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MidtermW07 - solutions

# MidtermW07 - solutions - University of Waterloo MATH 119...

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Unformatted text preview: University of Waterloo MATH 119 — Calculus 2 for Engineering Midterm Examination '- er 2007 February 5, 2007. J Dura 'on: 7:00 — 8:30 pm. Family name (print): I.D. Number: Signature: Initials: Please indicate your section: [:1 001 D 002 I] 003 D. Harmsworth (ELEC) A-M. Allison (SOFT) A—M. Allison (NANO) No Aids Permitted Note: Your grade will be inﬂuenced by the clarity of your solutions. MARKS _uestion Marks Available Marks Awarded MATH 119 Midterm Examination Winter 2007 [MARKS] 1 [8] 1. Consider the function y(:v) = —. Calculate the second ﬁnite difference A2310 using :3 A2 x0, x0 + Am, and x0 + .2sz as nodes, and verify that for this function, (A502 —+ y”(:v0) as Air —> 0. SQ} U»? *‘R *ob‘c 0" ¥‘|(\-\\Qa AﬁiUU‘ceY. J. 1; i - _\_ / ‘Hu‘: T) A33“ @ i 1.313% 3L. \ - 1 k _\_ 10* N \ \ 16" an 1371)); ‘10 ‘ 1:351, 19" [>1 1a 1A1. 9‘ 1 (13be - .1 (1.) QDM‘L, Jr (at. ’v 351d (19* D13 U = f’M—W A3) 3L, (am DWWJLJ my») 3. = ‘13 ¥ 1°51 - 213 ~Lixobx+xf + 33MB ’r 3151 10 (1a AﬂCLa 113;) G) 3. 2 Ax 10(1‘3 buobmlbﬂ . was Ego - a W): - 1. (1.,» 5111324») M \$6 \$30 a 49—23 0.5 A1“? C). C.) <be1 1° I ‘ _, n ' 3- PM“? 3, Course, ‘3“‘35‘2, has \5(L\= 335’“ “A \3 ()0- :3 so w ce5u\3< 33 611.} . {9 X “‘3 y \ condwﬁoa '6 oniM- Page 2 of 7 ’3 ""9"“ N we} mun-nan» :Vfrlvgmzmmssxgzv » , MATH 119 Midterm Examination Winter 2007 [9] 2. Solve the difference equation Ayn+1 — Ayn — 4yn = 0, subject to the initial conditions yo = 1, y1 = 1. Find the value of y4. AW“ - ban -- Lian : o .5) 850*; "’ 903nm " 3‘3" = 0 1k ck“. av. is m 4'“, «,0 9 ‘\.e. Q'VBKMH): 0 3° wggpgm 9 ‘ w=éﬁ+§i—\i=%+i=w. CD (You. (0AA o.\5° Sp}; 3% Aimed: gird“ m (gunman; mink“ 3 we know an. 5.4, m *0 um; 33~\*33~ 5° 33: 13943”: S 93 = 399*3‘3» = ‘3 ‘3‘ = 233*35;= L“ > Page 3 of 7 MATH 119 Midterm Examination Winter 2007 [13] 3. Determine which of the following series converge and which diverge. Justify your conclusions. @2211 % Tl'ﬁs AiuerseSJ- Kl's ax €°5U125 wjh‘ leg < \ GD I b) 220:1 x/ﬁ This Converts“, b3 Hm, All-armhuj Series WU)“: (3) —_..7o 04 0—700 {7‘ 1 a") é‘< é; ¥or a“ n. c> 22:3 IDKI‘J‘S] This ACNUSZSA lab h Teal Xw Bivdamcc: 5km Hang : mm A 213/: : Ham; 0’; n-‘Ioo \‘V'JA @ ”[ka Canvases} “‘5 G» SQON‘W‘R. sales 00?er W “s e: m m: £34 0. Page 4 of 7 _.,-‘MATH 119 Midterm Examination Winter 2007 I. Cos(1(1) ' [8] 4. a) Find the 4th—order Taylor Polynomial for f(x) = 'chntered at \$0 = %- ‘H-Q: Cos Nix £0413: Cos V; : 0 Wm «M m»: warm/4: -1: v0? ”'“lCDS my PW”: ~T\a(os'“/;= O (Q) huh)“ W3 sin-me Pup/n: “353'“11: ﬁ3 t M (ii = 10 C05 Tot. 3;”) (‘M = 0 mm = mm mes/m givmiu-‘mﬁ g! rims-vi? i 4, q! ¥(~)('/J)(1—VJ)‘1 I H ‘md. CD Lzr know“) ‘Ofnhk gr Tank" “damp-it‘s, even I; “-1- Co‘m‘on-s we. Wang -1\(1—‘/;\+ fan/,1”. 6 i. (9 Use, Tat-3b.}: Ineqykkb 5m . .. b) Find an upper bound on the error involved if this is to be used as an approximation to f (x) on the interval [0,1]. \Ak. \nme. V960: 'ﬁgg‘“‘“1) 5° \‘PSYH‘E “5 0A mintervd. “can: \qui/Jxﬂg 15 ‘1, ”l w’ .—-’ U1 0“ mum\\ S \ 1; wt 00043 a» Anthea; \oounk’ we, can 0:150 ohxnm “WV \1'%\ é 3; an [010) so \Q(r0r\é %\'(§)5 ' @ Page 5 of 7 MATH 119 Midterm Examination Winter 2007 [12] 5. a) Find the general form for an nth-order Maclaurin Polynomial for the function f(t) = ln(1 + t). (That is, ﬁnd Pn,o(t).) rm: 3L (mi “oh 0 l'lth ill: ‘f‘(0l= ("D We: a. F‘loiw (Ht? We: @353 mm: .1 --> alum: 0+ t - e1 m... (-ii“*’(n-\)'.t" 3 E n! ® (95,” rm 75 0K = l K=\ K % but" “231‘. shout‘k b) Use your result from part a) to ﬁnd an approximation for /ln(1 + (1:2)dz, with be 0 swell an upper bound on the associated error (we’re not asking for any particular level of accuracy here, so you may use just one non—zero term if you wish). Note: you do not need Taylor’s Inequality here. 3 =7 X-xm’rlal: 1a-}j+p_§f_+-o Cl) 1 3 z) Iii ) V; ”(H-1a d1 = ( 3- 1H 1‘ -+ 3A1, l 0 j 1 a ” i O 3 5' 1 "/4 z 3 ~ 1 l -4 l 3 \O * 2| 0 G) i 3 Q/ )5 Q: (/9) will‘ lerroq-l L i ”g 2 \0 I ‘ V; ) \ \ he. “(“33 3.1 = -— 1 —— Joy 2“! 310 % ( . 2 +®¥W w" c) If you were to use the same strategy to approxunate ln(1 — :1: )dx, you would kw“ 0 need Taylor’s Inequality. Explain. in 566 Cox— ‘\’Lc Sci-es ww\9~ “0* 5% ml’rernw‘ﬁrg. . - '1 (1: Wu ReggiwiQ/Q-“ > O) 3 to m Page 6 of 7 ...
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