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Final_exam_s08 _

# Final_exam_s08 _ - MATH 241 FINAL EXAM Instructions Number...

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Unformatted text preview: MATH 241 FINAL EXAM May 15, 2008 Instructions: Number the answer sheets from 1 to 9. Fill out all the information at the top of each sheet. Answer problem n on page n, n = 1, ~ - — ,9. Do not answer one question on more than one sheet. If you need more space use the back of the correct sheet. Please write out and sign the Honor Pledge on page 1 only. SHOW ALL WORK The Use of Calculators Is Not Permitted On This Exam 1. (20 points) Let A = (0,0,0), B = (1,0,0), D = (1,2,2), E = (0, 2,2). (a) Show that these four points lie on a plane and ﬁnd an equation of that plane. (b) Sketch the quadralateral C whose vertices are A, B, D and E. (c) Show that C is a parallelogram. ((1) Show that C is a rectangle. (e) Is 0 a square? Explain. 2. (25 points) Let A = (3,2,0), B = (6,1,2). (a) Find parametric equations for the line L containing A and B. (b) Let F = 2yi + zj + (ck. Find the work W done by the force F on an object moving from A to B along L. 3. (25 points) The position of a moving particle is given by r(t) =t2i+2tj+lntk for 2 g t g 4. (a) Find the velocity, speed, and the tangential and normal components of the acceleration of the particle for any t with 2 S t S 4. (b) Find the total distance travelled by the particle in the given time interval. 4. (20 points) Let f(\$,yaz) =2\$3+y—22 (a) Find the points on the level surface f (x, y, z) = 5 at which the tangent plane is parallel to the plane 24:1: + y —— 62 = 3. (b) Find the directional derivative of f at the point P = (1,1,2) in the direction of the vector a 2 2i — 2j + k. (c) In What direction is the directional derivative of f a maximum at P and what is the value of the maximum? 5. (20 points) Suppose that a ﬁrm makes two products, widgets and ﬂibbits, using the same raw materials. If a: Widgets and y ﬂibbits are produced then a: and y must satisfy the constraint x2 + 23/2 = 8100. (This expresses a limitation on the amount of raw materials available.) Each widget produces \$5 proﬁt and each ﬂibbit produces \$20 proﬁt. How many of each product should the ﬁrm produce in order to maximize the proﬁt ? 6. (20 points) Write a triple integral in an appropriate coordinate system for the volume V of the solid region bounded above by the sphere x2 + y2 + 22 = 49 and below by the paraboloid m2 + y2 2 Biz + 21. Do not evaluate the integral but, if you used the right coordinate system, you should observe that the integration is not particularly difﬁcult. //D(:j22—_yy>2d‘4 where D is the region bounded by the lines a; + 2y = 1, x + 2y 2 3, x — 2y 2 4 and :1: - 23/ = 8 by making an appropriate change of variables. 7. (20 points) Evaluate 8. (25 points) Compute f sz . ndS where F(9:, y, z) = xsi + y3j + 23k and E is the boundary of the part of the ball 1:2 + y2 + z2 S 100 which lies in the ﬁrst octant (a: > 0, y > 0, z > 0) and n is the outward normal. 9. (25 points) Use Stokes’s theorem to compute f0 F - dr where F(x, 3/, z) = z2i + 4xj + y3k and C is the rectangle of Problem 1 oriented counterclockwise as viewed from above. ...
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Final_exam_s08 _ - MATH 241 FINAL EXAM Instructions Number...

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