ReviewMath2224Fall2007_1

ReviewMath2224Fall2007_1 - Math 2224 Spring 2005 Common...

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Unformatted text preview: Math 2224 Spring 2005 Common Final Exam, Spring 2005 q __ Q \ '5 (1:3 3 Ci'r: '5- ») I"-~§-;_.3: —§ 6‘0 /;7 . . . . 0° "-1 . . . 3)” q l. Consrder the infinite serles Z (—3)"/5 . Which of the follow1ng statements is ' I 1 - ._ .\. . —. [7:] “=1 5"" T 5 correct? (1) The series converges and has limit 5/8. @T he series converges and has limit — 15/8. -1 («3) (12)“ (3) The series converges and has limit 5/2. 5 my (4) The series diverges. ‘ =) a ’— “3 i _ ‘ _ i r : - 3 2. Conmder the functlon f (x, y) = x2 + kxy +y2, where k is a real constant. Which of the 5‘ following statements is correct? . TN" " (1) For every value of k, (0,0) is a local minimum of the function f. 051— W M ~ r _ ‘ gum; j; 3 :2. $18,512) For every value of k, (0,0) is a saddle point of the function f . W l“r l—(é) (3)- For |k| > 2, the function f has no local minimum. 3 (4)2 For lk| < 2, the function f has a saddle point at (—k/ 2, —k/ 2). ' i 3. h The integral of f (x, y, z) = x2 +y2 +22 over the solid that lies above the paraboloid ale W 2' = x2 +y2 and below the plane 2 = 9 equals: (1') [HARM/idenwpdide @ (”2+?Z>rdzd7”d9 3 3 3 3 3 9 <3) / / / (x2+y2+22>dxdydz <4) / / / (x2+y2+22)dzdxdy 0 —3 —3 _3 _3 x2+y2 I l 4. Consider the infinite series 2 2 . Which of the following statements is cor- n=1 )1 + n + 1 I-ect? Hm o r121 n-\ I I: ' _ n—voa (n—H)Z«(n—n)—ll f —\ a- (l) The Ratio Test can be used to show that the serles converges. r 1% fl) ‘ r . _ . , hm 319,2 , 9,33...— -. a (2) The Rat10 Test can be used to show that the serles diverges. ’ n_; Ba {,1 a 7, (malt (M34 1 @T he series converges but the Ratio Test is inconclusive. : \. l: i god-Ct? Taxi [M vi 1 . . ’ . . . . . who it (4) The series diverges but the Rat10 Test is inconcluswe. E * \ir‘m’r' r. s n -I hum-4A i , \IL : COMpQ t o W i 5. Let F (x, y) = xyz. Consider the following two statements. a \57‘ 39%;, : 3/Zw “'24 ns/z 1 “" cowl-Maui " At the point (2, l) the function F grows most rapidly in the direction . At the point (2, l) the directional derivative of F in the direction —4i+ j is zero. Which of the following assertions is correct? (2n) I V, JFst mmh‘mfldbjwbx/k the cilircoiwn fig V1764) : < 1L\Ct'(1,l))+t/(2’n7 0 . flaw/7— ) 1Y2 axa ; ‘f’luzm Vita“) -: <&,I+7a-M:Li() DZ— fvm‘? '= Vigil) °<~Hu> :Zh'LW'aé’H H7: or Statements (a) and (b) are both true. (2) Neither statement (a) nor statement (b) is true. (3) Statement (a) is true and statement (b) is false. (4) Statement (a) is false and statement (b) is true. ‘ ; 2 2x gladclm PM WWW?) 6. For any continuous function f (x, y), the integral / /2 f (x, y) dydx equals: W7 ’ mkfimhw‘ 0 x . ’¢ 0>Af£3mwwe <§>Avwhmwma J?””Wmmmi .42 4f; (o_AAfmnaa e>A femaa AY 47d 'L _ 4,2 x s s 2x 7 D ’iosx ) V 7. The first three nonzero terms of the Taylor series for f (x) = v3 ~x expanded about 7/361 fimé’d“ the point x = —1 are: M M4} Mal (1) 2—(x+l)/4—(x+1)2/64 (2) 2+(x—l)/4+(x—1)2/64 (3) 2+(x+l)/4—(x+1)2/32 (4) 2—x/4—x2/64 )flyraygiM 3)a)-r 8. If the linear approximation (also known as the tangent plane approximation) of the function h(x,y) = sin(—2x+ 3y) + cos(—2x+ 3y) is used at the point (3,2) to approx- h a)(>< *3) , . . . HP Xb’ unate h(3. l, 1.9), the resultmg approx1mate value is: Jr (1171559.) 