mat135y-a09 - % UNIVERSITY OF TORONTO ' J35 Faculty of Arts...

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Unformatted text preview: % UNIVERSITY OF TORONTO ' J35 Faculty of Arts and Science .<: A 6% AUGUST 2009 FINAL EXAMINATIONS 2" a? E «% MAT135Y I ' ® Calculus I 2 FINAL EXAM, August 12, 2009. Duration: 3 Hours NO AIDS ALLOWED. Total: . 102.5 Marks Family Name: (Please Print) Given Name(s): ' (Please Print) Your Tutorial Section/TA’S Name: Student ID Number: Print “yes” here if you are writing this exam as a deferred exam (ie you are not enrolled in the summer course): READ INSTRUCTIONS ON PAGE 2 BEFORE YOU START! Page 1 Of 22 Final Exam: August 12, 2009. MAT135Y — Calculus I ___#____—___—————————————————#———l____n_._____.___ FOR MARKER’S USE ONLY Make sure that this booklet has 22 pages. Problem 2: Problem 3: You may not use calculators, Emblem 4; f7“I cell phones, MP3 players, PDAS, Problem 5. /7 or any other unauthorized aid ' of any kind during the exam. l—Pmblem 6‘ /7 i Please read through the entire PIOblem 71 /7 test before starting, and take l—Problem 8: /7 note of how many points each bpmblem 9: /2'5j question is worth. Carefully read _ the instructions to Part A and Part Pmblem 10' /2'5 B, and note in particular that full ‘EOblem 113 /2-5 justification to answers in Part A Problem 12: /2.5_‘ is required, while in Part B Problem 13: /2.5 justifications are optional but will P M 13. 2 5 be considered for part marks if m em ' the result given is incorrect. Pmblem 141 /2-5 Problem 15: /2.5 Partial credit will be given problem 16; /2_5 for partially correct work. Problem 17: /2.5 Do not remove any pages from the exam booklet. Pmblem 18: /2'_5J Problem 19: / 2.5 Anything you write on the back of Prob1em 20; /2.5 any page will not count unless P b1 21. 2 5 yourwrite clearly in LARGE Problem 2 / ' letters CONTINUED ON BACK OF r0 em 2- PAGE —— in which case anything PTOblem 23: /2-5 you write on the back of that page Problem 24: /2.5 W111 be graded' Problem 25: / 2.5 Problem 26: / 2.5 TOTAL: 502. Page 2 of 22 Final Exam: August 12, 2009. MAT135Y - Calculus I PART A Answer all questions in Part A in the space provided. Show all your work for Part A. Any answer in Part A without proper justi- fication may receive little or no credit. Use the back of each page for rough work. Marks for each question in Part A are indicated in brackets DO NOT TEAR OUT ANY PAGES. 1. Sketch the graph of the function _ ln|x| f($) — 3: indicating all 0 intercepts o asymptotes 0 critical numbers 0 inflection points and drawing correctly all 0 regions of increase/ decrease 0 regions of up / down concavity Page 3 of 22 Final Exam: August 12, 2009. MAT135Y — Calculus I (Extra page for Problem 1. DO NOT REMOVE!) Page 4 of 22 Final Exam: August 12, 2009. MAT135Y - Calculus I 2. A helium balloon is released from the ground and floats upwards. Suppose you’ve figured out how long it will take the balloon to reach any given height. The needed time, t, (measured in seconds) is related to the height h, (measured in metres) by the equation 1 t=-~h3 2 h. 300( +3h +3) What is the balloon’s height when it is going up at a speed of 1 m/s? Page 5 of 22 Final Exam: August 12, 2009. MAT135Y - Calculus I 3. Recall that an isosceles triangle is one which has two sides of the same length, or, equivalently, one which has two angles the same. ( See picture.) For a fixed perimeter P, which shape of isosceles triangle will have the maximum area? Justify your answer. can you write the perimeter and area in terms of h and 6 in the diagram?) Page 6 of 22 Final Exam: August 12, 2009. MAT135Y — Calculus I (Extra page for Problem 3. DO NOT REMOVE!) Page 7 of 22 [1.5] [1.5] [2.5] Final Exam: August 12, 2009. MAT 135Y - Calculus I 4. Determine Whether the following series are convergent or divergent. If convergent, are they absolutely convergent? Prove your answer. - 2+ (1) :3 (10 2:: gig") (iii) 2:1("Un1n1n (1V) Page 8 of 22 [7] Final Exam: August 12, 2009. 