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MATH 140 Dec 2007 - VERSION 1 MCCILL UNIVERSITY FACULTY OF...

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Unformatted text preview: VERSION 1 MCCILL UNIVERSITY FACULTY OF SCIENCE FINAL EXAMINATION MATHEMATICS 140 2007 09 CALCULUS I EXAMINER: Professor W. G. Brown DATE: Thursday, December 06th, 2007 ASSOCIATE EXAMINER: Dr. D. Serbin TIME: 09:00 — 12:00 hours INSTRUCTIONS 1. Fill in the above clearly. 2. DO NOT TEAR PAGES FROM THIS BOOK! All your writing — even rough work — must be handed in. You may do rough work anywhere in the booklet. 3. This is a CLOSED BOOK examination. CALCULATORS ARE NOT PERMITTED. Translation dictionaries are permitted; no other dictionaries are permitted. 4. OTHER CALCULUS EXAMINATIONS ARE BEING WRITTEN AT THIS TIME. THIS IS THE EXAMINATION IN MATH 140 ONLY! 5. The examination booklet consists of this cover, Pages 1 through 7 containing questions; and Pages 8, 9, and 10, which are blank. Your neighbour’s version may not be the same as yours. 6. There are two kinds of problems on this examination, each clearly marked as to its type. 0 Some of the questions on this paper require that you SHOW ALL YOUR WORK! Their solutions are to be written in the space provided on the page where the question is printed. When that space is exhausted, you may write on the facing page. Any solution may be continued on the last pages. or the back cover of the booklet, but you must indicate any continuation clearly on the page where the question is printed! 0 Some of the questions on this paper require only BRIEF SOLUTIONS ; for these you are expected to write the correct answer in the box provided; you are not asked to show your work, and you should not expect partial marks for solutions that are not completely correct. You are expected to simplify your answers wherever possible. You are advised to spend the first few minutes scanning the problems. (Please inform the invigilator if you find that your booklet is defective.) 7. A TOTAL OF 70 MARKS ARE AVAILABLE ON THIS EXAMINATION. PLEASE DO NOT WRITE INSIDE THIS BOX Final Examination — Math 140 2007 09 — V6181 OH 1 1. BRIEF SOLUTIONS [2 MARKS EACH] Give the numeric value of each of the following limits if it exists; if the limit is +00 or —00, write +00 or —00 respectively. In all other cases write “NO FINITE OR INFINITE LIMIT”. . 56+x2 <3) 332214502— ANSWER ONLY ,— (b) 11m M : ”“0 (sin 3x)2 ANSWER ONLY cc——>0+ $ (C) lim arctan (_l) = ANSWER ONLY lnE 3 3 ANSWER ONLY g (d) § l to g I (e) lim (x/u2+2u+4— V1.52 —3u+1)= 1.5—? —00 ANSWER ONLY Final Examination — Math 140 2007 09 — VGI' S1 011 l 2 2. BRIEF SOLUTIONS [3 MARKS EACH] For each of the following functions answer the ques— tion; if the object(s) requested does / do not exist, write “NONE”. (a) The horizontal asymptotes to the graph of g(m) = 2 arctanx — 1 are ANSWER ONLY 1 if a: 7A —2, 0, 2 6 if x = 5 if :c = 0 —4 if x = —2 asymptotes to the graph of f are ANSWER ONLY (c) Air is being pumped into a spherical balloon so that its volume in— creases at a rate of 10 cm3/s. How fast is the radius of the balloon increasing when the radius is 12 cm? ANSWER ONLY (b) If f is defined by f(g;) = , the vertical Final Examination — Math 140 2007 09 — V61” 81 OH 1 3 3. BRIEF SOLUTIONS [3 MARKS EACH] Evaluate each of the following, and always simplify your answers as much as possible. (a) gene) = ANSWER ONLY (b) (—7; cos(arcsin u) = ANSWER ONLY (0) An antiderivative F(:c) of f (as) = sinha: such that F (0) = —1 is ANSWER ONLY t2 3 :1:— —— f’(2) = (d) Where f(t):1—t (3+t)2’ ANSWER ONLY Final Examination — Math 140 2007 09 — VeISl OH 1 4 4. SHOW ALL YOUR WORK! (a) [6 MARKS] Use Rolle’s Theorem and the Intermediate Value Theo— rem to show that the curve y = 1 + 2x -|- £133 + 4905 crosses the :c—axis exactly once. (b) [4 MARKS] Showing all your work, determine the value of the con— stant K that will make the following function continuous at m = 0: K272 f(x) = 1— 0059: 8 if x3 0 if 5B>0 Final Examination — Math 140 2007 09 — VGI'SI OH 1 5 5. SHOW ALL YOUR WORK! The equation 3:5 + 56234 + y3 2 4y + 3 defines y implicitly as a function of a: near the point (9:, y) = (1, 2). Showing all your work (a) 3 MARKS] determine the value of y’ at (12,31) 2 (1, 2)‘ 7 (b) [3 MARKS] determine the value of y” at (:c,y) = (1, 2); and l (c [3 MARKS] estimate 3/ when :6 = 0.97 by using the tangent line to the curve at the point (as, y) = (1, 2). Final Examination — Math 140 2007 09 — V61“ 81 011 1 6 6. SHOW ALL YOUR WORK! 60 [10 MARKS] The function f is defined by f(x) = 1 + 2:2 . 20—43; for 2<3235 A rectangle with sides parallel to the coordinate axes has one vertex at the origin, one on the positive m—axis, one on the positive y—axis; and the fourth on the graph of f. Showing all your work, use the calculus — no other method will be accepted — to determine the maximum area of such a rectangle. for 031332 Finai Examination — Math 140 2007 09 — V61” SI OH 1 7 7. SHOW ALL YOUR WORK! For cc 2 0, define f (3:) = ace—2“”2 . (a) [3 MARKS] Showing all your work, determine the intervals of its domain where f is increasing, and the intervals where it is decreasing. (b) [3 MARKS] Showing all your work, determine whether f has local extrema, and classify them, if any, as maxima or minima. You are expected to base your classification on tests studied in this course. (c) [3 MARKS] Showing all your work, determine all inflection points for f. (d) [1 MARK] Sketch the graph of f. Final Examination — Math 140 2007 09 — Ver81 011 1 CONTINUATION PAGE FOR PROBLEM NUMBER You must refer to this continuation page on the page where the problem is printed! Final Examination — Math 140 2007 09 — V61“ SI OH 1 CONTINUATION PAGE FOR PROBLEM NUMBER You must refer to this continuation page on the page Where the problem is printed! Final Examination — Math 140 2007 09 — V61" 81011 1 CONTINUATION PAGE FOR PROBLEM NUMBER You must refer to this continuation page on the page Where the problem is printed! 10 ...
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