# pfsol - Math 16C Final Exam Part 1 Problem 1(12 points...

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Unformatted text preview: Math 16C Final Exam Part 1 Problem 1 (12 points) Determine Whether the sequence con— verges or diverges. If it converges, then ﬁnd the limit. (a)(4 points) an 2 1 ' W , ,_ \ H r \‘\N\ “‘0 “WW Ld (:70 \ \Ne week Q Avnnxoer N (Va/k ’Y‘MJY 2) . A2. f E .L. ~ He 5 AL s. 4—» A , .1, 3A ‘33 (GN ~ N2- ﬁvwa \q; —o\ f Q Are . 2_ (b)(4 pomts) {2243 30:0 ‘ ‘ “1‘! w \ (71,». A"i ! j _, «(m 1 J. 7.4. A300 ﬁnd P4C>Q\hl_ n 'E A” I \ \“I‘L ,1 '9 \4 J”?— y\ 0:: n \ «O : My _\ Jr 0 : l 1 2 (C)(4 points) an = (_1)n n 1+n3 1 . \ L A x .33 “I \W“ H“: M \W“ l. ' w ‘3 {\"§O:> n? \ A \/ ““ \ ,1}...— w t/KTK'K O " OM pOints) an : \m “ m ' a 1' Y‘c-‘r‘oc 1 law}: A a . _‘ ' 0L ‘ 0k \ L 1" ’(NW A T W‘ 7. ‘ 7“?” w . ,. w—mm 2— 1 1 Z .3. A) ' L V: x I 9" g \; 2\\M l l SIACE, 1th hﬁcﬂ A \M H .ném “ 0:3 Problem 2 (20 points) Determine Whether the series con— verges Or diverges. State the name of the test you are using in each case. (a) (4 points) 23°21 UK M ax“ Va” 1M ' (1AM)! \ \CM \ “"‘ \ 1‘ W \"Lacwwx 1“ “\$00 0‘“ Y\"‘>C>b : 0 < '1 So M4 «HRS Cars/U70 (b) (4 points) 230:0 371;?” r; 00 Igﬁh—lk <0Q) 1A “A30 6" : C. A: 0 6" A” T“ P. '- '\ : 2,33%?“ W] A. g X -« ES; 51)“ x :3 s) x A. ( * rm: &; seq ll U Svyx 9% ; kaxﬁrtc «(10 (c) (4 points) 220:1 % d 4 ' 0° ___3_tl__ ( )( p°1nts)zn=l¢%TJ:§;“2 We can um M N5 ﬁrm ‘VUJF ovx HA3 6mg 1 K J‘— . \W % .. \ ‘ ’ ~§A”L'3+“L ~ \/"\ ‘ w" {\"300 “\$00 S"r\°‘\-§fl “MM Sn‘i‘f‘hl \ 4 3+0 H5 1 \ 0) go “Vine, kw?) An/eaig (e>(4 points) 2,21% \ ____ 00” E 2 “SA “ “'31 {\AA“ rnzﬁ Y\.:\ 1%»! <5 a q’ﬂries MW Q 99 Jdalj Sent) Aivﬂrzd Problem 3 (16 points) (a) (8 points) What is the 24th par— tial sum of the series 230:0 ﬁT 7 ﬁkﬂ ’VHQC a ’\’o \(‘QQMR “Aaﬁ a ﬁzamd‘ric 50W)" ,\ . n '\ E 00 in n-\ '— Ob S 1 ﬁt 20b E L g E Ob EN “:0 '3, g — A50 3 gngm “" {\ﬁo 3 (YA i 3 A571”) "L Lek C f E" ’ g Vﬁ Vax’ﬁaA Svm U 3“:qu k. +VN>- I; ‘.5/ , Br“; VS“:LV* In ‘E‘ (N'h) 3 5' So SﬁQV’C) ;ib“’ VS“ Z¥\\v ¥\-“)_ k? \_ ,C YN\_VNH]‘§:L\”C + )‘5 Qt} :I \dr _ -- ‘6 7;, “ \dr ﬁl’W/S’ \$1“? 