MATH140 Dec 2009 - VERSION 1 McGILL UNIVERSITY FACULTY OF...

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Unformatted text preview: - . VERSION 1 . McGILL UNIVERSITY .__ FACULTY OF SCIENCE f ‘FINALEXAMINATICN f MATHEMATICS 140 2009 09 CALCULUS 1 EXAMINER: Professor w. G. Brown DATE: Wednesday, December 16th, 2009 ASSOCIATE EXAMINER: Dr. A.'Hundemer TIME: 14:00 — 17:00 hours FAMILY NAME=E11111111EIE111111 STUDENT NUMBER GIVEN NAMESEJEIIIDIIM E11111:E1 INSTRUCTIONS 1. Do not tear pages from this book; all your writing — even rough work — must be handed in. You may do rough work for this paper anywhere in the booklet. 2. This is a closed book examination._CaEcu|ators are not permitted, but regular and translation dictionaries are permitted. 3. This examination booklet consists of this cover, Pages 1—4 containing multiple-choice questions worth at ' most 50 MARKS, Pages 5—8 containing full solution questions worth 30 MARKS, and Pages 9-13 which are blank continuation pages. A TOTAL OF 80 MARKS ARE AVAILABLE ON THIS EXAMINATION. 0 Your answers to the'multlple choice questions must be entered on the Scantron form which will be provided. There is only 1 correct answer expected for each problem. Be sure to enter (1) Your student number (including the check code = the last two letters of your family name), (2) Your examination Version, (3) Your name. Please note that the Examination Security Monitor Programdetects pairs of students with unusually similar answer patterns on multiple~choice exams. Data generated by this program can be used as admissible evidence, either to initiate or corroborate an investigation or a charge of cheating under Section 16 of the Code of Student Conduct and Disciplinary Procedures. 0 Full solution questions in the second part of this paper require that you SHOW ALLYOUR WORK! Begin your solution on the page where the question is printed; a correct answer alone will not be sufficient unless substantiated by your work. You may continue a solution on the facing page, or on the continuation pages, or the back cover of the booklet, but you mtg; indicate any continuation clearly on the page where the question is printed! To be awarded partial marks on a part of a full solution question a student’s answer for that part must be deemed to be more than 50% correct. You are expected to simplify all answers wherever possible. PLEASE DO NOT WRITE INSIDE THIS BOX TOTAL NMC . /30 Final Examination 7 Math 140 2009 09 — VGI‘SI OH 1 _ 1 PART I: MULTIPLE CHOICE QUESTIONS Each of the following 30 questions is worth 2 MARKS. The maximum number of marks you may earn 011 these multiple choice questions is 50 MARKS; you may attempt as many of these problems as you wish. Show your answers only on the Scantron form. 1. The function defined by I .’L' + 4 If {I} < #4., fix): —4—:c if—4SxSS. 232—21 if5<sv. fails to be continuous (a) only at a: = ——4, (b) only at .73 = 5, (c) only at a: .= M4 and a: = 5, (d) nowhere, (e) on some other set. 2. Let fix) = $51n($). Then f”(1) is - (a) 7. (his. (c) 9. p (d) 3. (e) 2. V 2564 + 4 m is (are) best described by :E _ — 3. The vertical asymptote(s) of fix) = (a) a" I ~9, (b) m : 0, (c) a: = -—3, (d) two asymptotes, (e) no asymptote. 4. Let y = f(2:) be defined by the equation 142: + 313:2 + y3 = #16 near :13 = —1, y = -—1.. The value I of f’(—1) is ' (a) _5) (b) _7: (C) _4: (d) _11: (8) W9‘ 5. Let (sinmllf’ 1%) = 58 0 ifsc=0. ifsc750, Then f’(0) is (a) —1, (b) O, (c) 4, (d) 1, (e) does not exist. 