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UNIVERSITY OF WATERLOO
MATH 137 Midterm 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. Ml37Midterm 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.) (ac4) 2 Mia xii/o
magazl‘w wuﬁﬁm t<o<<;2, / 6.
W i’sz39<+9~\':. 'XLCHUFL 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 oneto—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 1»_ 2; :1
wit/ma ‘ix’Lr'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 gWH'
NM 0m {Xvét'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. M137Midterm 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. M137Midterm 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‘“ $503 “i / w+l\e—\«Hi : CNN)“ 0""): 33539 (________,..__. ~ x X x
/ [Md—M4 ——>>.2 Q/OKaO'
CM #7» Ml37Midterm exam Page 6 of 10 Name: [5] 5. (a) Prove that if a function f has a derivative at a number a, then f is also continuous at a. [3] (b) Give an example of a function f that is continuous at some number a but does not have a
derivative at (1. Give a proof that your continuous function is not differentiable at a. :FCX);[’X\ A), gmlmmct ammo Hill Wait/um ‘i—‘m—w
Cg M“ P— A i ﬂgo'l‘ ’fN
4,541.0 ’l’\ M137Midterm exam Page 7 of 10 Name: 1
4] 6. (21) Use the limit deﬁnition of a derivative to ﬁnd the derivative of f = ——2 at :1: = a timiNmamtwﬂwtimiﬂﬁmaw' ) 2 . L . li~ 1A7“ L Ht?“ pt
‘iffwmt are “gaunt t whqu /: ~aav—lh a ’L— ’L
(a +% ) a
/ ‘3&'% > hﬂlLI:—F—‘2‘3 L 2.
(OH/TA) 0\ a a
I m  l
\/ W J (/3) "W WE
Gt
[4] (b) Find all numbers between 0 and 7r at which f = sin x and 9(55) 2 cos x have tangents with equal slopes. M137Midterm exam Page 8 of 10 [5] 7. Find the equation of the tangent line to the function f e‘102 at t e 01nt   M137Midterm exam Page 9 of 10 Name: karctanxforzc > 1
5 8. Lt = — (a) e {l—lncc for$<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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