M138-MT-1111_solns

M138-MT-1111_solns - Last name (Print): First name (Print):...

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Unformatted text preview: Last name (Print): First name (Print): UW Student ID Number: Your course section number (see below): University of Waterloo Midterm Examination Math 138 (Calculus II ) Instructor: See table below Date: Monday, February 14, 2011 Term: 1111 Number of exam pages: 10 (including cover page) Additional material allowed: “Pink tie" calculators only H W“ Section: See table below Time: 7:00 pm. Duration of exam: 2 hours Exam type: Closed Book Circle your instructor’s name and section number Instructor Section Y. Cheng 001 S. Speziale 002 S. Speziale 003 D. Park 004 R. André 005 R. Andre’ 006 B. Charbonneau 008 A. Beltaos (P. Smith) 009 Instructions 1. Write your name and ID number at the top of this page. Please circle your instructor’s name and your section number up above. 2. Answer the questions in the spaces provided, using the backs of pages for overflow or rough work. 3. Show all your work required to obtain your answers. 4. Only “pink tie” calculators will be allowed for the test. 3 / 8 4 /8 5 / 8 6 / 8 7 / 5 Total / 55 Math 138 Midterm Exam, Winter 2011 Page 2 of 10 Name: E [3] 1. Using the Fundamental Theorem of Calculus, find a. function f and a number a such that I 6+/I$£St—)dt=2\/:E for allz>0. fkéfi \lfi : Pom ~76) {'6 £21) 2%X2 or, / ' X 3 ,3an # a a (aw: 5m.ng 3237721: fag/K (/3: 5 far) [15] 2. Solve tho following indefinite integrals. (For 3 points ouch) (a) /m4(1+$5)12dr :2: 99 . ‘r 0),“: s‘xL’LQ/JL : / +1, 7 V“ ELL = 134’“ 5. grl/wu _- 453m 5’ 5 I3 1:43. : “’1’”(l4~‘j~:_j Math 138 Midterm Exam, Winter 2011 Page 3 of 10 Name: C sf’yt x) V: i» /CL(A : V2’1'\_ (b) /sin(lnx)d:c _: k 7‘ J, z 15"” (“a”) '/%’@'7’*-1‘1-”~ u : cmé’fim) “AF: in gala : —$|M@X)d,L Vrk :xsm (flaw—[104641470 +m3x ’7“ 2x: : ASMCOML) —>Lw:{~47mx) H _.o— T f k (c) /tanxsec4xd:c 33' ¥ “X 3/zzvmx. $4927L9¢17LC13L : 7L. (1 {ring/fix) $941224 3K 2»; fihmd“ OLA 5 42(1de 5 +a2)&m :— :‘fi‘IJ-X/j kg 2/ “\- Math 138 Midterm Exam, Winter 2011 Page 4 of 10 Name: ,, x A —.—~ .a— dx , m ‘7; N 1” (d)/‘Mm ————~ J, ~1’ .—7_7\..-'/~:~ .— “FM +2“) .- (xlav-a: 2C4“ +5 4- ) 4“! 4 7‘ [/4 “am al.9- 474 Z r ,— w / 5 a“ \L / w— ; 4- " C144} \Wfi? glffl 8": : J (A d“: mils; W :2, L ‘ l ; w :}/@%m&:/cn& w¢75%‘* %W @& ’39-4—(‘1 WT’Y‘L')‘ a : Mcywhfi) +C/ 7// (fl / )Q A 7L +‘ 6‘ (ix. { 1 1+» '2.— 1‘+ L in: V1 éu’a- mL/SL :- l Jaclyn. 4» /VLV; §6b7é~&/9\ VU, Vangu- '2: M Q Wigflb 9— L, 4,“: 2mm d/u: xix :A&Ml+ swfiflri4$”9*9+g 2/ S<C ‘5L : (II-#29 +*C "7/ [8] Math 138 Midterm Exam, Winter 2011 Page 5 of 10 Name: 3‘ Evaluate, when possible, the following integrals. If an integral is not equal to a number explain why. (For 4 points each) B 4,. d3\ 2, Q ‘ 1) L e-w 3 I fly“ —’ )i— .I» w é-wx’ xq, I {»—7>+ 57“” t l .cvév/vw “I +‘._._/ \ a {W} 1?“..— 4—3wxf: -t {’7” \3fl7’ é-L V I, ‘w goo % (b) /m 2? 