5: 0 \ (‘5‘ 3 V113- (b)(8 points) Find the sum of the series 230:0 W. [\L») U a acaszXrlc. germ; \ .- ‘67 Ma“ N» OC) A :3 3%” ~ Em 2w ' <"" L A Z wafﬂu) __ \ N rift“ \“ \1 Problem 4 (16 points) For which values of a: does the power series 230:0 converge? What is the radius of convergence? H + cum 4 lM‘MXUy‘ 3A ‘ \w \ an \ : kiZA “KW” MMM‘ “60% ’ ‘ (MK) 3 ‘ (KWM‘ \ Thm \ A W1 mm“ “\$06 \ MK ‘s H a r m 7" SUM W‘Boo rm w new “\$09 a . \ /WW : k\*a)'3\x>¢1\ » .. \ - §\><*l\ "VI/xi SfFQCJ (OAVKr'éj/Lj szA Z \ V‘\\.0b A‘S/ﬂfng 'W‘LZA E‘AXNDA 31 Sn \JY Coﬂwﬁk"; XJof *‘ < Problem 5 (16 points) Find a power series representation for f = 1 +1362 and determine the interval of convergence. \ \ \* Hx‘ “ \‘ (‘Lixl‘i OK) '\ ~ 20F \“4x1\< 1 d Era-a (A 4x11) 'l<:; ar NH? ‘ " ' ‘ _. i so *irxﬂ in\\{(\/(;\ 3% (Omvefﬁqmc/Q ‘5 ( E ) % Problem 6 (16 points) Find the Taylor series for f(:z:) = centered at c = we, Wm *0 mm; sax : if: \onbvd“ A?“ <14<_{4o\(.\'€/AT \é' ,\ S , I; :31) ,V. £03k) a ;. y t X 1 I ~ Ry TGVXOKJ W‘Lafm) V‘K “a”: \0“ M 53400 : f\(x) 1 ﬁincx) Part 2 Problem 7 (16 points) Find a general solution for the follow— ing differential equations. (a)(4 points) 3y2% = 1 (1‘71 7- 1 (AK '7: Mg (b) (4 points) % + % = 3x + 4 m3} ‘5 4 {Nil "Jrlcf \lszar‘ Shadow} gar“ l \f I ¥ 3)? : \_ 1..., \My ML {Vie Lt 1th 1 , A’ 1 L/K ( bftllvj< 3" PGJJ vél‘ (V'Xl' 19%- + late-ﬁle 7X 7' (2x1 thMlx i: Xx" «1X1 JV C. 10 (c) (4 points) my’ + y = x2 + 1 (\LJ {5 ﬁ, QMW’arAV \mfar 212k. 20m» 7 ’ i '12)“:- >< * :1 ngdx AX M,_§ec"¢c.l V“ 1W“ 6 T Q 7“ Md. \o- 104‘. y’X ‘V y j )C‘ H \Zamixcz '. w xY : x‘ +\ 3 imam yx : {(xtumx : % “(Ac C (d)(4 Points) 3/ - 2933! = 21“ TM; \1 a XVA’OFJ’V \«M-V (XLV‘UP 7"“ 1x77“ IMAX/ X W: Le\‘ 2100 : c: 3 e M1 IV “' Problem 8 (20 points) (a) (10 points) Find fx, fy , and fgcy for f(a:, y) = ln(\/:):2 + 3/2). —\/~ x ‘: O ,L. /,\‘\\f\ Loﬁ’, ix I“ ‘ A m» \ , . -- 4. i" 1“ 1 I» r 7/ . S<~7 ’ \EX‘ kxfx TAX * y \ K ' " XI_\_7‘L ‘ '1X\ 1 ,, _ L (:1, \ 3. / x7 : : 3yk XI’VVI» “ X“ * ) 1/ (XLV7 (b) (10 points) Find fmfy , and fmy for ﬁns, 9) = 3:4 — 3682212 + 1/2- 384: m5 a 5X71 R7 2' ’éxlx/ +l\/ _ 9 \ ﬂ . Ry 2' My 1‘ 3;(‘*X“é><7 ) *“RXV 12 Problem 9 (16 points) A