1 % cos(:c)l 6. Find lim _ m—vfl (1)3 (a) 1/2, (b) 1, (c) 2, I (d) 0, (e) does not exist. Final Examination — Math 140 2009 09 ~ Ver81 OH 1 2 7. The minimum value taken by the function f (m) = 33in(22) — 2005(3) on [0, E] is (3’) _2a (b) ~43 (C) 5: (d) _12! (e) _10 8. Let f(a:) = :54 111(w2 + 7). Then f’(2) is (a)321n(11)+4/11, (b)321n(11)+16/11, (c)32ln(11), (d)321n(11)+64/11, (e) 321n(11) + 16. 4:1: 1+3: (3‘) (”26): (b) (—738): (C) (054): (d) (43—1): (8) (—6110) at (1, 2) passesthrough the point 9. The normal to the curve y = 10. Let f (at) = msin(:c) + 008(32). The number of values of a: in the range [0, 3071'] at which f has a local maximum is (n) 16, (5)15, (cm, (d) 30, (e) 29. 332+922+18 11. F" d 1' W. in wi193$2+7x+12 (a).0, _ (b) —2, (e) 2, (d) 3, (e) does not exist. 12. Let y = f (m) be defined by the equation '9' + 125::2 + $643. : 13 near 3: 2 1, y 2 0. The value of f’(1) is ' (a) _9: (b) _77 (C) _5: (d) T10) (e) _13 13. Let f($) 2 arctanmfi). Then f’(1) is ,(a) 6, (b) 3, (c) —4, (d) —3, (e) —11. 14. The function defined by sin(8$) f(w) = 23 acos(5m) if a: Z O. ifzc<0, Final Examination 7 Math 140 2009 09 7 VGFSI 011 1 7 3 15. Let f(32) =arcsin(:1:). Then f” (wé) is (33456/27, (b) —100/27, (c)—50/27, '(d) 95/27, 7 (e) 115/27.” 16. Let f(:1:) = $2: 5. Then m3) is ' (a)—13/196, (b) —11/98, (c)—2/7, (d) 5/196, (e),—l/49. 17. Let flat) = 2:3 — 9x2 — 273cc — 2. The function f has a point of inflection at :5 2 (a) —1, (b) 3, (0)4, ((1)10, (e) 13. 18. . Find lim SIM). -$—>0 I33| (a) 1/2, (b) 1, (c) —1, (d) 0, (e) does not exist. sin(:c7) , 19. Find lim (a) O, (b) 4, (c) 3, (d) —5, (e) does not exist. 20. Let f denote the function f (:13) = (1 + 43:)e“9’* defined on [0, 00). Which answer best describes the location where f takes its global (zabsolute) maximum value? ' (a) J; = 5/36, (b) a: = 0, (c) a: = 4/9, ((1) :13 = 1/9, (e) maximum not attained. 101 21. Let flat) = sin(:r). The largest interval containing a: = —-—%--7r on which f is concave up is (a) [—147r,—137r], (b) [0,00), ((3) [716%,f147r], (d) [~157r,7137r], (e) [—1571', —147r]. $2—4 2 + a: (a) (40,46), 0)) (37-3), (C) F5741), (d) (1774), (6') (-2141)- 22. The tangent to the curve y = at (—1, —3) passes through the point 23. Let fig?) 2 ln (2 + 6). Then f’(3) is Final Examination — Math 140 2009 09 — V81" 81 011 1 4 24. Let y = f (m) be defined by the equation y2 + 362: + 7553 ln(y) = 37 near :6 = 1, y z 1. The value of f’(1) is ' (go—9, (b) —7, (c) 411, (d)—4, (e) —10. _ , 4w—1 25.. Fmd $113.16 3:2: _1. (a) 1¢g34, (b) men—111(3), (c) 1, (d)31n(4)—4ln(3), (e)4/3. $2 26. Let f denote the function flat) = $+14 location where f takes its global (=ab'solute) minimum value? defined on [0, 00). Which answer best describes the (a) :5 = 14, (b) m = 0, (c) a: = 11, (d) as = 28, (e) minimum not attained. 27. Find .lim WW. $—*CX3 (I: '(a) 0, (b) 00, (c) 4/3, (d) 8/3, (e) does not exist. 28. The graph of the function f (x) z e” - (3:3 — 6:1: + 12) has all of its inflection points at V (a) a: : O, (b) :1: = ~6,. (c) a: = 0 and w = —6, (d) 3: = 6, (e) none of the preceding. 