1dr : «g/w #1 ix .0 F + faoc 0 VaZ‘DK’Ff __ é ' :- K I ell/t r. “809’ %<:L4;:l‘dxk, 14%700 O and +goo {ecu ; gw [WWQ v w)” {J‘iou h T W” T Math 138 Midterm Exam, Winter 201] Page 6 of 10 Name: ‘1“ [8] 4. The ellipse given by the equation (T-ZV y2_ g_1 4 + is the ellipse + § 2 1 shifted to the right by 2 units. (a) Use the discs method to find the volume of the solid of revolution obtained by rotating this ellipse about the x—axis. Clearly justify pts) L— 9/ V: each step of your answer. (5 — 1’ \ —-» VIA/-17 7" 13 :2; - 51:!— J :3 a 1+ 4 a 4‘ Make“ ” 1' Jf/Tgrw 2g :: (’ ’ ll 9 IL— 0 3 ~~ r» O , - —1 Joan—2i) :"z’H/Lv—E: ' (“0* 713 \ 71'] , s (b) Set up, but do not solve, the integral which uses the cylindrical shells method to calculate the volume of the solid of revolution obtained by rotating this ellipse about the y~a.xis. (3 pts) 50L): 3 [/ ~®m3lflp L)»— MW: Math 138 Midterm Exam, Winter 2011 Page 7 of 10 Name: M [8] 5. Solve the following initial value problems. (4 pts each) d3: 1 __ — = > (£1)de (1+t)2, x(0) 07 t 0 f7; a? l+t " r + :2; a —> I F lf't k—I Dale): 0 c) O 1 Z: ’; +C_' :3 C : a - —2, >1: ff;+9.. an I- .— 711+; d (b) d—Zzwry. y(0):2 9/ (1)? ~ ‘1. 6W4€,W\ ?($\)= _, adj”: 7K ' ’ ffo)d“ f—Idsp _3k ‘/ “q : "a, 2‘ .fl' : we, [8] Math 138 Midterm Exam, Winter 2011 Page 8 of 10 Name: E 6. Suppose that y(t) is the population of a species at time t, and satisfies the equation @ = woo e y) aft 100 (a) For which value of 1/ does the population grow the fastest? (3 pts) Mx W 7 SM fl ,‘5 MA gr; )4 ' ‘mAflJZ ch‘ 36W 1;.) u _?L—yL-IU’U?_ /LS‘D 4 7 M. a {Sap-LII,“ /av clerk” mom/i v; MAM 0047947” ML Wax (fa/aw 4+ m Pf +11; M)? 0 M4 /UD SV 33%- IS WAY/Wag flJaJWTZP :50 d 100 — (b) Solve the differential equation i = explicitly producing an expression for the population y(t) at time (5 pts) ‘1’ A a, , at , I 2‘ : luo l3) :— J» — w”— MAN’Q at fl/‘o > auger?) hm % o 5.21; 1’ :— H: fl( '“'2) ’m IDD . ~’-—- 3" £4.25; 3 l=<3r1¥la+w°fl~ ~ m" — r 3:; 9/!0‘3 \a a ‘2’ 9 fl 7L” ) lOU r _.L~ ‘ if; a + mew; - fi/mg-vgj' 4-CJ /m) _ ,1- Vamp/a) +c. ' ’00 /UD’2/ ‘ 4,. )1». _ + +—C,- )3. \zZL-iL—C/ NO IUD"? To? W'b :> .2;- : “dill/{— 25 "m" ’- £20.47 mm 9 L __ CL ’1. ' 9% ;au ,;,:,2,+ 27> LCD—«VA kl b” "" ’ ’2? 7 __ /50 r. C 6R (é C2 *6 9’ 2' Math 138 Midterm Exam1 Winter 2011 Page 9 of 10 Name: [5] 7. A sphere with radius 1 metre has temperature 0° C. The temperature T(r) at a distance r from the center of the sphere satisfies the equation £11102 drgv‘z Find an expression for the temperature T(r) at a distance r from the center of the sphere a7— dr M: g) m” :2: " M C” C/(L pf. J, C //7.J::Jr\r :3 “L C :I, + C. 2.69 w»: /“ 77/13 : ~23“ * C :5 ' J'L flar" e wzwwfis a: 7A7”) 72A): -251 #2073 J1— Total = 55 pts ...
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M138-MT-1111_solns - Last name (Print): First name (Print):...

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