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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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 Spring '08
 Wolfe
 Calculus

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