Final Fall 2005 Answers

Final Fall 2005 Answers - Math 122 Final December 13, 2005...

Info iconThis preview shows pages 1–11. Sign up to view the full content.

View Full Document Right Arrow Icon
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 2
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 4
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 6
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 8
Background image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 10
Background image of page 11
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Math 122 Final December 13, 2005 KEY Name 1 l ‘2 L 3 4 5 6 7 8 9 10 Total E / 200 Directions: 1. No books, notes or passing go and collecting $200. You may use a cal— culator 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 our work. and ex lain how vou arrived at our an— y , ., 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 aésk. . 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) MW dm‘ 5 , Gib “3:? mm mm. A? u‘ a ——~—-—————‘ "3: V2; , MW WWWWM ()/emvczm+4 U gig? Hm", M‘ w H‘ t. / du _ __\/u2 i0? “LO “1‘ wm “ ‘ m ‘ WWW-“W . m K ' mm m H , d w 8de use“ Oh? @fi?2§§_ . 7 w 24: (b) m+l d1. 3 g ":kdfixx (ii) m “in” 3 ‘ "’WWWM w r 2“ \/$2+2$—16 W 3331+?” a}. '3:- U'fii X +&E&MEQ2 “5%; m. '75 ’2» (L dwfla‘xépu d% W E.— (ai‘x-EVMKQW “3“ 4'1; Chg N, fig“ afi+& (c) /£E$€C232dili “*3 XWNVW SWNX d‘fi' % W1 Um‘fi finghfixdx 2;» X‘WN‘K. WQMV§E<A¢E 4» Cg; ckwrfiwz \1 mem “3 2 - (d) fsinSmcos3xdzc. "—3: EU CQKE$ fig“ UT: Sxmv‘x as: MD?) 6““; dun): Cwm‘é QM ‘ __ L‘\ {a . Li G WE: gkfwwg ch”) 3 w E)“ A: MSWX “erg: M Q. Li «:3 ” 2. (20 points) 1 + sins: (a) l ~sin23: m 3 w Sggélxw gcaegxmmufi {35% EESsW Ca 41' >4 m ' % 7% m (b) /(se03 3m)(tan3 33:) d5); 3 S U “rflng “WWW k.) 7:: $‘gc; WA 0w $E§g£%%TRN%X (£334? .3. A 3 fix, 3 gag-gm” 3S€L%‘I~Tfiw%i g ’2 m A,“ Law m ' 1 m _, Lia w W ‘3 Qéaamfi ~+ fiai >£m<3 1a w fim '3? fig “:2 MW flflw‘ a “*1 3&4“ W M 5: a: *2, S at, £29 CM? ((1) /(4+:2>2/2 d1" :3“ j I E M Mrmw {if} “75; 2:: QT‘AN £3 ‘ 42% fie? M Rfi‘éfizg—gfi &@ “3: Wm?“ ESQ” M 65% w a €559 Q fffi, 2&1 ‘ E . \ m Us H gm; mmwxwfimx+£4 3. (20 points) i (a) For the differential equation (67% : F (:L', y), Where F(:L', y) is given below: Use Euler’s method with h 2 1 and y(0 = 0 to approximate M4). h?(.~:<p~ifi “i it in Rafi? i '. . \kwk 3 XS!%‘}’% 9:. 1 31(0) = 1 Q" Raw dié at" 2:) 4. (20 points) A tank contains 30 gallons of pure water. Water containing 2 pounds of Grape Kool—aid mix dissolved per gallon enters the tank at 5 gallons per minute. The well—stirred mixture drains out at 5 gallons per minute. (a) Write a differential equation (with initial condition) for how much Grape Keel—aid mix is in the tank at any time. will dim r, :L ma» fagwifl ggg ‘3}{03 C) i i W W \l 1‘ Mag; 3:0 if: "t C” magi“ Q ' ., rm?“ \ attime a 5% km e5 Mil Home“? * k ‘ CV23 “gm “MW—k if”: @Qtfl Q (a mini; #3:ng WW 8% (c) HOW much of the Grape Kool—aid mix Will be in the tank after 10 minutes. w. lg e \Hm‘) mega meoe «w ((1) Find a second order differential equation that has general solution: y = ale/53’ + (22643” R i; 5 gave: “:2... Z r m “Q; m» aem in; \sz w £33k Slaw-3:) W” was “3‘1: mm?! $9 5. (20 points) (a) Find the equation of the tangent iine(s) to the curve x2t2—2t+2 y=t4w4t2+5 at the point (2,5). f2“ Cian W mark?“ a mafi*1”& ex em?» - 1 5&3 Wm t Matt’s; Q etew-ai‘iefl \g «:2 5 43230 ~ ‘1. mete-«w e (we) (c) Set—up an integral to find the area, of the region outside 7‘ = 3 ~— 25in6 and inside 7" = 2. Do not solve the integral. 