PracticeMidtermSolns

PracticeMidtermSolns - Math 21C—2 Practice Midterm...

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Unformatted text preview: Math 21C—2 Practice Midterm So’Ul‘ionj Name: I Signature: % / Student ID: 123%‘61Mm — U12 c There are ten (plus cover and bonus) pages to the exam. 0 The exam totals 100 points, plus 10 bonus points. 0 You will have 90 minutes to complete the exam. 0 No calculators, notes, or books allowed. 0 Good luck! 1 1. (10 points) Definitions and Examples: a. (2 points) Write the definition for a sequence to be bounded above. (An example is not sufficient for full credit.) ASquvmce {an/S is modal above gF them add; an M Such F— WJ, q” 6 M For all n, b. (2 points) Write the definition of an alternating series. (An example is not sufficient for full credit.) A {cries i) an QHCFM‘HA’ {aria gm “Hernunltly Fosi‘Hx/e and neaq+lve. if HI. 'erm; c. (2 points) Write the definition for a sequence to diverge to infinity. (An example is not sufficient for full credit.) {m3 (livery; ‘lo lniplnflll («C “For Ever) M, flm i; an N IoLl‘ find" I) > N :5 4n 7 M . d. (2 points) Write the definition of the cross product of two vectors. (An example is not sufficient for full credit.) no; "Asa; 3“ CE \7‘ a} d X \7 = go 9 > 7;, where 7: l5 “1 Unl'l’ Wtj'or perpendicular JrJ (71 flqhe muffin-Ina (1’ m4 7/ and fallhfyfv He. Rial“ ‘th Alp e. (2 points) Let f = f Write the definition of the Maclaurz'n series for f. (An example is not sufficient for full credit.) 49 Th chlaurin series 19v 35 Z 4“”)(0) h 41 n', h=0 2. (10 points) Short Answers a. (5 points) State the Integral Test for infinite series. LU" {Gin/g be “ $833.1th 0‘; FoSle/Q +Qrmf' Sumac” ‘F If A (oni’inuwj) Poffflve, defiant? wrung/73" 0T/ 16 v{or q” M Z ( N m f°$f+iw Shh; 2f) Cindi 3 4,, For AH n7IV. / x) n0" Z 4" «ml f Quidvé Comm/fl 0r diver/6 +70%?”- n:y\) \ b. (5 points) State the nth- Term test for infinite series. I4: “"1 an Joe; noWL QXiSi‘ or‘ 1) 00+ (Bf/It, 7% Zero} N900 \ \, TLQH 4n Awergef‘ n3! 3. (10 points) Determine whether the following sequences converge or diverge. If a sequence converges, find its limit. a. (3 points) an = 1n(n + 1) —- 1n(n). FM A0“) livin) : hm flw(n:\') hm an - “404 “.50.; n-‘N" \ 9:; ‘ (it; “5 = NH b (3 pomts) bn=n2'l“(”) -ln" -1,‘ --An fl“(h 2 ) awn 4- A 2 4 lm b“ ’ hm n2 -— (m a :: (m a new n90” h‘yb‘ have (my I “*3 a .‘ TC? is; (e e Am I-Ml) " ‘w W “ "A" n a W {true 4 1. MW § | c. (4 points) (1n = ( m)“. Th, (MW doe} MT” QXB'IL. | MN" [in : Q59 n-V" n! +11. ‘?m»+ hm qufnmlC) bdwwn by)? P0$i+;w(7 (47,4, n—va (.3) n 4 (and mifl‘flw/y lap/’4. 4. (10 points) Determine whether the following series converge condi- tionally, converge absolutely, or diverge. (—1)" (In In)“. a. (5 points) 2 71:2 «4 W H)" - f7 CW" m M)". We m koo+ T€)+ *0 “\L‘ 00 0‘ l 53 LthQ’VS- AJ 51 h Abram Comer/ante 76*, {a (_1)n+1 no.999999 ' b. (5 points) 2 =1 3 [a L 0"” 4” I ~ ‘— ~ > conglder Z 0.41%") r C, 0.611111 ""' h k1! n . \ ,,_ “rho Se.er gin/erg?) (:1 ML P‘)€fl€j ’957L. H°WQVQF 9° '1” I ——7 _, ‘ : 1%1 3) an AHQ/n'tflnl fen‘ej/ mm “q n=| h an > 0 ‘Car 4” I’\ an '2 an. 4M” n “A (in “'9 O a; n—JWO. 5 go \71 TL AH’QIYM‘HAJ {ma T€5+l .90 (")MI 2”“...