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Unformatted text preview: SﬁmPLE TERM TEST 2 (Sai FALL 2200:} Faculty of Mathematics
University of Waterloo Math 137
Term Test 2 — Fall Term 2006 Time: 7:00  9:00 pm. Date: November '20. 2006. AIDS: ‘PINK TIE’ CALCULATORS ONLY Family Name: _—.._____._._— Initials: ID. Number: _.____._____.__ Signature: Check the box next to your section: 001 B. J. Marshman (11:30 am.)
002 O. Winkler (12:30 pm.)
003 L. Marcoux (1:30 pm.)
004 N. Spronk (2:30 pm.) 005 B. Ingalls (1:30 pm.) 006 D. Park ( 9:30 am.) 007 YR. Liu (1030 am.) 008 S. Sivaloganathan (10:30 am.)
009 S. Gupta (2:30 pm.) 010 S. Gupta (1:30 pm.) 011 C. Struthers (9:30 am.) 012 D. Harmsworth ( 8:30 am.)
013 D. VVolczuk (8:30 am.) 0000000
[100000 Your answers must be stated in a clear and logical form in order to
receive full marks. Reference any theorems or rules used by their name, or the appropriate abbreviation. Useful abbreviations for this
test include LSR (limit sum rule), LPR (limit product rule), LQR (limit quotient rule), LCR (limit composite rule for basic functions).
the corresponding continuity rules (CSR, CPR, CQR, CCR). and
derivative rules (DSR, DPR., DQR, DCR), as well as the Intermediate
Value Theorem (IVT), and the Mean Value Theorem (lVIVT). Note: 1. Complete the information section
above, indicating your instructor’s
name by a checkmark in the appro
priate box. 2. Place your initials and ID No. at the
top right corner of each page. 3. Tear off the blank last page of the
test to use for rough work. You may
use the reverse of any page if you
need extra space. MATH 137 TERM TEST #2 PAGE 2 [4] 1. 3.) Find if y = (22 + I)“2 + 62 — marctanxt
LBXW ~0ncTamX ~ 11 l
[4] b) Find f’(z) if m) = ln($3e"). W"
C 3 — i]
[4] c) Find the tangent line to the curve cos(zy) = —z + 1 at the point (11 7r/2). Ix 13: 1; + (NV/73 (aci} [4] d) Given that the points on El. curve satisfy the equation .1” = y“ for I > 0 and y > 0.
use logarithmic differentiation to ﬁnd 3/ in terms of z and y, and hence Show that y’ is
indeterminate at. the point (e, e]. [ "QM‘ﬁ‘Zl/x The graph of Plx) shown at left represents the“
proﬁt a certain manufacturer makes by selling 2:
units of product. Sketch a graph of P’(I) (the
marginal proﬁt) on the given axes justifying the
behaviour of your graph at $1,122. and 3:3, MATH 137 TERM TEST #2 PAGE 3 [10} 2, [5] :1) Evaluate each limit. Justify your method by stating what rules/Theorems are used (e.g.‘,
limit quotient. rule LQR, l’Hépital‘s Rule L‘HR, etc.)
(i) 11m .1” 2—1 sm(1rn:) [‘ Vrr] (ii) lim mac—’2
I—‘m [0] (iii) lim(1 + 3x)!” 1—0 1 b) Use the Squeeze Theorem to ﬁnd 1m}J £2 cos(;).
E 0 1
tans: if I aé O
0) Consider the function f(x) = f
1 if :E = O (i) Use the deﬁnition of continuity to show that f is continuous 31. z = 0‘ (ii) Use the deﬁnition of derivative to show that f‘(0) = O. MATH 137 ——————————————______—_ l2] l3] TERM TEST #2 7f 3. Consider the fuction f(.'II) = 21 — cos(§:r)i PAGE 4 5:.) Using an appropriate theorem, Show that there is a. c in [0.1] such that f(c) = 0, b) What is the smallest number of steps of the Method of Bisection required to estimate c with an error of at most 0.01? Explain your reasoning, c) Use £1($), the linear approximation of f at m = 1. to estimate f(0i9). d] Explain why f has an inverse f‘1 on R. LEM: boggy“)
=" 593% rib431 e) Find f'](2) and f'1I(2) and hence state the equation of the tangent line to 1 = f‘l($) at x=2v H l 4. \ (x—z
,#
2 417/2 >1 IVIATH 137 TERM TEST #2 PAGE 5 4. Consider the function f(1:) : f—i
a: [2] a) Find lirn f(3:) and lim ﬁx), z—o+oo :c——oo [2] b) Find lim ﬂat) and lim f(:c). r—~0"‘ 1—0” [3] c) Find all critical numbers (points) of f.
['x = *1
['2] cl) Find the intervals on which f is increasing, and on which f is decreasing.
men. on x > \3
[2] B) Show that f”(m) > 0 for :c > D, and f”(:r) < O for a: < 0. Mcvo M 3" (1
[4] f) Use the results of a) — e) to sketch the graph of y = f(a:), indicating any local extremes.
points of inﬂection, or asymptotesi :15
X
ellH:
[1] g) Show that, for all a > 0‘ b > 0‘ 2 52. ab MATH 137 TERM TEST #2 PAGE 6 l2] [3] [3] [3] 5. Label each statement as TRUE or FALSE in the blank provided. (Use the space between questions to justify your answer. Guesses will not be graded.) a) If f(:c) = ln(x3e‘z). then the domain of f’(z) is {zlr aé 0}. b) The function f(:c) = x3 is differentiable at a: = 0, c) If f'(x) > 0 and g’(z] > 0 on an interval 1. then y = f(a:)g($) is increasing on 1. cl) __ The function f(:1:) = cos(arcsin ‘1') — V1 — 1:2 + 2, is a constant function. e) If f (x) is differentiable on R and has exactly two real roots, then f'(z) also has
two real roots. f) If the positions of two horses are given by differentiable functions f(t} and g(t) for t 2 0, and the second horse runs faster than the ﬁrst (ie, f’(t) < 9"(t) for every t 2 0)
then if they start at the same point (i.e., f(0) = 9(0)), the second horse always leads the
ﬁrst horse (Le... f(t) < g(t) for every t > 0). ...
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This note was uploaded on 01/25/2010 for the course MATH MATH137 taught by Professor Oancea during the Fall '08 term at Waterloo.
 Fall '08
 Oancea

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