MY EXAM 2 SPRING 2010 SOLUTIONS

# MY EXAM 2 SPRING 2010 SOLUTIONS - Math 172 EXAM 2 April 7...

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Unformatted text preview: Math 172 , EXAM 2 ' April 7, 2010 Name: [2 e “A (9717 (ID#: _______— Section#: , SHOW APPROPRIATE WORK or EXPLANATION on each‘problem for full credit. Box or circle your 1 ﬁnal. answers. Calculators/ note sheets are NOTallowed. Numbers in the [ ] indicate what each problem is worth. Check the board for useful info. Determining series convergence or divergence: Known Maclaurin series: 00 =an=1+x+\$2+m3+... n=0 Geometric Series Telescoping Series Test for Divergence Integral Test p—series Comparison Test Limit Comparison Test Alternating Series Test Ratio Test Root Test ‘r J x J ﬁ.w.m...i.v i-..,,_.<.,...u.v.u 7%; c 4 “9% HVWMM’Z“ ”7/: 5 C 6 C : DE: / MMD>0 , __3%_ -316 3+7 :- j DE' : (<6 / where. Kié 1L “- ——3 r; a games) ' 7: -3+Ke [/éw’e’wiﬁy 54%; ham, S'IKCa ty- 5 _/ ' - A’ppiy 11\$de (wdczﬁbu; 30);] =2) /: -'37‘-/(€ ;.—7 /<-_—_ "/3 a -—‘ﬁ 3% . /" Mu ax: :—3+(‘/e)e W 7:»3+ﬂe ' 2 1/?4 ) __,,/;m../ 2. For the curve :3: %t3+1, y: 5—t2, OS t g 2, do the following: ' a) [8] Find the length of this curve. 3?)le (3191' = {WV} {ad‘s WW5 {a Z: WM 1/): ‘H‘H =§W¢é [Zf117iﬂﬁzd‘t twat = 6 b) [6] Find an equation of the tangent line to this curve at the point Where t = 1. d . dz -:: 7454?; ”2% “e "2/2; 6741/ 3 ”a! n , 3‘) Em 3. [10 each] Using appropriate tests, determine Whether each series is convergent or divergent. °° 2 n=1 , “2 Aim . Z4; Ofﬁcial: o< new 4 "q «34; ﬁwa/lnzl/ M 90161. , y, B awwqeuff [wsmég 771 fly?“ \$4025; if 4/{0 (mm/7.4.»? 5}, 7% [m4 {trigm Térf Q, 7W[7zf+—:;TT‘) 7 ”Liz*?m Why v] 0Z7S ....__._.,.." / :: WL‘ («Lima / 3 V / ”LAME“ W . “*1?“ ML [1—qu 73.. 9mg {’35, it /Cw;m 7/0 4% (WW.z/€l}£9€f/ 7L 7iuéa germ-’6; Wu9+‘a/% be— awn-i» a m M [ﬂaw w Wm.“ 8)n+1 f)“ :2 i " h an M w raw Kw/gai/ 21:: Mayo“ C“ gynf m) KW“)! K e)” , : szM (i) ;; 0' 4 Z 55 safe; 5; ﬁéQ/‘ﬂlﬁ/j) (wwwigwf l; @750 +24% VHI M490 *0 “71 063 ' A , (”ﬂ : (“0N7 5M1 , 5 ,. K Z“! 4' Wm? M! ‘ 2 M g7» / (,JLKLA 11 ﬂ 4/ N 57 gnﬂ ’ ‘ wt“ En: ,, §1MLL éﬂL :V’yTé—f/ (“£wa bh-H {36W PM HZ?) ’4‘ bn 61M S‘MCL {lam} I: 5014MB! ﬂM{ Mammlmxe {1411’ ”2712’éh 3Wf+ Cmve‘f‘yh FMaj/ 711440er 3 7"“ M): Mo Zhié::’) 7M (é) : 0’ f0 72”! gumés Mace Md“ VFW" (aways L; [/4775 Sam; 7725+. 9m wsind 3. (continued) 0> \$623)” n=0 govt/”66”“; 7/0‘1W : 7Mn/ 225:3)“; =7}M 23:):g 5/. (/1400 {4-900 "906 €57 71‘?! W25 11\$ M 4? ﬂwf 7155' , 0R: 3/1444 "a” ZWH " 7N1 ((ﬂ-Ljn} ._ 00/ 71'4"” gym; 7c aljug'ﬁ/cmf 1'? 