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Unformatted text preview: C. UNIVERSITY OF WATERLOO FINAL EXAMINATION
FALL TERM 2005 Student Name (Print Legibly) Signature Student ID Number COURSE NUMBER COURSE TITLE Calculus 1 for Honours Mathematics COURSE SECTION(s) 001 002 003 004 005 006 007
008 010 011 012 013 014 DATE OF EXAM Monday, December 19, 2005
TIME PERIOD 9:00 am.  11:30 am. DURATION OF EXAM 2.5 hours
NUMBER OF EXAM PAGES INSTRUCTORS (please indicate your section) 01 C. B. Chua (11:30 am.) [I 08 J. Verstraete (11:30 am.)
02 D. McKinnon (12:30 p.m.) El 10 D. Wolczuk (1:30 p.m.)
03 A. Nayak (1:30 p.m.) D 11 C. Struthers (10:30 am.) U E! U MATH 137 04 X. Liu (2:30 p.m.) 12 D. Wolczuk (8:30 am.) 05 C. Small (1:30 p.m.) 13 P. Balka (8:30 am.)
06 R. Khandekar (9:30 am.) 14 A. Chau (2:30 p.m.) 07 B. Marshman (10:30 am.)
EXAM TYPE Closed Book ADDITIONAL MATERIALS ALLOWED ‘Pink Tie’ Calculators only DDDDDDD Notes: Marking Scheme: 1. Fill in your name, ID number
and sign the paper. 2. Answer all questions in the
space provided. Continue on
the back of the preceding page
if necessary. Show ALL your
work. 3. Check that the examination
has 10 pages. 4. Your grade will be in
fluenced by how clearly
you express your' ideas,
and how well you organize
your solutions. o. MATH 137 — Final Exam. Fall Term 2005 Page 2 of 10 [3] [4] [7] 1. a) Given ﬂan) = cos(a:2 + 1) + {ECx, ﬁnd f’(0). . d
b) Given y = $51” , use logarithmic differentiation to ﬁnd 3%. c) Find the equation of the tangent line at (2,0) to the curve deﬁned implicitly by
y2+$3+tany=8 (1) Evaluate each limit, or show that it does not exist. , , 2$2+m—3
<1) Bit—$1— .. . Irv—2
(n);1_13 z—2 (iii) Inn) 11:2 ln(m2) MATH 137 — Final Exam. [2] [4] [3] [4] 2. (1) [(23 +1)1/3 dx 4 1
(11) 1 556% dx Fall Term 2005 a) Evaluate each integral or ﬁnd the general antiderivative. Page 3 of 10 b) If g(x)= f: v 1 — t2 dt, use a suitable theorem to show that g(a;) is an increasing
function on the interval (— 1 ,.1) MATH 137  Final Exam. Fall Term 2005 Page 4 of 10 1132 [2] 3. a) Evaluate 11m 2;. Justify your answer.
{3* DO [1] b) Determine whether the function f (x) = 3326‘“ is even, odd, or neither. [6] c) Find the intervals on which f is increasing, and on which f is decreasing, and the
intervals on which y = f (m) is concave up, and concave down. [3] d) Sketch a graph of y = f (as), indicating any intercepts, extremes, and asymptotes. MATH 137  Final Exam. Fall Term 2005 Page 5 of 10 An offshore oil well is located at W, 6 km from the nearest point A on a
straight shoreline. Oil is to be piped from
W to a reﬁnery at B, 8 km from A, via
an underwater pipeline from W to P, and
then to B by an overland pipe along the
shoreline. [2] (a) If the cost of laying pipe is $M per km underwater, and $0. 5 M overland (where M
is a constant), show that the total cost C (cc) in dollars 1s C(x) = M(\/x2 + 36 +4 — E), where x = AP. Specify the interval of values of a; relevant to this situation. [6] (b) Determine the location of P which minimizes the total cost C(m). Justify your
method. MATH 137 — Final Exam. Fall Term 2005 Page 6 of 10 [10] 5. Find the area A bounded by the curves y = em, y = sin(7r:z7), a: = 0 and a: = 2. Your
answer must include an appropriately labelled sketch of the region, and a brief explanation
of how the deﬁnite integral for A is derived, starting from a regular partition of [0, 2] and
suitably chosen approximating rectangles. MATH 137  Final Exam. Fall Term 2005 Page 7 of 10 1 [3] 6. a) Sketch the graphs of y = arctana: and y = F on the given axes, showing any asymp
totes. '5
1
[1] b) Consider the function f (at) = arctanrz: — ——2. Explain how you know that there is a:
only one point c in the domain of f such that f (c) = 0. [HINTz Look at your sketch] [2] c) Use a suitable theorem to show that 1 < c < x/i
[3] (1) Find the linear (tangent line) approximation £(m) to f (:r) at 1 (Le, £1(m)).
[3] On the graph, show how Newton’s method would use :31 = 1 and £1013) to compute
the next iterate $2. Then use the formula
for Newton’s method to calculate x2. MATH 137  Final Exam. Fall Term 2005 Page 8 of 10 [3] 7. a) (i) State the Mean Value Theorem. Be sure to include all the hypotheses. [4] (ii) Use the Mean Value Theorem to prove that if f is differentiable on an interval I
and f’ (3:) > 0 on I, then f is increasing on I. [4] b) (i) Draw a sketch on the given axes which illustrates the Right Riemann Sum R4
(i.e., 4 subintervals) for f(x) = a: + 1 on the interval [0,4]. Will R4 give an 4
underestimate or an overestimate of / f(m) dm?
0 ‘5 ‘X 4
(ii) How does the Mid—point Rule M4 compare to / f (2:) dx?
0 MATH 137  Final Exam. Fall Term 2005 Page 9 of 10 8. Determine whether each of the following statements is true or false. Justify your choices
by either citing a theorem, providing reasons why the statement is true, or providing a
counter example that shows why it is false. Answers with no explanations will receive no marks.
[2] a) If f is continuous at a: = a, then f is differentiable at :1: = a.
[2] b) If f is continuous on [(1, b] and f has an absolute maximum at some point c in (a, b), then f’ (c) must exist and equal zero. [2] c) If f”(x) exists and y = f (x) is concave upwards on R, then y = 6“”) is concave
upwards on R. [2] (1) There exist differentiable functions f (w) and g(:1:) satisfying f(:13) + g(:13) = a; for all
:1: E R, and f’(0) = 1 = g’(0). sin(:r2)
[2] e) The function f(:r) = x for 93 7f 0 has f’(0) = 1.
0 for a: = 0
[2] f) The curve segments y = e” for 0 g x S 1 and y = Ina: for 1 g x S e have the same length. [2] g) Suppose that F’ (x) S G’(.z) for all z E R. Then F(a:) S G(z) for all a: E R. MATH 137 — Final Exam. Fall Term 2005 Page 10 0f 10 For Rough Work Only... (you may tear this page off and discard when the exam is over) ...
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