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234-wilson-08faex02 - page 1 Your Name Circle your TA’s...

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Unformatted text preview: page 1 Your Name: Circle your TA’s name: Achilles Beros Nicos Georgiou Ben Otto ‘Christelle Vincent Mathematics 234, Fall 2008 Lecture 2 (Wilson) Second Midterm Exam November 20, 2008 Be sure that we can tell whether something you wrote was intended to represent a vector or a scalar! You could use an arrow over a letter to designate a vector, or some other symbolism. We will recognize the standard unit vectors f, j; and E, and hence that linear combinations of these do represent vectors. Please write your answers to the eight problems in the spaces provided wherever possible. If you must continue an answer somewhere other than immediately after the problem statement, be sure (a) to tell where to look for the answer, and (b) to label the answer wherever it winds up. In any case, be sure to make clear what is your final answer to each problem. If you came up with several possibilities, don’t depend on us to choose one! Wherever applicable, leave your answers in exact forms (using g, x/S, cos(0.6), and similar num— bers) rather than using decimal approximations. If you use a calculator to evaluate your answer be sure to show what you were evaluating! There is a problem on the back of this sheet: Be sure not to skip over it by accident! Thereis scratch paper at the end of this exam. If you need more scratch paper, please ask for it. You may refer to notes you have brought on one whole sheet of paper or two 6x8 index cards, as announced in class and by email. BE SURE TO SHOW YOUR WORK, AND EXPLAIN WHAT YOU DID. YOU MAY RE- CEIVE REDUCED OR ZERO CREDIT FOR UNSUBSTANTIATED ANSWERS. (“I did it on my Calculator” and “I used a formula from the book” (without more details) are not sufficient substantiation...) Problem T Points 1 Score 1 16 2 10 3 12 4 12 5 12 6 . "‘ ' 14 7 12 l‘ 8 “ 7‘ 12 "J [ TOTAL f 100 page 2 Problem 1 (16 points) Let f(m, y, z) = 3:2 + y sin(z). (a) Find the gradient Vflx, y, z). (b) Find the gradient V f at the point (2,2,0). (c) Find the directional derivative Dfl’f at the point (2, 2, O) for the vector 11’ = f+ 2f— 21;. (d) Find a vector 17 giving the direction in which D5 f at (2,2,0) is largest. What is the value of the derivative Dg f at (2, 2,0) in the direction of 27? page 3 Problem 2 (10 points) Set up (you do not need to evaluate it!) an integral in spherical coordinates to calculate the mass of an object that fills the region below the plane 2 = 1 and above the cone 2 :2 r, if the density at any point (:13,y, z) in the object is 6(33, y, z) = m2 + 342 + 22. page 4 Problem 3 (12 points) Find an equation for the tangent plane to the level surface 1'2 + y2 — 22 ~‘— 2333; + 4:102 2 4 at the point (1, O7 1). page 5 Problem 4 (12 points) Find the average value of the function f (cc, y) = «3:2 + 3/2 on the plane region that is inside the circle of radius 2 centered at the origin and is in the first quadrant (a: 2 0 and y 2 0). Hint: The integrals might be easier in coordinates other than rectangular! page 6 Problem 5 (12 points) 2 M 4—x2—y2 Convert the integral / / / 1 dz d$ dy —2 0 0 to cylindrical coordinates and evaluate the cylindrical coordinate version. What does this integral compute? Use Words to describe what the resulting number means. page 7 Problem 6 (14 points) Find the absolute maximum and minimum values of f (x, y) = 11:2 + 2233/ + 3y2 on the triangular region bounded by the lines a: = —l, y = 1, and y = :c — 1. Be sure to give both the maximum and minimum value(s) of f aid the points ($,y) where f takes those values. page 8 Problem 7 (12 points) Set up and evaluate an integral to compute the volume of the region in the first octant (a: Z 0, y 2 0, and z 2 0) beneath the plane 25E + 6y + 3z 2 12. page 9 Problem 8 ' (12 points) Find the maximum and minimum values that the function f (27,11) :2 3:2 + 23/2 takes on the circle 2 2 ac +y = 1. ...
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