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math_172_(fall_2008)_(schumaker)_sample_final_exam_(spring08)

# Math_172_(fall_2008)_(schumaker)_sample_final_exam_(spring08)

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Unformatted text preview: Math 172: Second Semester Calculus Spring Semester, 2008 Final Exam “nun-un— "II—Elam "Innnn—_ mmmmmmm-m-mm Name: Section: ID Number: QUestion 1. (10 points) Set up, but DO NOT'EVALUATE, an integral for the arc length of the curve 7 y—4-si11(x), OSxSx. Question 2. Find the volume of the solid obtained by rotating the region bounded by y=0, x=0 and the line y= —2x+2 about the y axis. -(a) (9 points) Use the method of cylindrical shells (integrate with respect to x) V; 10’ meflz) J06 ; f0, 211.1:_[~Zz+ 2) J1 .- 2:: I; ~2z1+zz)d.x 3 21f [ ‘32}: {'11)}; ;,2n[/“§ 4‘!) O] @ (b) (9 points) Use the method of cross-sections (integrate with respect to v) V :Vlfrrxl 4y Y= -2*+Z O > y-g.="2x . 2 (1)4 , LE/lrftqy y g x: {43.1 [95y : [z/y_1«y7+’l y3)lé :M/Z’Z+:) e Question 3. (6 points each) Evaluate the fol-{owing integrals. A. fgt/ZZxcosxdx :1, ’IA ~= 1 Chi: (oyfm)dm 8w: da: 1/: yin/x) i I c 1 gmfawf “fﬁ/ zS’ib/I) (186 Z , : g-Iwmw ”(war/a2»)? =— 5+.m/wdw/og) Efﬁmdx = J ' 4 Hard Meant}: {at 1%1‘m/e)’ 3+C . Easy: :24 w; {ma 1‘: 114, Au : mm W W’Z—wzm 3 A [amt/H 2%. [or—z) + c ' Question 4.’ (8 points each) Determine, by evaluating appropriate limits, whether each of the following improper intervals converges or diverges. In case an integral converges, detemine its value. A. folx‘4/5 dx‘ = I {Ham / 3 [1M gZI/‘S'li {3.401‘ U ‘ I a [1m §[/—£/Zr) 640+ l) *3 g. ”1“ N\. \J E. , m l \ \J Question 5. (6 points each) Determine, with justiﬁcation, whether each of ’ the following series is absoiute’ly convergent, conditionally convergent or divergent. °°5" M an; 5” /Q:i’/;5"“ :1 ; 3: (b) n40; , h, 01., (MM), Sh ”H Question 6. (20 points) Find the radius of convergence and the interval of convergence of the following series :J—4” Cluck end Faint! WW 1:—3 : ratho tact ‘ J-l n q . n 5: z( ) "Z(I) QM! - [14]“, 6:4 ' n=l f4” n: -N” 7 : +l - h , 0 W4” {2 II AﬂLema/th Sud-u Tuf , __ _L : (31;?! lg! V A”: f? /‘/ Ah+l< 5" :: pm: 911:: : ’1’” I, ' L nq‘m/ an] T 0/) £2 6": 0 If L( II 5:18 abiot‘ﬁiy yZMaIW ‘ (O'hvmenf ‘ MW 24‘ Z ’L'W/ a? [1 we ’ , y- »:W::: I”. F 4 : “4 < :64: d : [St (I “a danﬂw‘ [é‘ﬁ'eV/Pf : V P~z .L (I _ ihTiyVa/ c6, (mus/70m, [*3/57 Question 7. (20 points) Let f (x) = 1n(2 — x) and let * ln(2—-x)=co +c1x+c2x2 +C3x3 +... be its Maclaurin series expansion (also knoWn as its Taylor series expansion centered at 0). I‘Wernaﬁ' Soluﬂm Find the coefﬁcients 60,01,02 , and 63 . {60: Z L, {ﬂu/MK” //—-* = /1‘X+X21‘X34w PM n /"’X ‘ V APP/la: x A X/; , 3 f/x): ln/ZvX) ff”: ah) 7:: = Hf +§2+§ 1“" FM: ~{2_x)_, {70): ﬂ} Muﬁ‘P/V [)7 3" I {0/}?22 —{2‘)<)-2 301/0) = /y :‘2/2'k)—3 flu/0) : .24 :L — 71 +5 * fig”) (W Question 8. (20 points) Find the solution (assuming y < 2) of the differential equation gr‘sz—y); that satisﬁes the initial condition use A ll-er-‘ZL+C o= '%*‘C C“: 5-0 —-Lx14"- Question 9. (10 points for A; 5 points for B) Consider the curve generated by the parametric equations below, where 0 S t s 21:: A. Find the shape of the tangent line when t = n/4. (Recall that , 1r . n «i ""2 ,_-- I. C034 51114—7) d)’ 3': 2'0: H) :. _ l J; 425. ‘ -—3 («3/0 {1}: If) 3 (NH) yin/1‘) , ’ 4+, ' . ' h of the curve. 7 ‘ 7 o, ‘I l A d? - l - L ‘C J" n 2:)2‘f/Zg) [It ' 4;: 37/«2/{km/1‘) V ‘ _ J} : 2 . __ t L C f0 11 9a: /{)rm?/{)+(o:2/t) 4'6 4* Car/H Question 10. (10 points for A; 5 points for B) Consider the curve in polar coordinates r = 2cos (0/2). A. Find the slope of the tangenfline to this curve when 6 = 1t/2. (Recall that cos; = sin-145 = g.) A d), Jy x= maze) « z. " c} . dx é: (a a ﬁg male)- rim/e) X A43 )2; rah/9) 4,» d? +Hos(e) .6 3 d6 {la/9) #372): an'VLd‘VE ‘ 4‘s = -z/%)9"’/§)/: “/5 ‘ ”5 ' ’sze ‘7; ' a: s rmwz)+r/%)«es/%)=-§ ,2: i w 9 W” ‘ rl/2)c«rr%)~r/Z)m/w = .o—ngz).:= {z B. Write dawn _, . 573943 Question 11. (5 points each) Consider the vectors 3 = (1,2, —3) and 3 = (1,2,3). A. Determine, with justiﬁcation, whether the angle 0 between a. and 3' IS ' acute (0 s 6 < n/Z), right (0: 3/2), or obtuse (It/2 < 0 < 1:). Circle your response below. 01:57.: [+4'ﬁ 4 0 (o:[e)<0 z? Wile 0513“" ACUTE RIGHT OBTUSE E. Find the unit vector in the direction of Ii. aI =- J? (2)1423“ : JM 5x32 . [I 3’ E I 2 '3 / z 3 a a ['3 d, I Z wag-ww ,1 v M 2-2) : Z(é’/,é))-d/Z/3))+E[ San Question 12. (5 points each) A. Find an equation for the plane through the three points (1,0,0), (0,1,1), (1,1,-1). tel: HMO; 6V0; 1,1) “*4 R/1,1,—1) FI/ld 0 harms! recW F5; 631?}: (0,4,1)— (1,0,0) : <~1,1,17 ‘—5‘ “#3 ' P2= OR—on (s1,.1,-4)-<1,0,o7 ~= <0/1/17 a. PE: X PEQ / 2 I“? 15/ 2 Z/ll‘ll/‘41/;{-/// ’ . B. Write a vector equation for the line passing through the points (1,1,1) and (2,0,0). ' , C. Determine, with juStiﬁcation, whether the line in part B intersectsthe plane in part A or not.) Circle YES or NO below. [3.17: (“22"/’/>'<‘/’I/’I7: “Zﬁ‘lj‘l-f-O 7 ’5' [Java/“I *0 “9 P1056 0/111 kWh/val mam FT? 722 II Le PM?“ MW)‘ 43/77/010); ’7 Hm} an ‘fhe flame . C I, (1)07 (Z/olo>'_' {Lao}: , .<,oo>="2 f0 )2. (41) 01°>‘<IIOIO>); (he/’Z/wl; I I No YES . ...
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