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# Exam 2 - AMQWW EXAM 2 MATH 2153 SECTION 2 FALL 2010...

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Unformatted text preview: \\ AMQWW EXAM 2 MATH 2153 SECTION 2, FALL 2010 INSTRUCTOR: WEIPING LI Print Name and Student # SHOW WORK FOR CREDIT 1!! SHOW WORK FOR CREDIT H! (1) (Spas) The curve 1; = 4 + 22,0 S w 5 1 is rotated about the y-axis. Find the area of the resulting surface. I 3 = j 271x :13 i d3: 1+ (\$5?)z AX B= “Xi o d)’ v i =.5H4x* Ax a;th ‘8 217xJH—4x2 4X 2‘ clu=8xdx ,2, = 5,271“ 9; :3: éMS/zfs : 2: (55/2... 4- 2 l 1 (2) (8pts) (3.) Find the limit of the sequence an = - lvl X . J" X—ﬁw X X500 5X: “M an : O 9 W'>oo (8pts) (b) Find the limit of (Ln = by using the squeeze law. ,L ~ .L <+ O 5 Q 5 7. ‘rwv 2 :— O 1 n ' n ’ n l \m m“! :0 1m ago. 17%st 2w W- k—wo ’ 1 mm» W ‘~1SS('VIZI4 5‘ - 9/— _I_ 5 M (.L‘ (‘mtﬁl=o f/rﬁwaéféw. 91133520 h: “ I17— hﬁw WEIPING LI (3) (8pts) Determine whether the series 2‘” (——1)“1“Tn is convergent or divergent. n=3 ‘ . " Inn A'famd‘mj genes [66; b":- n I LX' ’"X x r; x (n )9" decreasfvj; frx) ‘ a ﬁx) X; ‘ :; ’“IMX Hana +02 H25) ’9" I‘S decreasinj x2 ‘3“ > bit-H . ﬁ" X > e. ' ~ ‘ '“X .. .‘ '1; , (1) JLMMf(x) *1/34“: X Fig/loo l ’- D E" : o ' Hence 'H‘L Series 15 Convetjed'. 0O (4) Show the following series are convergent and ﬁnd their sums. mmmz%%r ac am! e boo en" 8 . :Z shE'E'QT) I Y:~.<[. 14-! H1; 4‘ 4a __ e - 4M “1—3: ' 4-6 4 , '+ Geomet’t‘c Sal/Q95 wF‘W 1Y[= (I I is v Ctnvefjw+- . .. 3 3 (11) (1013175) 220:1(m ‘ (n+1)(n+2))' _. .. .2— -_§. §'—a'"t-z z. 3 3 ; Dir}. 3 v... + r ‘1“ -< . 3-4:. gramm=.z as :55‘4 '3 5 g 3 - i a; _. ~— 35 :7; "" 3"!— + g (L \$.1— [-Z 4-1.. F 3 3 _. S" =a,+az + +61” " 7—; (M00117) 3 ‘ 3 .. 3.. Lil"! S " "— v "M p ("+2 2. ‘1 z “7&01-D ) CALCULUS III - 3 (5) (Spts) Determine whether the series is convergent or divergent. You must provide reason for your claims in order to receive any credit. 00 2 —1 (a) 211:1 3+1 ' lim a“ =1 [5M 11.),» moo V‘ﬂ h—no "HA .L (6) Determine whether the series converges or diverges. Must provide the right test to have any credit. 2 L (i) (5pts)z:;°=1ne—" - k, x1 J5 vx L ' v = (‘m X 8 dx M = x 11?er E51, X 5‘ 4X J?“ l L dqzzxdx ‘3 ' b —u . ’ ' b -( _. I'M Q Q :: (m e _ E. z, .6... M L [0904 t 2 19m 2 7- 3 1 —l4 2 v1 Q .3 Comij “17 14:: (ii) (513%) 230:2 {Kimmy Comfdﬂ‘fq: b" = M4 : n: ’ Rd, ‘ an a! [’11: :[f‘W'l —--—--/'TF :1 >0 w‘oo L,” hvno n‘ﬁc4/n4 “*7 179 9Q 60 1 it + g hZi En 2’: ’15 j / , n, “a , 3. Hence I an '5 deaf ' n=< fvseriés . g WEIPING LI (7) (15pts) Find the sum of the series 22:1 ((54%; Ah'wnatr‘uj 3068.4 error esff‘m'o—C With the error of less than 10‘2. (Rn, 4 bVH-l '2 1 So bn+[\$("1+7)z < '0 ’0 < (“7) , n: 4- ‘ Thus S :-._ Q.+ a2. + + 04L 4. - I ("3 J... (') —-—--~ + f" + z (8) (15pts) Determine whether the series is absolutely convergent, conditional convergent, or divergent. h +0 ’ 2n (1)2113}? R “(7‘ “ﬁt .- {I‘M an” :3 iw1 ( ’ «v “7‘0" 5‘” WM» 2" "I =- lim 1:1... :. be > I Irma z ‘/n i A]: (11) 220:1(1+;11-)—" hr‘ J. oo't Tés‘t 1m, Jmnl =l;wnoo[('+ n ] hf>oe ‘ -‘q ’ ’11") ‘ :Z— l I :— ‘. ___‘ “L: (I + n ) I’Lmoo[ n ) MEN}? :_ f," < I Hence W Sane; 75 (iii) 220:1(—1)”ﬁ “it'ij saws 1—95-1— E l ‘ I '- ' w- ..L— : 'W) "a (Mm Ln:J—‘:~‘ b":"J‘l‘q—_= WT bx!) Prim 9:1,)!“ + “W mi“ ‘5 CW5?“ , M" Z (6')an / :5, 72’ 4W5” CALCULUS III 5 (9) (Bonus 10pts) A sequence {an} is given by a1 = V5, an+1 = V3 + an. (i) Show that the sequence {an} is bounded above by 3. 133—qu, {MA/“0+;9ﬂ inucipie, Q‘L'J-g‘ < 3 . Surfose 0”. < 5 ‘ Hence Q“H;}5+qn < 15+3 =f—6_ <5415; Theefme an<5 1can n. (ii) Show that the sequence {an} is increasing by induction. a.=J'5‘, a; 45:3 > 5w ‘53:“: “W {MA/Raﬁﬁn 73"?"53’9‘6 J arr-l < an. 5 if. an-’ < 3 + “V! . QMA <.15+0\n‘ ya Lnyiui'Hm a, 4374“ < Jsﬁh ' "H- (iii) Find the limit unlwm an_ I ~ I’VL fa } '\$ L and Menﬁf‘onical9 IMCMQSJ . n I d u 53 To, Hence In ((1975:; H (m an = L emf“ m m" a /' 4 . ,= 7" 'm .4 . ‘ :9 5 “w 1;me + w W 00 . A :1 JS-I‘L ’- 3 . ’ L, ; ...
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