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UNIVERSITY OF WATERLOO
MATH 137 Midterm Examination SOLMTION Â§ 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. Jeromeâ€™s S. Speziale 10 117 Instructions 1. If you are in Prof. Stastnaâ€™s Section 8, then you are writing the wrong exam. Go to Prof. Stastnaâ€™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
instructorâ€™s name and your tutorial number up above. . NO ELECTRONIC DEVICES other than your â€Pinkâ€” Tieâ€ 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
ï¬‚ow or rough work. . Show all your work required to obtain answers. . Your grade will be inï¬‚uenced by how clearly you exâ€” press your ideas and organize your solutions. 118 Calculus 1 7â€”9 p.m. Circle your INSTRUCTORâ€™S NAME and your TUTORIAL number. Ml37Midterm exam Page 2 of 10 N ame: [3] l. (a) On the diagram below, sketch the graph of the function y = 1x2 â€” 3m + 2, indicating all
intercepts. Give reasons for your sketch. 34W x3'3x4â€”9. =2 bun.) (ac4) 2 Mia xii/o
magazlâ€˜w wuï¬ï¬m t<o<<;2, / 6.
W iâ€™sz39<+9~\':. 'XLCHUFL Wk {01â€™ Â«.21 ~(x1â€”3x+.2â€˜/ Wk.â€œ 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 Liâ€˜ Wm} W 5 Â« Mal ,
W â€˜7 a / t qxw Â«+355 5Ls~9<+1 / â€™ {9100â€˜ 5Mva â€œVâ€˜OiÃ© l)~X$g,
/ â€™ â€œ>00 3;â€; ~x+~asiygÃ©e<+t / o 7M9 wgÃ©lï¬‚jlÃ©g.
a, Ml37Midâ€”term exam Page 3 of 10 Name: 2. Let f(x) = Vac2 + 1 fora: 6 [0,00). [2] (a) From the deï¬nition of a onetoâ€”one function, prove that f is onetoone 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 1Â»_ 2; :1
wit/ma â€˜ixâ€™Lr'l gwï¬‚ "'5â€˜ +1)? l [i] (c) Show that lim v 3:2 + 1 â€” a: = 0. (There is no need for an eâ€”based argument.) 3 WH/hm CW' W 1 â€˜2. râ€”Â«y ' L//
xâ€œ+(Â¥tb+lÂ¥)Wï¬‚â€œ+y x+â€˜x awakeâ€”m {LN MC V W+x gWH'
NM 0m {XvÃ©t'ob W Oil/WWW W W
Aim \iï¬‚1+l'â€â€™)6 3â€”. O â€œx4 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. M137Midterm exam Page 4 of 10 Name: 3. Let = 53: â€” 6. Using the formal 6â€”6 deï¬nition of limits, prove that ling f = 4. Your proof should be written so that it reï¬‚ects a full understanding of the limit concept. M137Midterm exam Page 5 of 10 Name: Tb
[V 4. (a) Evaluate ta: :8 , and justify your answer without making use of Lâ€™Hopitalâ€™s rule.
WL Â«We (;aM.><Â» â€œL, ! My (/
â€™32" cm x x
(arm/Vex ï¬lo/\X â€”â€”â€”'>l .Lâ€”â€”â€”>_L:J.: /
x W 96 Mo 0 \
W? cm M M Â«
WY...) iâ€{:â€™i MACâ€”goâ€™i/
X
[3] (b) Find Inn) 952/3 cos , and justify your method.
H x revth
$wlw({)31M03WW~;:Â§ Â« 1W
3 â€˜/ O 5., \ 93/5 â€˜/NW â€œ3/3"?â€ awkdm 0â€:70 :w â€œw \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 â€˜Â¢+"â€™fâ€™Â«â€˜>* Anvâ€˜
We Â«440 Wâ€˜ " I , and explain your reasoning. Mm 1Â«+I\"Â¥H W Wâ€ qLX/JWâ€”ï¬ 0*
W.,%m â€™x 96â€˜â€œ $503 â€œi / w+l\eâ€”\Â«Hi : CNN)â€œ 0""): 33539 (________,..__. ~ x X x
/ [Mdâ€”M4 â€”â€”>>.2 Q/OKaO'
CM #7Â» Ml37Midterm exam Page 6 of 10 Name: [5] 5. (a) Prove that if a function f has a derivative at a number a, then f is also continuous at a. [3] (b) Give an example of a function f that is continuous at some number a but does not have a
derivative at (1. Give a proof that your continuous function is not differentiable at a. :FCX);[â€™X\ A), gmlmmct ammo Hill Wait/um â€˜iâ€”â€˜mâ€”w
Cg Mâ€œ Pâ€” A i ï¬‚go'lâ€˜ â€™fN
4,541.0 â€™lâ€™\ M137Midterm exam Page 7 of 10 Name: 1
4] 6. (21) Use the limit deï¬nition of a derivative to ï¬nd the derivative of f = â€”â€”2 at :1: = a timiNmamtwï¬‚wtimiï¬‚ï¬maw' ) 2 . L . li~ 1A7â€œ L Ht?â€œ pt
â€˜iffwmt are â€œgaunt t whqu /: ~aavâ€”lh a â€™Lâ€” â€™L
(a +% ) a
/ â€˜3&'% > hï¬‚lLI:â€”Fâ€”â€˜2â€˜3 L 2.
(OH/TA) 0\ a a
I m  l
\/ W J (/3) "W WE
Gt
[4] (b) Find all numbers between 0 and 7r at which f = sin x and 9(55) 2 cos x have tangents with equal slopes. M137Midterm exam Page 8 of 10 [5] 7. Find the equation of the tangent line to the function f eâ€˜102 at t e 01nt   M137Midterm exam Page 9 of 10 Name: karctanxforzc > 1
5 8. Lt = â€” (a) e {lâ€”lncc for$<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 :Zfï¬‚) â€˜1â€˜ 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: â€” 1)H(â€”:I3). Determine whether f satisï¬es the conditions of the Intermediate Value Theorem over the in
terval [â€”1, 1], and justify your answer. 84W lbwâ€”t 1H):ï¬â€˜â€â€œ4 (XHZl $40 Â«90 â€™ H
M lï¬>=WV*l=â€”l> A0" \
\/NCDâ€˜Tâ€˜ ('msâ€˜CiW (m m ["213 ., m V M m W+â€˜ W2 [1] (0) Does the function f have a solution to the equation f = O for some a: in the interval [~1,
Explain brieï¬‚y. 1}? BQAQHH) W) ï¬t MAM/d '\ M137Midâ€”term exam â€™ Page 10 of 10 Name:
â€”â€”â€”â€”_â€”_________â€”__ J BLANK PAGE ...
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