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Final Fall 2006 Answers

Final Fall 2006 Answers - Name Math 122 Final KEY...

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Unformatted text preview: Name Math 122 Final December 12, 2006 KEY Directions: 1. No book, notes, loud snoring, steriods, or delinquent activities. You may use a calculator to do routine arithmetic computations. You may not use your calculator to store notes or formulas. You may not share a calculator with anyone. . You should show your work, and explain how you arrived at your an— swers. A correct answer with no work shown (except on problems which are completely trivial) will receive no credit. If you are not sure whether you have written enough, please ask. . You may not make more than one attempt at a problem. If you make several attempts, you must indicate which one you want counted, or you will be penalized. . You may leave as soon as you are finished, but once you leave the exam, you may not make any changes to your exam. 1. (20 points) d x/ugj: 2 (a) Use the formula: / fizq: agua +0 1 U: ax to evaluate/m dm do: adx ow I d0 2. U2 z 2 :’ v *0) = —- 14%?er LT 0-1-01 q U . ..__._. q (fix) 4a: 2‘ : 0‘: RXdX (b)/(1+m2)2dm U \+>< d ' DL 3* U 0+1") lllxd —; - ‘29—“: —\— S .11. (C)/F “3 I d x 7< = )4 0W :W x du= .1.— dH \1 .— ——‘_ x 7< X X 2 2 (d)/C9”dm= j\‘5”"" olx sma: S‘Nx Seacx — 6\N>< dx = Qh\cscx~co~rx\ + cosKH-EQ 'Z 2- (20 points) U __ I ’L (a) tan3 :1: dm = j "m BIZ/x 0‘ U m ' {0‘7 \rrm U: SECX «3U: sccx flwx dx 1 E :2: V —\ __ Jé 3'5 7' ,L J o‘v- JU—v _ 2 3%. -'-2 ' ‘3- (SECXU + 2 (sec >43 + C b ___£...__ ()/2m2—3m_2dm :: 5’1 + \ Dad—'M. $4.; 5 = 43* _k B ‘ - *1 = RC!-‘-D-3+G(ax+l) 7c 3. 3=\ X=__|__ S:~.E-P| a 9",.17- (c) 9m2+16 561$ __ j \ dfi ‘ :L‘ + (sz‘m = S L d1 = J— NICTAN L). = —‘—— RECTRN (afll U7?” :5 Q; a. k L (a Q \ 2:3 (d _. 3 /{x ) Wears - 3‘ 9m 6 d6 1 M ‘4 X— S Né ‘ ‘ Uroose CH:- cosede dw—S‘Ne d6 C56 ._ dU -SIN6 ’L :3 I S\Ne dU 3 -~ __ 2 __ ’5 “M \Y \ U “ —:—)— -D 7: (.056 __ C096 3 3. (20 points) (3) Use Euler’s method with h = 0.50 to approximate y(1) initial value problem for the \/(0 «’1: W95 (1+m2)&;—4xy (HM (m8) 6 (‘3 ~7— «qr-K") d_y_ _ 3:1:- dm 331—6 811123 3X 3—3, (900:6 wa Jadx —7.>’< e”? = 9 | (3 '1‘! av \ A ,,..'..~ "7" 9" \Nx OX +9 (9‘3" Je e s ‘3 4. (20 points) (a) The country of New Zealand has $ 300 million in paper currency in circulation and each day $ 15 million comes into (and out of) the country’s bank. The government decided to introduce new currency featuring a certain Chemistry Professor because a smaller portrait was needed. The banks will replace old bills with new ones, whenever old currency comes into the bank. Let y = y(t) denote the amount of new currency in circulation at time t, with y(0) = 0- 5 i. Write a differential equation for how much new currency is in q circulation. \ 91=\5—\51=is—:/. BOO d'b 300 30 is ' fl \/ (0) =6) ii. Solve the differential equation from part 5‘6 -b = _I_ ma:- \5 [OCH = 6 fl + i : \5 20 . Lt dd: 90 4.4, ‘ 2° C1 \/= LIT ‘56 6° at +0] = are 3006’ + e “ e -3251: —%0t \l:zoo+ce \/=3oo~'3©oe iii. How long will it take for the new bills to account for 90% of the currency in circulation? _ 19.3 1: 8—10 = 300 - 300 E | y” —- 51/ + 6y = 0 y(0) = 5, y'(0) = 12 5. (20 points) (a) For the curve given by x = gig/2, y = 275, 1. Find the equation of the tangent line at (18, 18). t=01 O“) ; '——' ’2. ow=gz= T=g \;—\a=g(x—Is) ax em 1:. 3*? ii. Find the area of the region that lies between the curve and the m—axis for 5 g t S 12. $2 \2 R \1 ;2 a t A: jydx= S atfidt= j 3+. (it: 35. 