fruit fly population starts with 20 ﬂies. After 5 days, the population size is 100. Assuming that the population grows at a rate proportional to its size, ﬁnd the population after 10 days. Let yet) ’5 it 9% MR)“ at ’lamﬂ JC (CA ion/j) §a yCO) ’3 ‘10 am} 703—) '3 \00 (51M 4” "(Le Mn: 3 (in W51 Va) 0 Woewmm‘ *0 7% M um «7 7+) :: K 7 swig Wylie K AV ‘ k f \I‘~ .4 Sag/ink“ (a. \$47» :- 5' ME 9M7. : Ki *c‘ :9 7H): KHQ :Cem We ACiA +1: K N C‘ \/(o) :10 2) :10 1 Cam 3 C J}/‘ C‘Ké ' 115 So YCLI ’ ‘10 j \00 :, :LOQSK© QR .: :9 SK:'&§:D KT - j“ ‘00 Problem 10 (16 points) (a) (8 points) Find all critical points of the function f(:c,y) = \$2 + 3/2 + 22% + 4. Tc gun/l Aim (F’i‘llcql Psﬂk 0% g» 3d gx “*6! £7, ﬁts/Al +92%, 0 15% :;X 3*le 0/.(7 : 175‘x1 O: 3 x50 (Dr Vi) w S0 (010) \3 Q ((Vllkrfasl “pm/(l Y5—\ , Oak/ml E) xii-{i :3 xctﬁ g0 (ﬁfﬂ “NJ (“ﬁfw art (nHml Vac/\\ﬁf 14 (b) (8 points) Classify each of the critical points you found in part (a) as a minimum, a maximum, or a saddle point. a Q”: \$17 ‘ I szw : a A as”- m7) XXV : KO,O) ONO D) j‘ (JK’O) (nl gal-1-0:(fya __ ,,,,,, A, gmkoio) Z“, .150 gu (0,0) (’1 MVM‘MVM W1 ,«0 Alﬁﬂ) : \l-l)'l * (203.) 1‘ 02 ~ 9 : — g <0 30 (\$1 )'\) \S a SMXR gen/er (“\E *9 : (QCAMQ‘v («16,1 f « Xéo _/——’/ ‘ 15 Problem 11 (16 points) Find the extreme values of the func— tion f (51:,y) = x2 + 2y2 on the circle 3:2 + y2 = 1. (Hint: Use Lagrange multipliers.) \ , i \Nt WM *0 wximikv Row) Svlage JV ‘(0 TLL 44%)”va QlXp/l ’3 O) w m 7 ix») : X1 ‘\\/I ,\ “\‘W «le‘rw Odcvv/ aft (r:\‘}(a\ gal/Cl! aSK iba Sfixn¢5§be\ \:(><Z,‘~~//>\ : [7) J ly)\/) : X1 397' l ﬁ/Q C( (ml Quiméf)’ P " >\lX\ iyx—l) 3'0 {MO} aw} (4)03 m Vos’fibu exit/5M ROM: illom : g; («Ml imoi 39b\,0)~:1 / D. \‘S MAX. d :l. U ‘VL\C_ Mil/H *917ISchfcvllagu,y)ro >0 Problem 12 (16 points) Evaluate ffR a: dA, Where R is the region in the my-plane below y = \/ 25 —— x2 and above the :c—axis. «ea. f y: 1 :-« (XV—1, “7 A" +7“ 1, 1"; \‘/\~/ I (A C‘T-:.\Q A (“a/Hui SI (4an a}? S 21‘. g 7' 1:3:( DS~X1)/ \Xfﬁ :0 - 0 1156 make; sense mu SUD/)ZX M ’K \r SVMMY/eg mm 36% YWM (Su M Amati/Le garb (Nanak We. (mt/q 17 ...
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