2:2 ' 29. For the function firs) = a: - e"? the intervals of decrease are (a) (71:1): (10) POOH-1), (C) (1,+'00), (d) (—oo,—\/§)I and (Vi-too), (e) (—00,—1) and (1,+oo). mm 3.0. The graph 'of the function f (2:) = a: - (a: — 4) is concave downward on the interval I (a) (—0071): (b) (la-+00): (C) (031): _ (d) (—0010): (e) (0,+OO). Final Examination 7 Math 140 2009 09 7 V6131 on 1 5 PART II: FULL SOLUTION QUESTIONS These questions are together worth 30 MARKS. Begin each solution on the page where the question is printed; a correct answer alone will not be sufficient unless substantiated by your work. You may continue a solution on the facing page, or on the continuation pages, or the back cover of the booklet, but you must indicate any continuation clearly on the page where the question is printed! To be awarded partial marks on a part of a question, a student’s answer for that part must be deemed to be more than 50% correct. You are expected to simplify all answers Wherever poSsible. Final Examination — Math 140 2009 09 — V61" S] OH 1 . 6 1. . SHOW ALL YOUR WORK! (8.) [4 MARKS] Let 9(32) = 1 + sinh 2:13. Showing all your work; find a linearization of g at a = O. ' ' - (b) [4 MARKS] Showing your work, use your linearization to approximate g(0.005). (c) [7 MARKS] Use either the Mean Value Theorem or Rolle’s Theorem —— no other method is acceptable w to explain why the graph of g crosses the line y = 1 exaCtly once. Final Examination — Math 140 2009 09 — V61” 81 OH 1 7 2. SHOW ALL YOUR WORK! [6 MARKS] Showing all your work, determine the function f such that f”(:c) : 2 ~ 122:, f(0) = 9: f(2)=15- Final Examination — Math 140 2009 09 LR V81” 81 OH 1 ‘ 8 3. SHOW ALL YOUR WORK! [9 MARKS] Showing all your work, find the point(s) 0n the curve 3; = Zfi which is (are) Closest to the point (2, 8). Final Examination 7 Math 140 2009 09 — V6181 OH 1 CONTINUATION PAGE FOR PROBLEM NUMBER C] You must refer to this-continuation page on the page Where the problem is printed! _' ' VERSION 1 1 MCGILL UNIVERSITY — FACULTY OF SCIENCE 3 FINAL EXAMINATION MATHEMATICS 140 2009 09 CALCULUS 1 EXAMINER: Professor W. G. Brown DATE: Wednesday, December 16th, 2009 ASSOCIATE EXAMINER: Dr. AfI-Iundemer TIME: 14:00 — 17:00 hours INSTRUCTIONS I ‘ 1. Do not tear pages from this book; all your writing ~ even rough work — must be handed in. You may do rough work for this paper anywhere in the booklet. 2. This is a closed book examination. Calculators are not permitted, but regular and translation dictionaries are permitted. 3. This examination booklet consists of this cover, Pages 14 containing multiple-choice questions worth at ‘ most 50 MARKS, Pages 5—8 containing full solution questions worth 30 MARKS. and Pages 9-13 which are blank continuation pages. A TOTAL OF 80 MARKS ARE AVAILABLE ON THIS EXAMINATION. I- I 0 Your answers to the multiple choice questions must be entered on the Scantron form which will be : provided. There is only 1 correct answer expected for each problem. Be sure to enter (1) Your student number (including the check code = the last two letters of your family name), (2) Your examination Version, (3) Your name. Please note that the Examination Security Monitor Program detects pairs of students with unusually similar answer patterns on multiple-choice exams. Data generated by this program can be used as admissible evidence. either to initiate or corroborate an investigation or a charge of cheating under Section 16 of the Code of Student Conduct and Disciplinary Procedures. 