MidTermsolutions_Fall_09

MidTermsolutions_Fall_09 - PRINT your last name Signature...

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Unformatted text preview: PRINT your last name Signature ID# UNIVERSITY OF WATERLOO MATH 137 Mid-term Examination SOLMTION § Initials Monday, October 26, 2009 Instructor Section Tutorials J. Nissen 1 101 102 F. Zorzitto 2 103 104 A. Nica 3 105 106 J.Nissen 4 115 119 M. Eden 5 109 1 10 M. Eden 6 1 l 1 1 12 B .D. Park 7 1 13 1 14 C. Struthers 9 St. Jerome’s S. Speziale 10 117 Instructions 1. If you are in Prof. Stastna’s Section 8, then you are writing the wrong exam. Go to Prof. Stastna’s exam in RCH 204. . Put your name, signature, and ID number at the top of this page. To prevent loss of your exam, circle your instructor’s name and your tutorial number up above. . NO ELECTRONIC DEVICES other than your ”Pink— Tie” Faculty Approved calculators are allowed at your examination desk. . Answer the questions in the spaces provided, using the backs of pages, or the blank page at the end, for over- flow or rough work. . Show all your work required to obtain answers. . Your grade will be influenced by how clearly you ex— press your ideas and organize your solutions. 118 Calculus 1 7—9 p.m. Circle your INSTRUCTOR’S NAME and your TUTORIAL number. Ml37Mid-term exam Page 2 of 10 N ame: [3] l. (a) On the diagram below, sketch the graph of the function y = 1x2 — 3m + 2|, indicating all intercepts. Give reasons for your sketch. 34W x3'3x4—9. =2 bun.) (ac-4) 2 Mia xii/o magazl‘w wufifim t<o<<;2, / 6. W i’sz39<+9~\':. 'XL-CHUFL Wk {01’ «.21 ~(x1—3x+.2‘/ Wk.“ tux <1. [5] (b) On the diagram below, sketch the set of points (2:, y) in the plane that satisfy [ml + lyl g 1. Give reasons for your sketch. i201, W J W Li‘ Wm} W 5 « Mal , W ‘7 a / t qxw «+355 5Ls~9<+1 / ’ {9100‘ 5Mva “V‘Oié l)~X-|$g, / ’ “>00 3;”; ~x+~asiygée<+t / o 7M9 wgélfljlég. a, Ml37Mid—term exam Page 3 of 10 Name: 2. Let f(x) = Vac2 + 1 fora: 6 [0,00). [2] (a) From the definition of a one-to—one function, prove that f is one-to-one on the interval [0, 00). [3] (b) Find a formula for the inverse function of f (x) = \/ x2 + 1 as given on the interval [0, 00). State the domain of the inverse function. x 1»_ 2; :1 wit/ma ‘ix’L-r'l gwfl "'5‘ +1)? l [i] (c) Show that lim v 3:2 + 1 — a: = 0. (There is no need for an e—based argument.) 3 WH/hm CW' W 1 ‘2. r—«y ' L// x“+(-¥tb+l-¥)Wfl“+y x+‘-x awake—m {LN MC V W+x gW-H' NM 0m {Xv-é-t'ob W Oil/WWW W W Aim \ifl1+l'”’)6 3—. O “x4 W [ii] (d) Use the information obtained above to sketch the graph of f over [0, 00), and of its inverse & function f *1 together on the same diagram below. Indicate any asymptotes that may be present. [h 6. M137Mid-term exam Page 4 of 10 Name: 3. Let = 53: — 6. Using the formal 6—6 definition of limits, prove that ling f = 4. Your proof should be written so that it reflects a full understanding of the limit concept. M137Mid-term exam Page 5 of 10 Name: Tb [V 4. (a) Evaluate ta: :8 , and justify your answer without making use of L’Hopital’s rule. WL «We -(;aM.><» “L, ! My (/ ’32" cm x x (arm/Vex filo/\X ———'>l .L———>_L:J.: / x W 96 Mo 0 \ W? cm M M « WY...) i”{:’i MAC—go’i/ X [3] (b) Find Inn) 952/3 cos , and justify your method. H x revth $wlw({-)31M03WW~;:§ « 1W 3 ‘/ O 5., \ 93/5 ‘/NW “3/3"?” awkdm 0”:70 :w “w \ao \/ Ba W wwai Wmaw W /e> W (iii 1 W,W xii/30w ~>zO 0mm who, 0, [3] (C) Find hm w aim/x [amaze M MW' 4(+|>O W ‘¢+"’f’«‘>* Anv‘ We «440 W‘ " I , and explain your reasoning. Mm 1«+I\"¥H W W” qLX/JW—fi 0* W.,%m ’x 96‘“ $503 “i / w+l\e—\«Hi : CNN)“ 0""): 335-39- (________,..__. ~ x X x / [Md—M4 ——>>.2 Q/OKaO' CM #7» Ml37Mid-term exam Page 6 of 10 Name: [5] 5. (a) Prove that if a function f has a derivative at a number a, then f is also continuous at a. [3] (b) Give an example of a function f that is continuous at some number a but does not have a derivative at (1. Give a proof that your continuous function is not differentiable at a. :FCX);[’X\ A), gmlmmct ammo Hill Wait/um ‘i—‘m—w Cg M“ P— A -i flgo'l‘ ’fN 4,541.0 ’l’\ M137Mid-term exam Page 7 of 10 Name: 1 4] 6. (21) Use the limit definition of a derivative to find the derivative of f = ——2 at :1: = a timiNmamtwflwtimiflfimaw' ) 2 . L . li~ 1A7“ L Ht?“ pt ‘iffwmt are “gaunt t whqu /: ~aav—lh a ’L— ’L (a +% ) a / ‘3&'% > hfllLI-:—F—‘2‘3 L 2. (OH/TA) 0\ a a I m - l \/ W J (/3) "W WE Gt [4] (b) Find all numbers between 0 and 7r at which f = sin x and 9(55) 2 cos x have tangents with equal slopes. M137Mid-term exam Page 8 of 10 [5] 7. Find the equation of the tangent line to the function f e‘102 at t e 01nt | | M137Mid-term exam Page 9 of 10 Name: karctanxforzc > 1 5 8. Lt = — (a) e {l—lncc for$<1 Find the value of the constant k that makes f be a continuous function for all a: in (0, 00). On the diagram below sketch the graph of f for the value of k that makes it continuous, and indicate all asymptotes that arise. / Wk, M (x) : ~ {rho :Zffl) ‘1‘ o< l T l . . . . . 1 if a: Z 0 . [2] (b) Recall that the Heav1s1de function H 18 given by H = 0 .f < O . Now con51der 1 a: = (:1: + + (a: — 1)H(—:I3). Determine whether f satisfies the conditions of the Intermediate Value Theorem over the in- terval [—1, 1], and justify your answer. 84W lbw—t 1H):fi‘”“4 (XHZl $40 «90 ’ H M lfi>=WV*l=—l> A0" -\ \/NCD‘T‘ ('ms‘CiW (m m- ["213 ., m V M m W+‘ W2 [1] (0) Does the function f have a solution to the equation f = O for some a: in the interval [~1, Explain briefly. 1}? BQAQHH) W) fit MAM/d '\ M137Mid—term exam ’ Page 10 of 10 Name: ————_—_________—__ J BLANK PAGE ...
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MidTermsolutions_Fall_09 - PRINT your last name Signature...

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