7L Eiﬁ ﬁne A46, 00 _1 n 4.-[10] Use appropriate test(s) to Show that ;( )1 is conditionally convergent. 0f+ 2:4” (”NI/VS barman: rts «walﬁrm’fiv unkr wﬂ 52% 2a,“ 774?; 12,, dam/l7 cent/194% @174 h «Ni A4; 7M1 [4“) :0, §o m Maw 7’W’Ih‘ éxvieg (“W’YC M4§ gym M7lWM75V'h) QM?!“ Tet—9+”! ' pa 2:10“) NM éefanga Z/a»l 22—71—27] 4Mz: 0 . ,L. . M 7% (5:7)*7 ‘ ”4:: %W+/)*’7 mine /+#);/1 9M Z253 Mad” 1:. at J‘s/vow!" [)v set/1E9. W. km 6:) 77¢ Mm? [av/79mm fes’f VIM?“ as m ‘ M ~ 0, Z {5‘57 a/{o- o/Méwtf/ whck was g0?” “’9,“ng g6) 7'2"Q Mjomé 41ft?! I; (mdﬁsol/MN: Cmvcﬁfnf" Z ( 2)n(:r—1)". 5.[10] Find the interval of convergence for the power series 2 n20n+1 . ‘ QM ZJWH Misﬁ<h ”M L“): 7% W . w 7714 [ﬁzz/m) ‘ V W—‘aw 'M Vl—ﬁoo m1, (’11MKX’4)” [new n+3» ' 2-27x'd. s; ﬁhtwkw&\éfﬁkli%rM/fs 7/rd/4/ =9lardei w) i 4 X4 g ., “<3 h h . 1 '62) 'i‘ 90 ’ 1 I n A114, 7t W, new; 5 MC} .1 i: 175/ ”M. g a f (/13 7k 7,711/ £16771 Aavmsc {41025) M XT’“ 3 77/»: uwé; germ; Z:fL~—,-----v-—-------*FL)(11/)IA 32%;, _/ MAMA (”C (mmmbtf‘ 77 14—1—1 4? 7% A47!- iéWZ-f 7;”?! (4": ’L‘ labial” 5/2021"?- Jen/wees ﬂV‘J A“ Zﬁgﬂ"): VH'j/ 6. [10] Find the Taylor Series centered at a = 2 for the function f (3:) = % . I . . . , w. ’ "l , ”£151 ‘ ’F‘W" I“ ﬁzki {0 411/123 1; . 2 C 3/”.7 1X71)” ‘ {7);}: (-0)?)2—3 “fl/l2) "; :11; C {___£:?MM%3; 11111 V: l“; ............. a4 (—04 , ( 2 -’L {//[)e):€-I)(1)); 3m,{//[293 —g:§-l 3:: AZ; 2W?! (X )j n) a n/ {n —I ”ml {N (x), (NW/X *) .1 f 2/2): C ) M , 0””er 42/sz) V\ atmg, , .J;n\; 4( / 1&4? if 2% ’ X—«l +2 a /—'i 'ZJ1:§;[ ) ). 7. [6] Using an appropriate series, compute a decimal approximation for 6—0'1 with Ierrorl < .001. ”ad, (ﬁr/he'll?! méw .7051-‘334 4,3 ‘4 ,.?05']* M ,70’13364 éaléﬁob.) 8. [2 each] Fill in the blanks. Unsimpliﬁed answers are ﬁne. No work necessary. 00 n. ’Iivﬁkvm: ﬂag a) The sum of the geometric series Z; 3n n+1 is [/Y‘ /— 3* n— bThfh'ooi=1111..." ) esumoteserieszn! +—1-!+2—!+3—!+ is C :.-l 0) Give an example of an alternating series which is absolutely convergent: . 7. M d) If the Maclaurin series for f(\$ )is )an”, then the Maclaurin series for f’(x )1s ' 2 V1 x n: 1 M=l d) If the Maclaurin series for f(a: )is Z nrr" ,then the Maclaurin series for /) f(\$ )dm is Z ”+1 n=1 —1 n 1 1 1 1 Bonus [4] What is the exact value of the inﬁnite series Z< 271+) 1 = 1 — g + E — 77‘ + 5 —- . . . ? .. aw) V1 MI >< +C ...
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