MATH-137-1049-Midterm_exam

MATH-137-1049-Midterm_exam - MATH 137 Print your last name...

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Unformatted text preview: MATH 137 Print your last name Initials 1 Signature w ID#_‘— UNIVERSITY OF WATERLOO WATERLOO ONTARIO Monday, October 25, 2004 Instructions Midterm Examination Calculus 1 7—9 p.m. Circle your instructor’s name and section number Instructor J. Verstraete S. New P. Balka V K.G. Hare l B. Marshman 'F. zdrz’iua J. Munroe Section \laUl-PUJNH Instructor ' Section B.D. Park 8 N. T heriault 9 S. Campbell 10 C. Struthers 1 l P. Balka 12 S. New 13 X2. Liu 14 1. Put your name, signature, 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, or the blank page at the end, for overflow or rough work. 3. Only the SHARP EL—531W Faculty Approved calculators are allowed. 4. Show all your work required to obtain answers. 5. Your grade will be influenced by how clearly you express your ideas and organize your solu- tions. , M137 Midterm Exam Page 2 of 8 Name: \ Pm {at} 1. Using the known graphs of basic functions, sketch the graphs of the functions below on the diagrams provided. You can do your rough work on the left, or on backs of pages. Clearly indicate the domain of each function, its intercepts, and asymptotes (if any). [3] (a) y = I222 — 32: + 2| *5: \tm-fiM—x y N w» a \ ‘5=\’*1'?>’Y~*7‘\ A ‘ i {ol‘n} we a ‘ (Web . M137 Midterm Exam Page 3 of 8 Name: » [4] 2. (a) Solve: cos(2:r) = 00833 when 0 3 a: 3 7r. Q55" cosh t0 = cman WI) " someosm—s‘mmemm e“- emsvx — sm‘AL ,W hwamn,aW1MAb-thu C0319 T S\ n19 = = CU? AL - 0- LoELHQ filfi‘E 1“- \—-v to? 8 y 2. Ce? )L-\ + LOQW’K v cos/mm) ’2 QBSZ AL u Log/t —\ = O cues. l‘k = \ t M ABEL-13 a. 4 [4] (b) Solve the inequality: ln(x + 1) — Inst 2 1. {Eur lawman) Singm+h—«QA/x>/\ M‘H > 0 HAW 7 .‘./~A>-'\ Sm< 7:? /\ I‘L-l- 45m” Qn/‘L/ Chm 7:» >0 ‘ + '3' 9" ’Y‘ "flzglmrr—o Ml, M ‘ (st-H 2 am M -tie. [4] (c) Find all points (3:, y) in the plane that satisfy the inequality Iy — ml 3 cc. Sketch our solution set on the diagram below. ’ , < l’ifggg'm away \7 «NH I “M0 Wb Ix M 3m“: nausea? 04‘ ‘ \ 9W yer”) U f~’¥ é, M “j a M137 Midterm Exam Page 4 of 8 Name: [2] 3. (a) Give the precise definition of what it means to say that a sequence has a limit 19. wkgf‘xmmes gig” Mm m 9 0.32 mm W (5% W A 276 We u‘xs‘w‘s K ‘Smh 4500‘}: her a“ 7/ “wk \M—pka. [6] (b) Using the precise definition of limit ofa sequence prove that lim n+2 = n—m 2n + 1 2 “an” M5: \«M’vx‘fl < i am i .. ' ’nfi-‘l __L Wham. i m“ " '2.\ <i =’Y\+2 L 7m Ztm+23-L2n®\ £2 ‘3 \>= -% "242nm \ 2n+A~2n—\\ \<-—-"€.‘7— C 2(sz (a A} 0 -' a . gm ww K \ Zam“\ 4 CL 4H7. \ *— ~ L 42. \ mu 2\ AWN—2. "' —_ > E. .3) or \Aqu wP\<<&_// GED‘ 4% g. -r > E ,1) '3». 0k > “€41 M137 Midterm Exam Page 5 of 8 Name: 4 [4] 4. (a) Evaluate lim V712 + n — n using any suitable and valid method. 71-400 1 4W,an .. n3(\h’bk1\ Mn (b ‘ 1 “ER (Wm 'Vh :. m- 4 ‘3 “hr-0o fi+flmmt \+\ LCR (W-Vfi *"Y\ 7 1., ___\_ '5 \\M w W V K “W m w __ \H‘fi l! ¥ “"500 U-‘Sa’: +\ W A “m \ _0 / “Ft—5m "n “" xx/ [4] (b) Evaluate 1i1%x2/3 sin using any suitable and valid method. 