follows that the average temperature over D is given by T avg = T(x,y,z)dV 4π = 4π D ˆ ˆ 6 ˆ π = + 4π 4π = 6 4 ˆ 6 z +dz rcosθ +rsinθdθdrdz + ˆ π 4π ˆ ˆ 6 z +dzdrdθ 5. (7 points) Let F = x+3y,x y. Use Green s Theorem to calculate the work done by F on a particle moving counter-clockwise around the triangle with vertices (, ),(, ), and (, 5). Solution: Let T be the given triangle and R be its interior. Then ( (x y) F dr = (x+3y) ) da T R x y = ( 3)dydx R = Area of R = 5 Page 4 of
MTH 34 Solutions to Exam November 9, 8 6. Consider the integral below and answer the questions that follow. ˆ 4 ˆ (a) (3 points) Sketch the region of integration. Label any relevant intersection points. +y 3 dydx x y = x 4 (b) (5 points) Evaluate the integral above by reversing the order of integration. Solution: ˆ 4 ˆ +y 3 dydx =ˆ x = = 3 ˆ ˆ 9 +y 3 ˆ y y +y 3 dy u du dxdy = 3 ( 9 ) = 4 3 7. (6 points) Find the surface area of the portion of the paraboloid z = x y that lies above the plane z =. Solution: Let R be the disk of radius centered at the origin. Then Area = +(x) +(y) da = R ˆ π ˆ = π = π 8 ˆ ˆ 9 +4r rdrdθ +4r rdr udu = π 6 (93/ ) = 3π 3 Page 5 of
MTH 34 Solutions to Exam November 9, 8 Multiple Choice. Circle the best answer. No work needed. No partial credit available. 8. (4 points) Let F = x,xyz,yz. Which of the following is true? A. curlf = x y and divf = yz xy +z B. curlf = x,xz,y and divf = z, xy,yz C. curlf = x,xz,y and divf = z xy D. curlf = z xy,,yz and divf = x+xz +y E. None of the above. 9. (4 points) Let P = P(,5) and u = i 3j. If f(p) = i+4j, then D u (P) = A. 7 B. i j C. D. 5 E. None of the above.. (4 points) A. 8sin4 B. 4sin C. sin4 D. ˆ 4 ˆ ˆ 8sin E. sin6 y 4cosx z dxdydz = Page 6 of
MTH 34 Solutions to Exam November 9, 8. (4 points) The surface φ = 4π 5 can be described as a A. Cylinder B. Plane C. Paraboloid D. Half-Cone E. None of the above.. (4 points) Let R be the set of points (x,y) that satisfy 4x +y 4. Then the absolute maximum value of the function f(x,y) = 4x +5y on R is A. f has no maximum value on R. B. 4/4 C. D. 6 E. None of the above. 3. (4 points) Let C be the curve whose ˆ parametrization is given by the vector equation r(t) = ti+t j, t. Then 6xds = A. 3 ( B. 5 3/ ) C. D. ( 3/ ) ( 5 ) E. None of the above. C Page 7 of
MTH 34 Solutions to Exam November 9, 8 4. (4 points) At the point P(,,3), in which direction does the function f(x,y,z) = xy z + y x increase most rapidly? A.,, 6 B.,,6 C.,,3 D.,, 6 E.,,6 5. (4 points) Find the work done by the force F = y,4x along the straight-line segment from (,) to (3,3). A. / B. 6 C. 3/ D. 3 E. 3 6. (4 points) Which of the following vector field plots could be F = (x+y)i xyj? D. Page 8 of
MTH 34 Solutions to Exam November 9, 8 More Challenging Problem(s). Show all work to receive credit. 7. (4 points) Let f(x,y) = x xy +y 4 and observe that f has a critical point at P(,). Consider the following statements. A. Only (a) is true. B. Only (b) is true. C. Only (c) is true. D. Only (a) and (b) are true. E. Only (b) and (c) are true. (a) f has a local minimum at P. (b) The second derivative test is inconclusive at P. (c) f has a local maximum at P. 8. ( points) Let R be the region in the xy-plane enclosed by the circle x +y = 4, above the line y =, and below the line y = 3x (see Fig. ). Evaluate the integral below in polar coordinates. No credit for any other method. Solution: Notice that the region R intersects the circle at π/6 θ π/3 and radial lines from the origin enter the region at r = cscθ and exit at r =. R da = = ˆ π/3 ˆ π/6 cscθ ˆ π/3 π/6 rdrdθ (4 csc θ)dθ = (4θ +cotθ) π/3 π/6 = π 3 3 y = 3x y = rsinθ = ( 3,) R Figure : Region R Page 9 of
MTH 34 Solutions to Exam November 9, 8 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 4 3 4 4 4 5 4 6 7 8 9 4 Total: 6 No more than points may be earned on the exam. Page of
MTH 34 Solutions to Exam November 9, 8 Vectors in Space Suppose u = u,u,u 3 and v = v,v,v 3 : Unit Vectors: Length of vector u Dot Product: Cross Product: i =,, j =,, k =,, u = u +u +u 3 u v = u v +u v +u 3 v 3 = u v cosθ u v = i j k u u u 3 v v v 3 Vector Projection: proj u v = u v u u Partial Derivatives 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 Curves and Planes in Space Line parallel to v: r(t) = r +tv Plane normal to n = a,b,c : a(x x )+b(y y )+c(z z ) = Arc Length of curve r(t) for t [a,b]. L = ˆ 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) = and f y (a,b) =. Let D = f xx (a,b)f yy (a,b) [f xy (a,b)] (a) If D > and f xx (a,b) >, then f(a,b) is a local minimum. (b) If D > and f xx (a,b) <, then f(a,b) is a local maximum. (c) If D < then f(a,b) is a saddle point. Geometry / Trigonometry Area of an ellipse x a + y = is A = πab b sin x = ( cosx) cos x = (+cosx) sin(x) = sinxcosx Page of
MTH 34 Solutions to Exam November 9, 8 Multiple Integrals Area: A(D) = da D Volume: V(E) = Transformations D E dv Polar/Cylindrical r = x +y x = rcosθ y = rsinθ y/x = tanθ f(x,y) da = f(rcosθ,rsinθ)rdrdθ f(x,y,z) dv = E f(rcosθ,rsinθ,z)rdzdrdθ E D Spherical Additional Definitions curl(f) = F div(f) = F F is conservative if curl(f) = Line Integrals Fundamental Theorem of Line Integrals ˆ f dr = f(r(b)) f(r(a)) C Green s Theorem ˆ P dx+q dy = (Q x P y )da C D Transformations x = ρsinφcosθ y = ρsinφsinθ z = ρcosφ ρ = x +y +z f(x,y,z) dv = E f(ρsinφcosθ,ρsinφsinθ,ρcosφ)(ρ sinφ)dρdφdθ E Page of