2. { BEE; me- ~38qu :1 “TV (a it 6. (20 points) (a) For each of the following series, determine if it converges or di— verges. For each test you use, you must name the test, perform the test, and state the conclusion you reached from that test. , 00 5”(nl)2 1' (270‘ l l e? t “32‘ ml "‘2 z N”) N M M Ma Q, \ a (My Lagwm 2: M a. List t ; WW rm ‘ . ‘ N 4% m (aémflyfi 5;; Us 33 N “‘§ m (1N «MIQN VEQ @\V, 11. ‘2 fl , 7 n + 1 n:l W > 5”” will.“ N g Maze 2: m ‘ . WW , N a... L‘QT, Q/vm Ntl «3., Jle WW “l . b5 em a NEW N M m ' re: w;~~ w N N m N“? (b) For each of the following series, determine if the following series R converge absolutely, converge 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 (—1)“ 1. g nvlnn m “We ls F3 \ \l rumeraeu g “is, d a ~==~ Emmi , 3i ill/WV \ ‘\ WWW v” . AM“ m, 0 \f WWWM‘ m { N " W "" m M W N mid {jam \l é CGN O .. °° (—1)" u' ; 1+ 4‘n , i m N em “Wm QM “:14 W X if 33 D w 7. (20 points) (21) Consider the power series: TL 00 (cc—#1) 1. Where is the power series centered? if: WE ii. Find the radius of convergence. \CMikgm mwfi\w hiefl 045% \em‘rél Wm‘ev‘m‘" 2L eke iii. Find the interval of convergence. (Hint: Check the endpoints.) , as Wg (:2 L13)?» {jam ‘3 Cami) m Emu?) 2“) ,a/ \ W $3 \ \2‘ 5 2 m (b) If the Maelaurin series of f is x3 .735 m7 f(33)~33+~5—+1-6+T5+.H 1. Find 134(1):), the fourth degree Maelaurin polynomial for f % ‘ W V “3‘ ii. Find f”’(0), the 3th derivative of f(3}) at a: : 0. .. M WWWWW PM . "i" $3.?(03 ¥%Q’W}K€Q €33 e. E3" in. Find hm ———f(m)3“ 5” CC—»0 (L’ 3, 5 ‘ D errrrwww~r ,Q/vm g 50 \ slat?) 8. (20 points) (a) For vectors '6? 2 (l, —2, and 79> : <3, —1, 1) find: 1 3.77 3+1+E x % 11 if X 5 ,U 3 in in . $5" awe. 3 ‘1”1 élfigjv % M l %~fi iii. Find the area of the triangle with “CT” and .5) as two of the sides. i Wm w W m We 63% “r 55 mg: iv. Find a unit vector perpendicular to both if and 5 finale} WWW V. Find the value of a: so that ‘3 x (2,217 —-8) and “a7 are per~ pendicular. affine“? (43%)“??? H gmeflqu$<3 f"? (b) i. Find the distance from the point (l, -2, 3) to the plane 33: — y + z r: 1 mgr it? :35“: «mm/93W: ll m QM W ‘ M ii. Find the point of intersection of the line: :1:=—l+t, yzl—Qt, z=3+37€ and the plane 23: — y + z = 7 éliwl-WE‘) w (imam «in (gt—gm a» “"7 Wgww'g§£wl+&%+q§dgt 3:? <Q}M\}(;) West ear 9. (20 points) (a) Find the equation of the plane through (1,2,3) that is perpendio ular to the line $=1+2t, grime‘xg wgfi M m y=2+t Wt zz~3~5t (10) Find the equation of the line of intersection of the two planes: 393% 4% text V11 9.Wi a let m+2y+zzl (C) Determine if the two lines: and a:=1-4t, 3::2—t, are parallel, intersect or are skew. \th‘t flfimfi ere-M 2x~y+z=2 PQ‘M‘T (ha-39:? )4 33‘ tevéfbt \i fr: :3»? "h E 7:: y:2+&, zz4~2t «(MHJ%)W§> y=1+a zzz+m <“333)G§ Not“ QQRRLLEL Wflwwjj‘f t H '% cg:er fizfiakg [Mmmm 10. (20 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 Zm—mjzi The Méaclaurin series for cos 2:1: is 4 164 2m——25£!—+—47§——... The two planes: 2:13 + y — z :- 5 and a: + y + 32 z 7 are perpendicular. The sequence an 2 {ln(2n) —- ln(n + 1)} diverges. ~1+1~1+1—l+...=0 in; 71:1 1+1:2 ...
View Full Document

This note was uploaded on 02/17/2010 for the course MATH 122 taught by Professor Butler during the Spring '07 term at Case Western.

Page1 / 11

Final Fall 2005 Answers - Math 122 Final December 13, 2005...

This preview shows document pages 1 - 11. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online