0 Full solution questions‘in the second part of this paper require that you SHOW ALL .YOUR WORK! Begin your solution On the page where the question isprinted; a correct answer alone will not be sufficient unless substantiated by your work. You may continue a solution on the facing page. or on the continuation pages, or the back cover of the booklet, but you M indicate any continuation Clearly on the page where the question is printed! To be awarded partial marks on a part of a full solution question a student's answer for that part must be deemed to be more than 50% correct. You are expected to simplify all answers wherever possible. PLEASE DO NOT WRITE INSIDE THIS BOX 1(a) , 1(b) 1(a) . 2 3 /4 /4 / 7 /6 /9 TOTAL NMC /30 _ Final Examination '5, Math 140 2009 09 — .Vers1on 1 , - - , 1 PART I: MULTIPLE CHOICE QUESTIONS Each of the following 30 questions is worth 2 MARKS The maximum number of marks you may ' earn on these multiple choice questions is 50 MARKS; you may attempt as many of these problems - as you wish._ Show your answers only on the Scantron form. 1. The function defined by . . :c + 4 if :1: < —4, fix) = —4—a: 11451-55. 2m—21 if5'< 3:. fails to be continuous . '(a) only at a; = —4, (b) only at :1: = 5, (c) only at a: z —4' and a: = 5, (d) nowhere, ' ‘ (e) on some other set. ' 2. Let flan) =x5 ln(a¢). Then f”(1) is - (a) 7, (b) 5, - (C) 9, (d) 3, (e) 2- v2$4+4 _ 3. The vertical asymptote(s) of fins) = m IS (are) best described by (a) :L' = —9,’ (b) :1; = 0, (c) m = ;3, ((1) two asymptote-s, ' . (e) no asymptote; 4. Let y = f($) be defined by the equation 14:1: + ya?2 + y3 = —16lnear :1: = —1, y = 1-1.. The Value of f'(_1) is y ' _ . - ' (a) —5, (b) —7’,. (c) —4, (d) —11, '(e) —9'.. r 5. Let fa): {(5111.37)} ”“501. 03'”- if$=0. The‘n f’(0) is _ ' (a) —1, (b) d, (c) 4, (d) 1, (e) does not exist. 1 — cosh). , ' 6. Find lim 3 m—-0 a: (a) 1/2, (b) 1, ‘_ (c), 2, I. (d) 0, (e) does not exist. ' Final Examination __Mntn 140 20097 09 —«- V er SI 01] ' 1 , . ' ' 2 7. The. minimum value taken by the function f(a:) = 331n(:c)7— 2 005(3) on [0, g] is (a) —2, (b) *4, (c) 5, (d) —12, (e) —10. 8. Let f(:c)= 41n(n2+7). Then f'(2)is" (a)32'ln(11)+4/11, (b) 321n(11)+16/11, _(c)'32'1n(11), (d) 321n(11)+6-4/11,7 . (e) 32‘1n(11)+ 16. - - 41: I , _ 1 +3 (3‘) (72:6)? I (b) (—758): (C) (014): (d) (4,“1)’.(e) (—6: 10) at (1, 2) passes through the point ' 9. The normal to the curve '3; = 10. Let f (as) = xsin(m) + cos(:r:). The number of values of a: in the range [0, 3011'] at which f has a local maximum is' ' ' . . (a) 16, (1).).15,I(c)0, (d) 30, ((2)29; -:c2+91'+18 11. F' d ' —_. . 1n 31331332 +7113+12 (no, 7 (b) 42, (c) 2, (d)3,' (e) dOes not exist. 12. Let y = f($) be defined by the equation 3; + 125':2 + me“ = 13 near 5:: = 1, y = O. The value of f”(1) is A . - - ' ' ‘ (a) 49, (b) —7 u (c) —5, ('d) —,1iol, (e) —13._ 13. Let f (3:) = arctan(a:6). Then f’ (1) ie (a) 6: (b)3a (C) _43 (d) _3: (B) "11' 14. The function defined by sin(8:r:) ms): :23 , ‘a cos(5a:) if a: 2 0. ifz<0, is continuous if and only if a is (n)5, (ms; (94, (d)-1, 77(e)70.7" Final Examination — Math 140 2009 09 — VGI'SI OH 1 ' 3 15. Let 'f(x) = arcsin(:c). Then f” (_g) is (a)—56/27, (b) 400/27, (c)—50/27, (d) 95/27, I (e) 115/2‘7j7 , a: .16. Let f(x) = $2 +5. (a)—13I/1916, '(b)I—11/98, (c)—-2/7, (d) 5/196, (e) —1/49. Then f’ (3) is 17. Let f (rs) = x3 — 9x2 4 273m — 2. The function f has a. point of inflection at .1: = (a) "-1: (b) 3, (C) 4! (d) 10: (8) 13' 13. Find lim sm(a:)- H, Ix! (a) 1/2, (b) 1, (c) —1, (d) 0, (e) does not exist. . ' 7 _ 19. Find lim 3171“” ) m—eoo 2:4 (a) 0, (b) 4, I (c) 3-, (d) —5, (e) does not exist. 