~ o.‘i‘M‘M‘\ (om/€781 (conclH‘ioM/A/J H) " W (MW/111 (mWw‘e’y) . W 0.4%“? h 5. (10 points) Determine the values of a: for which the following series converges conditionally, converges absolutely, or diverges. What are the cen- ter, radius, and interval of convergence? ()0 Znnx" n=2 00 n h We consider In a! ) final afrly Quo‘l’ Tef/fi n33 " Inn/{W 2 M74 QM Unleff 45:0‘ 5:) Cenlef Oi ("WV “Q. U Tl-t Nah!) o\c (onVQK nL'e {J AAA {IL lalQrvnl a": (unwrian I) 4520 \ \ ‘ M wing ',»’< I“ Ns‘lfw. “ml ‘6’ ll,“ malt“ if Gill/14f ’0 (M70) NW" 0/ duo ml exb‘l“ (46(0), 5o H» Eerie) sliver“; 4390450 $1 {L 0% Term Tgf. TL jade E; (anoll+l°h4l/ (M W ML nowhere, I 7 fl 6. (10 points) Compute (no shortcuts”) the Taylor series centered at 1 for the function new; My) = W “‘3 2 Whhirfi Hmi Mac) a 7944"} 7cm) ‘ ’13.? PM - i;- 4/; PM = 37' PW) 2 4% [Lia N‘U) = '3' v. m) (20'3)0~h‘5)w3‘] if“ f‘ ’0) :95) ’77... (“Um (2n'3)(2n-;) ---5.3 -‘ I I‘LTHVU + Z: a?“ ’ Owl),3 7. (10 points) Using what you know about familiar Taylor series, write a power series for f = 642. Use the first 3 non-zero terms to estimate 1 _2 fezdx. o (Hint: you should be familiar with the series for em.) ¢ 2’91“. Quill: e ’ W M w n 2 1 W (will ‘7 (“ll ‘4 a ’54 __ ~: ———-""'" S e ‘ T m M ‘f 2 K _ _,, — l“¢ +2| 8. (10 points) Let P be the point (g, g, —1), and Q be the point \/§ \/§ (“T’"_4"_1) a. (3 points) Find the component form for ——\ .5 E—E~—E_—~—l Q‘<‘q“«1, ‘1 H,’()§ —) b. (3 points) Find the magnitude IPQI. \m = ‘4‘ gr +o* U 32 «2 q‘Lq *0 :1 c. (4 points) Find the unit vector in the direction of PE 3) a vhf} VCCfOP, 90 Pa U 71 UN} Vec,+0/ [A fit dtredLl‘Jn 0‘; 9. (10 points) Let a = 277+ 37-— 13, and v = 32*— 2j+ 2013. a. (3 points) Find 11‘ - 17. (a: +33%)‘(31‘ '35 WW = W3) +(3>(’3W"W°’ a 6 ‘(3 '20 2. ~20 b. (3 points) Find 11’ x 17. Q g t 5‘ 1: :1,(3(m)_(4)(_;)>—~§‘(2(10)‘(—M3)> +L(2l—2)-3(3)) Exv z 3 ‘ ‘2 ’ _, 0 -q-q 3 -2 20 =?(4"’3‘3 WM +3) HM ) = Sa’z 4+3; 4373 c. (4 points) Flnd {II-(11x17) a: a (a “7) : (2‘: +3; ~?)-(§76’Z‘ All; "32) .. QM?) + 3(43) 1* (4)08) “'0. A H21 anch’) Mk K PKOAUCQJ R V€L+OK WL‘RA i) 0r+A070h4l i) 10 w! V‘, -‘ 50 EC ' O} bi flk Flare/He) MC bo+ PAM/v61: 01m) Deiint‘floh o4: Ole‘D/Dhfilrly. 10. (10 points) Find parametric equations for the line which passes through (2, 4, 5) and is perpendicular to the plane 3x + 73; —— 52 = 21. U917 m Etwhw‘ f” 0\ Wm, M determine Hmf‘ K 1 <3, 7) 59> I) a nor/Ml vet-7L” 4° “WWW. T14", we . U59, )flL Pa/qun‘c [P4th {V 4 fine JIM,qu Po (yo, #10, 2») wk pawl/cl +0 <Vx) V4, V3 «awn-WV. , ‘3220+sz , Z=zo+fv3~ 50 fl» [twflmf 14” ’t hm. “mics/[k (RH/IL End -* 9046 (W. 11 Bonus. (10 points) Let pn denote the nth prime: p1 = 2, p2 = 3, p3 = 5, p4 = 7, p5 = 11, etc. Determine whether the following series con- verges or diverges: 00 ' De I so Z «a w 7:0 2 .. n" P“ W, A2 12 ...
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PracticeMidtermSolns - Math 21C—2 Practice Midterm...

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