5 s 5 55 D» %_ _ EL 'b \ = 5- ( \1 - 6 5 5 iii. Find the length of the curve for 5 S t g 12. '5 ll ’— (I: ‘1 214%? >2 — s = j W ova = _._. \, - 1 S '2: s ‘1 “1 r2. «4... an 2' __._ @i “’ 3 2: (b) Write an equation in polar form for the vertical line that goes 5 through the point (2, 0). I i X = a 2 l : l RCOFG : a Q C069 l. l (c) Find the area inside one petal of 7" = 3 cos 30 T TT 7. ’2. ‘— 3 '3‘ G 0 TT Ge S \N 3 32.5.2? do =E—(e+ Tl EC \ 1 H O __ 2. (“\e "IT ' 7T ' Li 6. (20 points) Determine if the following series converge absolutely, con— verge conditionally, or diverge. For each test you use, you must name the test, perform the test, and state the conclusion you reached from that test. 00 (a) Z(—l)”arctann 71:1 Nsm Team “QM NAG—mm N =1; 2m but NéOO °° nnl3” M ' Jam @W-"b film! = o <1 @ RAT“) New LaN'lam/(amfl a? (C) n; n_1 ‘ x l 1 __,§c, To . \ _____g g 0|” 0 Comp.”‘eer R q 3 m f m “6T 9' \ =0 \/ l 4 l v lCOkW. Com) 5 N900 m ~l -\ \lFt ‘l 00 n l 1 (d) gel) + name 0° V30 A, \ \ \ \N’T.\'1:‘5T S t — M\ ‘5 —-"—“ 0 N0 R85 3 x 0m? &mx A)“; :0 7. (20 points) (a) Consider the power series: 77. °° (93-3) 2 +1 n=1 n i. Where is the power series centered? C23 ii. Find the radius of convergence. 9% N“ \= \><~">\<l Rcl 14 NH; C)“ '53 iii. Find the interval of convergence. [Hintz Check the endpoints] .4 .43 54:11 i \ Com). COND- N-‘n A \i - \I ” 3(‘u i Nx\ 0‘ (b) For f(a:) = lncr i. Find P3 the third degree Taylor polynomial at a = 1. "LED =_§/ha( CD 3 "l 2 2: _ $7023): I V390=0‘H(><~D +":5(>1r0 +3!(>< \3 “gylx 2 :71? 1‘ ii. Find R3(x) the remainder for the third degree Taylor polyno— gu’hg : 2E mial at CL=1. X . ~(o 41”, 32350 = (E) (x‘oqvx : “in (#04 M. 8. (20 points) (a) For vectors ’c? = (1,2,3) and W = (2,3,1) find: i. 7-? 1 + c. a 3 = \ l 11 7? X _b> \ 1 “b ‘ 1 (an 5 - D 2 ) g '5 x '1. ’5 iii. The c056 Where 0 is the angle between 6’ and ¢ '81 \ b \\ C06 9 = " A A _,,_______ RH \b\ “’\ iv. A unit vector that is perpendicular to both 71* and (—7,6,—D lms v. The projection of I? in the direction é "a -b \\ p .I ‘6. = ——“ -—- = __ .—. {DO-3 \PQQJ E&\ \ -— "J b V" (b) i. Find the parametric equations for the line containing the point (2, 3, 1) and perpendicular to the plane 2$+3y+z=5 X: 2 Y= 3+34= 2" \—+ t ii. Find the equation of the plane containing the point (2,3,1) and perpendicular to the line a: = 3 + 2t, y = 2 + 375, z = 5 + t. \“i ll axerzfi Jr '2 9. (20 points) (a) Find the equation of the plane containing (2,3,1), (2,4, 5) and (3,5,8). R7 = <0 \ Ll> g l _) 3 N = q‘ = «gasp 6 —7<+A\/~'7'£= Oi (b) Let L1 and L2 be lines with parametric equations L1: cc=1+2t, y=2—t, z=4—2t L2: m=9+t, y==5+3t, 22—4—75 i. Show that L1 and L2 intersect. Find the point of intersection. H9¢=UH~5 ("ID-JD—R) ,._t —= S-‘YBS ~ ‘ (—1 3 1D “3‘7 —- S -3—h ' “*6 W =—- =1" 9 = mns 5 9~ ’6 3 ii. Find the equation of the plane that contains both L1 and L2. (afib’gb $3: (n)o)7§ <\) 39-” jKA-la X -\~ '2 '-' 5 (c) Find the distance from the point (2, 3, 1) to the plane 293+3y+z=5 \BC'aD -) 3(334 \CD ' 5\ DEV": ___ .____' W W 10. (15 points) Indicate Whether the following statements are true or false by circling the appropriate letter. A statement Which is sometimes true and sometimes false should be marked false. 00 If 2 0,7123" converges at x = 1, a) “:1 @ F 00 then Z an converges. n21 00 If E anx” converges at :0 = 1, a) “=1 T ® then 2 anus“ converges at a: = —1. 71:1 ) If the 7th degree Maclaurin polynomial for f is T ® C P7($) = 1 — m3 + x5 — 3:7 then f’”(0) = ——1 4 6 d) The Maclaurin series for a: sinx is 3:2 —— + . . . ® F If f is continuous and lim f = O °° T Q 6) then / f(a:) converges. o f) 71:1 00 If an, (7,1,0n Z O and an + bn 5 an and Z en converges m as @F then 2 an and 2 bn converges. 71:1 12:]. g) @F ...
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