20: Let f denote the function f(:v) = (1 + 4206—9“ defined on [0, 00). Which answer best describes the location where f takes its global (=absoiute) maximum value? (a) .1: = 5/36, (1)) a: = 0, (c) a: = 4/9, (d) :c = 1/9, (e) maximum not attained.7 21. Let f (as) = sin(a:). The largest interval containing a: = —%1r on which f is concave up is (a) [—147r,—137r], (‘0) [0,00), (0) [—161r,——141r],-(d) [—157r,—1i37r], (e) [—157r,-141r]. 332—4 2+9: (3) (410,45), (’0) (3,—3), (0) (ii-11): (d) (—7,-7): (6) (fa—“4)- 22. The tangent to the curve y i at (—1, “3) passes through the point - 23. Let fi(:c)I—-=ln (2+6).Then f’(3).is' ' mall—gig. egg. (0%, (d)—%, (cs—655.- Final Examination — Math 140 2009 09 — VeI'SI 011 I _ _4 " 24.” Let y = f(:c) be defined by the equation yz f36af+ 7x31n(y) = 37 near 3: = 1, y = 1. The value of f’(1) is 7 - ' (a)—9, (b) *7," (c) 411, (d)—4, (e) “10.- . _ 4” — 1 25. Find-3151:1333 _ 1., (a) log34, .(b) 1n(4)—1n(3), (0)1, (d)3ln(4)—4ln(3), (e) 4/3. $2 26. Let f denote the function f (to) = :1: + 14 defined on [0,00). Which answer best describes the location where f takes its global (=absolute) minimum ‘value? (a) :1: == 14, (b) at = 0, (c) a: = 11; (d) x = 28, (e) minimumtnot attained. 27. Find 11m ———"S’Jr:3’_3. (a) O, (b) 00,- (c) 4/3, (d) 8/3, (e) does not. exist. 28. The graph of the function f (m) = em - (2:3 — 6:6 4— 12) ' has all of its inflection points at (a) x =-- 0,’ (b) a: = —-6, I (c) a: = 0 and m = "6, (d) a: = 6, I (e) none of the preceding. . 7 . $2 29. For the function f (x) = :1: - e“? the intervals of decrease are _ (a) (__,1a1) a (b) ("00‘1"”1): (C) “Fl—90), . \(d) (—ong—x/s) and («an»), (e) (—00,—1) and (1,+oo). - (a: — 4) is concave downward on the interval WIU‘ 30._ The graph 'of the function f (I) = :1: ' (n(_,oo,1),j(b)(1,r+oo), '(c) (0,1), (an—com), (e) (o,+oo). Final Examination — Math 140 2009 09 — V6131 01] 1 ' ' y 5 PART 11: FULL SOLUTION QUESTIONS These questions are together worth 30 MARKS. Begin each solution on the page where the question is printed; a correct answer alone will not be sufficient unless substantiated by your work. You may continue a solution on the facing page, or on the continuation pages, or the back cover of the booklet, but- you must indicate any continuation clearly on the page Where the question is printed! To be awarded partial marks on a part of a question a student’ 3 answer for that part must be deemed to be more than 50% correct. You are expected to simplify all answers wherever possible. ' Final Examination — Math 140 2009 09 —_ Vers1on '1 -- ' ' 6 1.. SHOW ALL YOUR WORK! ' (a) [4 MARKS] Let g(:::) = -1 + Sinh 2x. Shearing. all your Work; find a linearization of g at a = 0. . _ I ‘ ' (b) [4 MARKS] Showing your work, use your iinearization to approximate g(0.005). (c) [7 MARKS] Use either the Mean Value Theorem or Rolle’s Theorem — no other method is acceptable -—~ to explain why the graph of g crosses the line 3; = 1 exactly once. Final Examination 4 Math 140 2009‘ 09 — VeI'SIOfl 1 ' . ' 7 2. SHOW ALL YOUR WORK! [6 MARKS] Showing all yburwork, determine the function. f such that f"-(3:) = 2 _. 123, f(0) = 9, fl?) = 15- ' Final Examination _ Math 140 2009 09 — VGI' SI 01] 1 _ 8 3. SHOW ALL YOUR WORK! [9 MARKS] Showing all your work", find the point(s) on the curve y = ‘Zfi which is (are) closest to the point (2,8). ‘ ' ‘ t , Final Examination — Math 140 2009 09 — Vers10n 1 CONTINUATION PAGE FOR PROBLEM NUMBER I: You must refer to this continuation page on the page where the problem is printed! ...
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