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Unformatted text preview: Faculty of Mathematics
University of Waterloo Math 137
Term Test 2  Fall Term 2006 Time: 7:00 — 9:00 p.m. Date: November 20, 2006. AIDS: ‘PINK TIE’ CALCULATORS ONLY I
V Family Name: MM 5 ID. Number: Initials: Signature: Check the box next tolyour section: 1:! 001 B. J. Marshman (11:30 am.) [:1 008 S. Sivaloganathan (10:30 am.) 
1:! 002 O. Winkler (12:30 pm.) [:3 009 S. Gupta (2:30 pm.) E] 003 L. Marcoux (1:30 pm.) ‘I:l 010 S. Gupta (1:30 pm.) II] 004 N. Spronk (2:30 pm.) 1:] 011 C. Struthers (9:30 am.) I:] 005 B. Ingalls (1:30 pm.) El» 012 D. Harmsworth ( 8:30 am.)
1:} 006 D. Park ( 9:30 am.) 1:] 013 D. Wolczuk (8:30 am.) 1:] 007 Y—R. Liu (10:30 am.) 1 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, OCR), and
derivative rules (DSR, DPR, DQR, DCR), as well as the Intermediate
Value Theorem (IVT), and the Mean Value Theorem (MVT). 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. [4] 1. [4] ' MATH 137 TERM TEST #2 PAGE 2 dy a) Find dag if y = (m2 +1)3/2 + 62 —— marctanx. b) Find f’(m) if f(3:) = ln(a:3€‘“’). c) Find the tangent line to the curve cos(:1:y) = —a: + 1 at the point (1,7r/2). (1) Given that the points (2:,y) on a curve satisfy the equation xy = ym for :r > 0 and y > 0, use logarithmic differentiation to ﬁnd 31’ in terms of x and y, and hence Show that y’ is
indeterminate at the point (e, e). The graph of P(:c) shown at left represents the
proﬁt a certain manufacturer makes by selling as units of product. Sketch a graph of P’ (the
marginal proﬁt) on the given axes, justifying the
behaviour of your graph at x1, x2, and £63. TERM TEST #2 : Valuate each limit. Justify your method by stating what rules/ Theorems are used (e.g.,
fnit quotient rule LQR, 1’H6pital’s Rule L’HR, etc.)
Ina:
. 1.
(1) mgi sin(7ra:) (ii) iim at 6“”
CE—>OO (iii) 1im(1 + 3m)1/x m—>0 1
[3] b) Use the Squeeze Theorem to ﬁnd , Inn) 3:2 cos(;).
» ‘ tana: if x 74 0
[5] 0) Consider the function f = a: ’
1 if a: = 0 (i) Use the deﬁnition of continuity to ShOW that f is continuous at a: = 0. (ii) Use the deﬁnition of derivative to Show that f’ (0) = 0. I [4] l3] ‘ ﬂ, , 7'
’ MATH 137 TERM TEST #2 PAGE 4 7r 3. Consider the fuction f = 2:1:  cos(—2—:c). a) 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 0 with
an error of at most 0.01? Explain your reasoning. i 1
l
l
g
i c) Use £1(:r), the linear approximation of f at a: 2 1, to estimate f (0.9). d) Explain why f has an inverse f '1 on R. l e) Find f’1(2) and f‘1 (2) and hence state the equation of the tangent line to y = f'1(x) at i5
" MATH 137 TERM TEST #2 PAGE 5
. j Wm— CB 4. Consider the function f = :3 [2] a) Find liIJn f(a:) and lini [2] b) Find 1118+ f and lirgr f
I
[3] c) Find all critical numbers (points) of f. l
[2] (:1) Find the intervals on which f is increasing, and on which f is decreasing.
I [2] e) Show that f”($) > 0 for an > O, and f”(;c) < 0 for a: < O.
[4] f) Use the results of a) — e) to sketch the graph of y = f (:13), indicating any local extremes,
points of inﬂection, or asymptotes. :3 ea+b > 62. [1] g) Show that, for all a > 0, b > 0, b _
a ¢ MATH 137 TERM TEST #2 PAGE 6 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.) [2] a) If f(:I:) = ln(:v3e‘x), then the domain of f’(a:) is aé O}. [2] b) The function f = [1:3is diiferentiablie at 23 = 0. [2] c) If f’(:1:) > 0 and g’(a:) > 0 on an interval I, then y = f(a;)g(x) is increasing on I.
[3] d) The function f = cos(arcsin x) — m + 2, is a constant function. [3] e) If f is differentiable on IR and has exactly two real roots, then f’ also has
two real roots. 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 (i.e., f’(t) < g’(t) for every t Z 0), then if they start at the same point (i.e., f (0) = 9(0)), the second horse always leads the
(ﬁrst horse (i.e., f(t) < g(t) for every t > 0). ...
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