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Unformatted text preview: page 1 Your Name: __________________._____._____________.—— Circle your TA’s name: Lino Amorim Jon Godshall Ed Hanson _______________________________________________
Mathematics 222,‘ Spring 2007 1 Lecture 3 (Wilson) Final Exam May 17, 2007 There is a problem on the back of this sheet! Do not accidentally skip over it! Write your answers to the ten problems in the spaces provided. If you must continue an answer
somewhere other than immediately after the problem statement, he 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. Wherever possible, leave your answers in exact forms (using g, x/E, cos‘(0.6), and similar numbers) rather than using decimal approximations. For example, sin(%) = %, and writing something like .499 may not get you full credit. If you use a calculator to evaluate your answer be sure to show what you
were evaluating! There is 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 up to three sheets of paper, and the class handout on
undetermined coefﬁcients, as announced in class and by email. BE SURE TO SHOW YOUR WORK, AND EXPLAIN WHAT YOU DID. YOU MAY RECEIVE
REDUCED OR ZERO CREDIT FOR UNSUBSTANTIATED ANSWERS. (“I did it on my calcula
tor” and “I used a formula from the book’? (without more details) are not sufﬁcient substantiation...) TOTAL { 200 I
I page 2 Problem 1 (15 points)
For each of the following equations, indicate its graph by ﬁlling in the blank with the number from the picture below. a__.___._ r=l+si110 2 2
U (CD—~— %‘y4—=0
(b)._____. —£3+L2=1 , 2 2 9 4 (e)————— %+1{g=1 page 3 Problem 2 (20 points)
Evaluate the integrals: (a) /x2cos(2:)da: (b) /tan3(a:)da: page 4 Problem 3 (20 points)
Evaluate the integrals: (a) /0 mile vii—9:22am: Page 5 Problem 4 (21 points)
For each of the following series, tell whether it converges or diverges. If it converges and it has, some posztiveencl seme negative terms, tell whether it converges absolutely or conditionally. Be sure to give
reasons Justifying your answers. 00 __1 12—1
(8‘) Problem 5 (20 points) page 6
Find the Taylor series at a = 1 for f = (103(132 — 1). Show the terms through the 3” degree term. Derive the coefﬁcients from the general form for Taylor series, do not just “plug in” to some known series.
That is, you should calculate a0, a1, a2, and (13 using derivatives, and Show how they are ﬁtted into a thirddegree polynomial. Page 7 Problem 6 (21 points)
Find all solutions of the differential equation y" + 2y’ + y = Gsin(2:c). page 8 Problem 7 (20dpoints) _‘
Lat ﬁ=27+i+k a11d17=i'—2j‘+2k, (a) What is I27}, the magnitude of 17? (b) What is the cos 6, if 9 is the angle between 11' and 17‘? (c), What is the scalar component of 1? in the direction of 17? (d) What is projgﬁ’, the projection of 11‘ on 17"? (e) Find two vectors {[1 and 2’22 such that (i) 1? = U1 +172, (ii) {51 is parallel to 73", and (iii) 112 is orthogonal
to 17. page 9
Problem 8 (21 points)
Suppose the polynomial 1 — + $22 is used to calculate, approximately, cos(x). If this will be used for
values of a: from —1 to 1, what accuracy can you guarantee will be achieved?
Your answer should use the remainder term from Taylor’s theorem in showing your answer is valid. If you
know another mathematically correct way to do the problem you can use that as a check on your answer
and get upto 5 extra points. But it will not substitute for an answer using Taylor’s theorem. page 10 Problem 9 ( 21 points)
Solve the initial value problem 53— + 2y = x3 (for a: > 0) and y(2) = l. page 11 Problem 10 (21 points)
Consider two planes, H1 and H2, given by ’ H1: $+2y—2=7
and H1: 2z+3y+2z=4. (a) Find parametric equations for the line of intersection of these two planes.
Hint: The point (1,2, —2) is on both planes. (b) The angle between two planes means the angle between a vector perpendicular to one plane and a
vector perpendicular to the other. What is the cosine of the angle between H1 and H2? ...
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 Fall '08
 Wilson
 Calculus, Geometry

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