Final_exam_ _(20)

Final_exam_ _(20) - MATH 241 Final Examination Fall 2002...

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Unformatted text preview: MATH 241: Final Examination Fall 2002 December 3, 2002 Instructions: Each question is worth 20 points. Answer each of questions 1—8 on a separate answer sheet numbered with the number of the problem. You may use both sides of the answer sheet. Write your answers to problems 9 and 10 on the sheet containing those questions, and hand that sheet in with your other answer sheets. Be sure to fill in your name, section number and your instructor’s name on all answer sheets, including the sheet with problems 9 and 10. 1. Find an equation for the plane that passes through the pointk(1,2,3) and includes the line parametrized by x=1+t, =5—t, 2:4—2t. 2. Let C be the curve parametrized by r(t) 2 ti + cos tj ~ elk. (a) Find symmetric equations for the line tangent to the curve C' at the point (0,1,—1). (b) Find the curvature of C at the point (0,1,-1). 3. Let 2 be defined as an implicit function of x and 3/ near the point (1, 2, ——1) by 2(1, 2) = —1; Z3 ~ 211322 + yz : —5. (a) Find—gandg—zatx21,y=2. (b) Find Du(z)(1, 2) where u = %i + gj. 4. Find the absolute maximum and minimum of my + 22 (a) On the ellipsoid 4x2 + 3/2 + Z2 = 8. (b) on the solid region defined by the inequality 43:2 + 1/2 + z2 s 8. 5. Let D be the solid region bounded below by the Qty—plane and above by the paraboloid z = 16 ~ 3:2 — y2. Suppose D is occupied by a solid whose mass density function is given by 22. Find the coordinates of the center of mass of D. 6. Compute the area of the region R in the first quadrant of the avg-plane bounded by the lines y = 490 and a; 2 4y, and the hyperbolae my 2 1 and my 2 4. Hint: consider the change of variables x 2 1w, y z 2 1) . 7. Evaluate 2 2 / (y2exy + 1) dx + nyexy dy + 322 dz, 0 where C is composed of the line segments from (-1,2,0) to (1,1,1) and from (1,1,1) to (2,3,1). 8. Compute the flux integral f IS F - ndS where F = (333 + y2)i + (y — xz)j — (mg2 + 2)k. (a) Where 2 is the boundary of the half ball 0 S 2: S 1/9 — as? — y2 and n points outward from the half—ball. (b) Where 2 is the hemisphere z = V9 — m2 — y2 and n is as in part (a). Problems 9 and 10 are on a separate sheet: see Instructions overleaf. MATH 241 Final Examination Fall 2003 Problem 9 Grade: Name: Section: Instructor: In the diagram below, the curves are level curves of a function f(x,y) and the arrows show the approximate size and direction of the gradient of f. Based on this information, do the following. (a) Give approximate coordinates for five critical points of f, ;and indicate their location on the diagram. (b) Classify each of the critical points you found in part (a) as a local maximum, local minimum, or saddle point. 7"?“ a v ’ i J i‘l\ / r \ ,Art’T’T‘i‘N‘kam ,/ A/ \ [L—L—«i—i—lWLNL Problem 10 is overleaf. MATH 241 Final Examination Fall 2003: Problem 10 Grade: Each of the following diagrams shows a closed curve C and a vector field F, represented by the direction and length of the arrows. In each case the signs of §C F - T ds and L F - 11 ds can be determined from the diagram, where T represents the counterclockwise tangent to C, and n represents the outward normal to C in the plane of the diagram. For each diagram, determine the signs of both integrals. Write your answers on this sheet. Grading will be as follows: four points for each correct answer, minus four points for each incorrect answer (except that the total score for the problem will not be negative), no points for each omitted answer, and a bonus of four additional points for four correct answers. ...
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