# ex3sol - MTH 133 ~— Seotion 14 ~— Test Three DEPARTMENT...

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Unformatted text preview: MTH 133 ~— Seotion 14 ~— Test Three DEPARTMENT OF MATHEMATICS Michigan State University November 37 2005 NAME: \ {El 9%? {35”} ID NUMBER: SIGNATURE: 1. This exam has 5 pages including this cover. There are 7 questions. 2. Include units in your answers whenever appropriate. 3. You may NOT use your calculator. 4. Sines7 cosines, and tangents of simple angles are to be written out. For example, if you have a C086», you should write 5. Present your solutions to each problem in a clear and orderly fashion. Include enough steps to permit the grader to follow your method. Your method must justify your answer. PROBLEM POINTS SCORE 1 30 2 20 3 10 4 10 5 10 6 10 7 10 TOTAL 100 Present your solutions clearly. 1. (10+10+10 points) Find the following indeﬁnite integrals. (a) fmsinzcdx é)? ﬁajﬂgb W y%{“{§3?§) ’ j” {£2ng a ﬁé‘iii’if W “a” Xéégx «% 5665 x agxi M 7<<¢3§§< mfgf‘ﬂﬁfyc 1 I 5” 3 (b) /_———\$(m2+1)dac { Qévir’éw‘g Tm¢hm 5€C§m§3§§f§ﬁa> % A 10+ _ fimva: X{5:6} w 7‘” X24»; €13 ééxzvéwi‘) +(gx1’ig74 "" (3&5) 9:2" fCrﬁ ‘% 7‘1. Wmhimiﬁﬁ {gégeh‘éwk f} 3"!”{3‘3 1% ii} 8: pg} Si? ‘ I _, jxéxi \$52K :1: /:~ ’ b y: x ‘ W} { x m?“ a 53% M '2?! ,3 ' m. ,, w i 5, 7 ~ <c>/x+1dm QM“ gﬁmixz‘ﬂ> ﬁwC x2+4 i gm :3“ 51:12:43» {oméﬁgﬁg’gmx} Z: i Z?‘ [\3 Present your solutions clearly. 3 2. (10+10 points) Find the indeﬁnite integral 3 x (156, ~1 < a: < 1 / V1 -— 332 ' ’ using the following methods: (a) trigonometric substitution; 6 aﬁiﬁv’si 6’? ggiegmg 2‘ W a Swazi»:— jii l, s‘ehﬁe /’ it in grew-g ; 2"» w ; was: {is} rm; g5»;(z ‘35?! \$2: x; ét‘éoﬁeafée j{§’{§§28)¢ gsw'éag iggjwiimﬁi F Siriwua} ago: ; w Frugal pm '3‘“; CA 3;)r’ré ‘1 fe§§+ées&%< ; W ti 3 w r i ' « \Q ,. ‘ ,L. m, 2 Z 3‘ >< 2» “i” 3‘: 5 ><~ > ii. < (b) back substitution. h” 95%5‘ 52% : Present your solutions Clearly. 4 4. (10 points) Compute the following limit: n , n 11m n—eoo (n + 1) 5“ :53 F”? ' hei/ } e»: Jig/:4 j’m n) m'FQG we; \ ; {fa/V“? V? w ﬂ ﬁgw QM 16w :3 Cami, I} 1:: IA m / ‘3 W39") m (to i m <22 we 4’14”? 5.5:“ W may? (“3” 5) ) 5 W :1; {3 1:34 W M J’W’“ ﬂ? {mega W aﬁw M; 3/;3, .a be an: 9;,5 So it” [53“ r" i may Vic/Mi} ' we we} VC 1 5. (10 points) True/False: @ F Q The function y(t) 2 O is a solution of the initial value problem Suppose f and g are continuous functions and 0 g f 3 9(1’) on the interval [7, 00). Then f7oo f dm converges if f7“) 9(5):) do: converges. ﬁg {3 W 95%??? 569:??? , +5.23% 1 i, /mdx:1“lf<\$>‘+0 5e; weddieo If an —* L as n —9 00, then lim ea" : eL_ 71"“)00 €er¢e xiii?) : (3 g {S €5e§§ﬁv€€m§j We we? arm/see 53>??th é Menage» 3% gvrgc‘hzm 3 2. 55%?“ “35 5:12: {leaéag kmagﬁ weasel-if s: é‘:’1~«§3}®§€4. %% =3t—y3, 11(0) =0- wm Fiwgﬁéé 33 +53 riﬂe £39? €621??? /, L??? game; 3;??? 3 @ gaffe?“ £53933; SIEEE : no?» giggide Present your solutions clearly. 5 6. (10 points) Use either the limit comparison test or the direct comparison test to determine whether the following improper integral converges. 00 1 d /1 Vas3+1 x ﬁg” 2x fl 7; Kg "gar all "11% {\iwj} U N Z {is} 2 x2 "tar all *sévﬂafeo} 5;; wry—1w“ % xi,“ 3 :5; £10!” all “7:5 1::wa §§?\¢Q g / , g c;qu w ,.3’ i 7 g a: x‘e it my} (g‘gsl )9 3:2“; pleat? , "K r ; MM?’ 6 "is l («0 latréé‘ﬂ ‘LQYEW; I ﬁle: &\§a {wtxvﬁ/iiﬁﬁ, % whet ‘3 7. (10 points) A hard-boiled egg at 980 C is put in a sink of 180 C water. After 5 minutes7 the eggs temperature is 38° C. Assume that the water has not warmed appreciably and that the temperature of the egg changes according to Newton’s Law of Cooling. How long does it take for the temperature of the egg to reach 20° C? L59?” Tig’} 19\$ %&mg> 6%? er? ﬁw {I if} mgb‘} ( I'm 3 The/ﬂ Newwﬁg Léiw <3“? {oagﬂji géagj 5i? ’ ~ r “ ,3; a; k (72. {52> 2, maﬁa, War—«355 W “W \l W\ WW Wy‘ \l 73% 4t ‘30 Q m “7‘ (“N w v m C(\ is i \ﬁ> JR 7“?“ 7‘” \a § x ‘ e13”? }%i/§~§ "2 <35?ng 'végf '5: 529Eer /_13_dx 2 égﬁaé ,,; \$(1+ﬁ) E5 :6 ‘t 3;... "be" gm“ :33;%§ ...
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