408DS09Makeup2_Sol - Exam 2 Math 408D Name Spring 2009 TA...

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Unformatted text preview: Exam 2 Math 408D Name Spring 2009 TA Discussion Time: TTH You must show sufficient work in order to receive full credit for a problem. Do your work on the paper provided. Write your name on this sheet and turn it in with your work. Please write legibly and label the problems clearly. Circle your answers when appropriate. No calculators allowed. 1.(a) (8 pts) Find the 2nd degree Taylor polynomial in a: — 8 for the function me) = arm. (b) (8 points) Use the polynomial in part (a) to approximate (801)—1/3 and estimate the error in this approximation. (You do not need to carry out any of the arithmetic in this part of the problem.) 1 2. (14 pts) Find an infinite series that converges to / cos(\/§)dac. 0 3. (14 pts) (a) Let a = 23— j +3 and 6: 33+ 41?. Find Projgd- (b) If 27 and 7.6 are perpendicular, what can you say about Projwff? What if they are parallel? 4. (14 points) Find the area of the triangle with vertices P(0, 1, 0), Q(—1, 1, 2), and R(2,1,—1). —-¢ 5. (14 points) Find all unit vectors that are perpendicular to both 17 = 2;- k and w :21 2j+ 2k. 6. (6 pts) (a) Find rectangular coordinates for the point whose polar coordi- nates are (736) = (—2, —§). (b) (8 pts) Find all polar coordinates representing the point whose rectangular coordinates are (3:, y) 2 (—2\/§, —2). 7. (14 pts) The curve given by parametric equations at 2 3t — t2, 3/ = sin 77?? passes through the origin twice. How many distinct tangent lines are there to the curve at (0, 0)? Justify your answer. Bonus problem on back. Bonus (5 pts): Assume that c? 7€ Determine whether each statement below is true or false. If true, give a reason. If false, find a counterexample. (a) Ira-Eza-athenb’za (b)If[i><b=EL'XE,thenb=5 (C)If(_i-b=c'i aanddxbzdxjthenbzé’ ‘ c3 'WM g macaw/x 3 ; J'MEX’L JX 0 [‘50 0 _: i/(_l) Y‘tH ’ L“, (2mm) 0 1' w (-0“ kw (20! (km 3 ,(aH’MSll =(a.g’ :(2.g+:,,).f+,.q>1 ((LH” 3w : (Z: (3E+'~(E> : 5:3(351ch) (63 If Cr+w w—f &&.JJ:0 so :6 I: (“Mb—5W pal/WW, «L»- 93:»; JA 34w Sula. A) 30 A475,; ' é Q F W (fi-SBW:(>MZ\w.-AJ kuz a :V" 4 40+ [new E2 —<"’)o)27 M 3 '{ FK’ 3' (2,0l—I7 " $055 FE=—-.¥ =11???» HPTZU W V '4: V? rims}. v , awo;-y/r395w:.oé+w:wesmo( S'VVZQ=}IKS—- a 5 QTQZ’C M (.72 :Z—lg-fll: Dim? = L i l" , '_ a l 0 —! , ~9L-ij -4», I ~2. z :4—Ql-§l-'~I7 .I'S +0 (30% C, +03, (To 3J0” ‘1 LffLY‘F“, __ [Cl—r =3V’r 3 SD ffiz‘at'fl‘47 M '3'LJ-Y4—Zl—S‘,—q7 W (,WJu—CJ'VL fMFwJ/leh4‘oLUH—Ot'l‘fi (a.(a\Kr,9\: (fig/ADA a =- -m c- a _, 2 6%} 91> a( Q) ' l galxt‘j)<—fg/fl> : '2 3W(’FDV\=’2—(~fi/L):fi (Xl‘335('3f3) *1» r:)(14r<31‘ :\)1‘L+q : WT: .. ~1 3 . "J.w9' If; E MSIW (>935 3'39 WW} 9,. 7170 ‘$0 (4,7717; 11m\ xfn WWM W49 (*4)“7613"“\ AV‘MWW Wfl/oSS‘LL. ...
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408DS09Makeup2_Sol - Exam 2 Math 408D Name Spring 2009 TA...

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