6/41) mm (1) h(3.l,1.9)%1.5 (2) h(3.l,1.9)wl.l 11.5.5131) 2: h(3)a)+ (3) h(3.1, 1.9) m —0.5. (4) h(3.1, 1.9) m 0.5 + bx (a )9) - o~ Helix/(3)2) "OH 9. A lamina occupies the part of the disk x2 +y2 S 4 that lies in the first quadrant and is bounded by the x-axis and the line y = x. The mass density of the lamina is p(x,y) : a; W WA x2 +y2. The mass of the lamina is n/4 2 n/4 2 '(1) / / r3drd9 (2) / / rzdrde ‘ 0 0 0 0 (3) ARM/ozétrdrde (4) /02/0x(x2+y2)dydx . _ °° (—1)” _ °° (—1)" . . , . 10. Given A — Z Inn and B — I; 1+3_nn, which of the followmg statements IS 11:2 correct? . . . . , .1 (1) Series A andB both converge. ’ior A Q‘Yanoft‘mj gal/“es mi, W‘H‘ bnéi‘n A Series A converges and series B diverges. ( “m b“ :0 2‘ b“ cifmmg I :j /—\ {/Onvvmyvg' . l ‘i'\ : ilm 7- ji-fHZ’FO (in/31,, 7 11"“ Hg‘v‘n n—vw H HO 43°03“ 4:3?) (3) Series A diverges and series B converges. (4) Series A and B both diverge. X ; 2x39 11. Consider the function g(x, y) = exzy. The set of all points where both ag(x,y) /8x = 0 ad! 331 and Bg(x,y)/ay = 0 is: % : X2 6 @he y-axis. \ " O ('3 X ‘ O 05' 7 (2) all points on the x—axis and y—axis. _ >0 (3) the origin, that is, (x,y) = (0,0). (4) the empty set. At least one of the two partial derivatives is nonzero at every (x,y) point. 3(620 23x30 run 3% :olrweufor‘ GUN:0)\1 r604 1 1 . . (1621/ 12. The value of the integral / / V 1 +x2 dxdy is: Ska—Rh HQ. 67 C2) 0 y Grok? m'iiow chafing. 2 m MM (2) zfi/3 (3) 1/2 (4) x/E fitmwimfim (1) (Ni— 1)/3 Consider the series 2 (2x — l)"/ The set of all x—values at which the series con— Radio vie/3+ 1 +u n= :7 lim o [R verges (the interval of convergence) is: n—wo mi 2!“ \ v r, :[I'm rx \2x~u\= (1) 0<x<l (2) 0<x<2 OSx<1 “\-‘.llw(lfl_)f)0<x£l “a” Wu ' __, 125),) 1 l4. Consider the following two limits. - _ . . ‘ (A 2 ' 2 2 - 2 LC lax—x )4: (91+ MA”- (A) hm [x -|2-51n 2y] (B) hm g [x2 Sin .4 (pom we; a; (x,y>~(o,o) 2x +2 (x,y)a(o,o) x +2y Comm?” Which of the following statements about limits (A) and (B) is correct? O<ZX<Z l N 'th 1' 't 't 6:) OLX (I ( ) e1 er inn ex1s s. V @Limit (A) does not exist but limit (B) exists and equals 0. chadwfizmwfl (3) Both limits exist and equal 0. 1:0 9% Xcl (4) Limit (A) exists and equals 1/2 while limit (B) exists and equals 0. tth Ml. 15/4 27: 2 15. T e volume of a solid is given by the integral / / / p2 sin¢dpd9d¢. Which 0 O 0 of the following integrals also represents the volume of the same solid? 2 2n \/4—r7 2 1/4_‘,2 4_’.2_,2 (1) f0 /0 / rdzdedr <2) / f0 ’ / V ‘ ’dzdxd) 0 —2 o x: e) Z J—cl‘sr. _r (Page; (3) AVE/()ZflAzrdI/dedz fAzn/rfirdzdedr walk 9:)2q) flint cow (0)} _ .1002 c135 2 “FY—f 3 4A, f , Qrifiicai POWH'S‘I; x:— ) 4% 4506 \ \C¥,\a,\=( oJcD \ K Ya, __‘. H , 1 J i ;"w‘:'=imwwwn "mmrm Trix“ m imfiwmmmm >VU‘ -L f _My.:/;,?me2_mwm:£:>$)};¢ m WW V. ;;;.fi.5...,w,,._mbffiifiéyzgmw ( x I ‘ D 1:; — —— ' 2" K' . 3 O . 7 y _—, r Sing y of? 379 3 "327‘ ram—.nxfa, Hwnzhflwnan , MW“. ‘.,;..‘..._._,"w;_;_;_“W..\LM_,.WW“bum,‘ 7. F , f ____ d a“, Hr, ‘ H "W , ‘, m “J - - -- w- V7 ,,7..D.A_=_,_._{,(c,_fie&>wtwo grapes?” ___ U. m .CE.,._._MW_A_-.__fi,.;___~,w_r __7.-.-_ v 7-- H ' , _ 0 0 ._ ., _ L .2 1 'I‘ \fl 7‘ , .. r. ‘\\.{/’Z/ ! ._ . _._g__¢.£ita\ 1 w r saw—72>} \L-g/Q ‘i p LE _‘_,_.-.__,‘._...~ M. - _.. MW _ ____.. _ WWW. ..‘.-_-._,_._ raw“. 22.-., ‘..._r___._m.§ j W m w «s cm :4: + I « F "“‘ “afaézj‘m‘ ”‘ ’“ fh‘y—L‘V‘” RWETFEE‘L {mm .7 Jam». -::r_.._..r...r_v_..A~_A___.,. .~ mu.- A—J .u mmwumm htdmmr V , X“? 0 91 x 7» + 0 ’2 Y~>P <2 jOz'fi \/ L U X L4 Slhl O ‘ ...
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ReviewMath2224Fall2007_1 - Math 2224 Spring 2005 Common...

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