1" a l . t .. m use know s‘mtfh X‘s mum bk. swim vs om ®$m\\o:\‘\r3 5?- \ . ‘ l . » . " v . a ‘ ‘ _\ é “0’ 6 Mb ‘k’ow ebalflo‘e To MUM w‘wwd W 4 2 ' gm“ 35‘? “‘S 3‘9“- M cm WWW “‘" “‘W‘é‘“ m “\QW ‘ 3 3 '1, M “Ikh é smU/Q "/1 5:- /‘/~ '5 W W 31" \lM ‘ '36- _O Um 7/2: _ /X~§0 “bfib M e [4] (c) Find a so that “lim —1— exists. Then find the resulting limit. Aim/MS D) m wQM’ mm «b mgwmr' Lt [5] C1) A%%W. “nth \5 4m =3) Mac Mm <\o mm. M137 Midterm Exam Page 6 of 8 Name: I \ 5. Let be the sequence defined recursively by the rule 1:1:2, $n+1=v3$n+10 forn=1,2,3,... (a) Use mathematical induction to show that the sequence is increasing and bounded above by 10. Wm+ Ac: prove '. Alm< Non-H 4m @bm amt; em mm App: '2 mm“: {lactwo :4” .‘.)&.“&>M is W “2,4144% g) PYCNQ QM“ \1 = \g+\ V [4] [2] glib-Nb 4 3MH\-\v\g L [*0 r) r 2 m 430»: Ks < lawns < EB < «256 < \Q \/ . m 3. $\r\m— 13¢ “3 WM (wuéwolb Ants) 4 "my; 4 \b Iii/z f. 13:; in {mm 6m KM m as) maiwade madam Mmé/flcmflgw //QED (b) Explain why the above sequence has a limit. Then proceed to find its limit. I L“) TM above. sechwm has a, \imK-‘r barman H- E o 5‘8de) mono-tom increasing Whmcxt % kam converges. l'm __ \im W - {am AL“ “" chfim 43wm+lo "L. L = 53mm L :t «'2 We m Serbwmm 4):» L2 :— 3L+m mmlom inmamns g M‘ = Z 0'31:- \(3=O m \xm‘ec ~03 5/ (L—SBQL+1\=Q / A“ I. L: S F"?— (c) Does the sequence {yn} defined recursively by y1 = 1 and yn+1 = 1 —- yn have a limit? Explain briefly. 25‘ = 3%“: Mg“ m «\N. exhumed a.“ dtMSW-V hum a \‘imkl an H bounds) K“ = O mullahs been and ‘éciHl/x 99‘ bellman 0‘2 \, ,L gafgmceg w/ (sums dun-Jr hm “to b. MOW. M137 Midterm Exam Page 7 of 8 Name: \ [2] 6. (a) Define what it means for a function f to be one-to-one on an interval I . < AVA is \‘-\ on an Miami 1 W? m. tom: wig firm m: Ml [at on a. e: m Poms“ i m H _ 2 Hv‘tmm 3*” l r . 2 [2] (b) Using the definition of a one-to—one function, show that the function f = 1 + ex is one-to-one on the interval of all real numbers. MC V 2; g EllisTl/xt ?3\£»(,I>Q‘ 15 am 3m cm. £94; KY]: KQfi mg L 9. at a. , g . e ts. Q ma mi 0M. mumsmg I Her H? (3 Qumhm 3Q. \~.\ Z'r’Lef = 2&6 mm r=s/ H’Zef = H1 @ \ / .3 {Lth \s \"\ 2e“ = 2&9 ‘ Cr 7* Q? J (L [3] (c) If f is the funétion of part (b) above find a formula for the inverse function f '1. ‘3 = “*3 = U- 6* W ___2 we“) 6‘5 = 2% " ‘ +6. / -M = Sbn a ~ lbn m MKHQQ=Z [5 L2 M} L» u ’2. ‘ , 7:41 H’e‘i) -~3{- , :‘ &‘Qy§: ((1) Give the domain and range of the function f in part (b) above, and then give the domain [3] and range of f ‘1. Sketch the graphs of functions f and f '1. You may sketch both f and f “1 on the same diagram below. when @:{Q 4 342% y 3191‘???) 910‘ DOW] 3 24'“ g - ’ M<7g m>0 =xf—Qm M3 M?! 4:12! as: o M137 Midterm Exam BLANK PAGE Sm». Lug“ “(‘m {6/2 {ill 0 W2 {3/2. Jr: 6/2 Fin/2- ‘ \ {1/1 (“f/2. {E Page 8 of 8 Name: \ab ...
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MATH-137-1049-Midterm_exam - MATH 137 Print your last name...

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