Exam 3 - V1 {WM/ EXAM 3 MATH 2153 SECTION 2, FALL 2010...

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Unformatted text preview: V1 {WM/ EXAM 3 MATH 2153 SECTION 2, FALL 2010 INSTRUCTOR: WEIPING LI Print Name and Student # SHOW WORK FOR CREDIT 1!!_ SHOW WORK FOR CREDIT 2!! (l) (10pts) Find the radius of convergence arid interval of convergence of the series 0° 3"(:l:+4)” «a ‘ n+1 ml [ivvq 2,91%). _ J7 yqjh'o 1795i— h'hx J I”! 5" (#4)" I J." = =Iim 51x+4| r—v—h: =3IX+44 < l R 5 h'Pm ’ J— -—‘—<x+4<"‘ ’4”$<X"4+5 ’ 3 ‘2? 0° 5"@'4+"L " be ’ A“; 1L Jane when x='4+§, “4: (75’ 5&5 “9'38 - When x z'4 — .L 9° 5"(-—4 —§+¢)“ _ “A "i gmwr J 19 Ahéwm‘fn s FEE—Mm -—,,2;Uu: je/Wg/ Ham? of Omvergénce L1 4—3; ’41? (2) (10pts) If 233:0 0,3" is conver ent, does it follow that the following series are convergent ? (a) Zf=o(—1)"Cn BS Theme)" 3, H— musf CKWV‘E/Yje fa: 4,1" [9457'- Yes. 194le ASCOWe’q-f - 3 < x < 5, I" Vfil'cwlal , H— Crn wage; Wk“ x t ‘2 . I+ Ja—EJ Mo‘l‘ 1’16“ch THAT ‘ch(»4_)h is necessaril/ _ (b) 21:10 Gn(‘4)n New Him. WEIFING LI (3) (12pts) All power series must have general terms for full credits. (a) Use differentiation to find a power series representation for f (2:) = What is the radius of convergence ? 9° 2 a “A”:an =t+x+x+x+_,, r’x ":0 Take AM;VJI‘V‘€ a" loo-H1 Sudes 3 '1’“ =fHKM’I=V+ZX+BX+4X+ U—x)‘ h: H I hm (“+I)X I'm = )X‘ <t ) “flog h x“ new (b) Use part (a) to find a power series for flat) = ‘ ' yer/oath a om T0103 42 jd-o n'Z, a 3H 2 . _ z +3-2X +4'5X+f‘4)<+ ' (I'X)5 hsz Dc ’2 . a .5’ 1 : ~ mum) " ;,+§.:X+£:x+—gx 5 2.. X .2. 2' (PX) h=2 Z (c) Use part (b) to find a power series for f(:r) = Z ' 0c 5 ulna H z. 5.2 5 4-3 4 5'4 I “’1’”: )X :X+"2*’<+.>_"*2“ 5 . (.I'X) “=2 2 (4) (Spts) Use series to evaluate the limit a: — tan-1 :5 lim 3 _ :r—bO (E , 7 +" 53+,5i- 5—+ RmX : X'— 3 5 7 *' x5 X5 X7- _ X :. —' —«—~ + —— X t” 5 r 7 >4 ‘ X—fmax ' [rm (J’FL:+:<:#*UV> L“ 3 h 5 5’ 7 X‘>O X “'70 a ,L CALCULUS III 3 (5) (10pts) All power series must have general terms for full credits. (a) Use a Maclaurin series to obtain the Maclaurin series for f (m) = cos(:c3). cos-t: §('l)m t2" :I—j:+:’§—q— [4:0 (2“)! 2'" ()0 2n H 6" JC‘XB 6050(3) = Xe)" (x3 ‘ ZN) x pup (2n)! ha, (2 h) I, 5 «a r» I —- -+ 22: 'P ' ’ £17 : dy/‘H- —— 2t %=2f—IZ dx 4X41: 25’” , J‘ y- : 2t d+ ’— dx‘ (IX 4" ix/d-(y A 2+: 2(2t—/2.)—2 2t : fi( 24:42) :_ (at—Ia)‘ “TC—7;— ztw/z ~24 4 WEIPING LI (7) (lOpts) Find the sum of the series 411. (a) 230:0(“Dngrf M, y. M " _ 4 a: («1) x4” _; Z CHM“) : 6 X h=o 14:0 ‘4! We“ T O (b) 211:0 my! Zn+1 i k’v (2H,)! v (8) (lOpts) Find the area of the region that lies inside 7" = 2 cos 9 curve and outside 7" = 1 curve in polar coordinates. " 2 Few! Hag When/Seaway! fond—5 +3 55(— (023,19 I :" Yc 49. A Z a e) Yr—2.Cps§= Y—"—| (359335 - .11. I 7T/3 1 9 ’ i a . 7r/5 Z I Z Arms (lease) —: )49 =;( (4&58'!) 49 .2 T, W ._ /3 3 l cg I r/s ._ :zj 2-()+CD$26’)'I46':;(6'+S”‘29) = £4» E 11/} 3.75 3 z (9) (10pts) Find the distance (arc-length) traveled by a particle with position (x, y) = (sin2 16, cos2 t), OStSn‘. L: f“ )fl dx: xii) d1,— =15’Ht6’os’t «it I O :’ :2 'Zarst Slln't (HT ' =» fl 2 a '2- L alt Co$f<0 j f4 5;...i-a75—f + 4 5714150751” \‘M Zh‘ \- V o Tr Tye J 7" , T (If \/——-—’-’J . : .‘ +§ 5'“ :Zfij {Smtcflt’ CH, “test- W/ 5 ° 2 CALCULUS III 5 (10) (lOpts) Find the points on the parametric curvem = 10 — t2, y = t3 —— 1215 where the tangent is horizontal or vertical. \ :ézé‘ "' Stz— [2 :7. DJ “MCI T = ——zt D, 5tz_'2=0 I t=i2‘ ix. ;_2.(-_t2)4—_O ! f*2._ x=é,)’.—.—(é. f=~2, X=6,3=lé. I (6) Me WMLS W;+L, “bnjed‘ fimmJQ/I' JC‘O X=.IO, 7:0. ((0, 0) is flu; fviwf th‘H/p M+l'(ap gfif;wn+4‘ (11) (Bonus lOpts) (a) Show that the area of the surface generated by rotating the polar curve 1' = f(9), a g 6 S b about the polar axis is b S =/ 27”” sin 6dr? + (gfidfi. . (b) Use (a) to find the surface area generated by rotating 7‘2 = cos 26 about the polar axis. b 2. z, __ ’- + y'L (a) BMW”), 3: Sazvry C‘s-W) cm I> ‘j 2n Ysine ~W‘ (“9, a Y: C0529 jOeS-‘flwmfih Hm role 4* 65526 3;. 0 ’ 29.: 9.: E. 11L ' The W {5 s WIMJ‘Wic % 4, 09“] “eds “I'D fit: 991 W -wrfa_fe_ +08: :1 f. ’2- .‘4—1 :— I £12.. flaw - W210 Y, 00326174; 2Y d9 2.81428 (49) _ r2 - 66523 ' . - DZ 3 3 2 I ZV-Ma529- ""‘9‘/£osze« + 912—. 4e D M; ‘ 60529 77/4 'W 2 47% Jags-sine-/“’“9+""229 do =47?! smede=4r‘("559) a I O O “NO—4:2) ...
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Exam 3 - V1 {WM/ EXAM 3 MATH 2153 SECTION 2, FALL 2010...

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