5. Evaluate f 2$“1d:1: $3 +30 Page 9 of 22 MAT135Y - Calculus I Final Exam: August 12, 2009. MAT135Y — Calculus I 6. Find the solution of the differential equation m3y’ : y(:z:)2, subject to the initial condition y(1) = 4. Page 10 of 22 Final Exam: August 12, 2009. 7. Find the arc length of the curve x = g + if, for 1 Page 11 of 22 MAT135Y - Calculus I Final Exam: August 12, 2009. MAT135Y - Calculus I 8. Find the volume of the solid of revolution obtained by rotating the area enclosed by the curve m2 + (y — 1)2 = 1 about the line a: = 2. Page 12 of 22 Final Exam: August 12, 2009. MAT 135Y — Calculus I PART B Answer all questions in the space provided. Clearly state your answers to the specific questions asked. You do not need to justify your answers, and you will get full credit if your result is correct. However, if you choose to justify your answer, you may get part marks even if your result is not correct. Each question in Part B is worth 2.5 marks. DO NOT TEAR OUT ANY PAGES. 9. Find the following limit, if it exists: limxno V $23641. $2 10. Find the following limit, if it exists: limxno 6 $2— 1. Page 13 of 22 Final Exam: August 12, 2009. MAT135Y — Calculus I [2.5] 11. Find the following limit, if it exists: 11mm tan‘zorm) sin m sin 2:5 [2.5] 12. Evaluate (14:32?“ Page 14 of 22 Final Exam: August 12, 2009. MAT135Y — Calculus I [2.5] 13. Evaluate f; sin(cos(ln [2.5] 14. Find a number (2 such that the following function is continuous: V 2 _ c x + 4, 1fx _<_1 fix) fl { 403:3, ifcc > 1 Page 15 of 22 Final Exam: August 12, 2009. MAT135Y — Calculus I [2.5] 15. A radioactive substance decays over time, such that after t years a block which started with mass of 1 kilogram will have mass of m(t) 2 7f” kilograms. What is the half—life of this substance? [2.5] 16. Evaluate: 12 eiézdx Page 16 of 22 Final Exam: August 12, 2009. MAT135Y - Calculus l [2.5] 17. Evaluate: f12x2ln(:c)d:c [2.5] 18. Evaluate: f: |az2 — 2a: — 3| da: Page 17 of 22 Final Exam: August 12, 2009. MAT135Y - Calculus I [2.5] 19. Is fig “firm convergent or divergent? Evaluate the integral if it is convergent. 20. Find the number a such that the line as = a divides the area under the curve 3; = $15 [2.5] (for 1 S a: S 5) into two equal-sized pieces (not necessarily of the same shape). Page 18 of 22 [2.5] [2.5] Final Exam: August 12, 2009. MAT135Y — Calculus I 21. The curve y = 3 — x2, for 0 S x S 17 is rotated about the y-axis. Find the area of the resulting surface. 22. Evaluate f tan3 23 sec3 xdm. Page 19 of 22 Final Exam: August 12, 2009. MAT135Y — Calculus I [2.5] 23. Some biologists are studying a predator—prey system modeled by the equations 45% = 2W + % and 9% 2 —7R + 2RW. Suppose the system is in equilibrium with the size of both populations strictly positive. Find RW. [2.5] 24. Suppose the fish population in a fish hatchery is initially 500, and one day later it is 750. Assume the rate of growth of the population is proportional to the population itself, what will the population be after one more day? Page 20 of 22 Final Exam: August 12, 2009. MAT135Y — Calculus I k: x. [2.5] 25. Find the orthogonal trajectories of the family of curves y = [2.5] 26. Determine the Maclaurin series for the function f = sin2 x. Page 21 of 22 Final Exam: August 12, 2009. MAT135Y — Calculus I (Extra page if needed for your answers. DO NOT REMOVE!) 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This note was uploaded on 05/29/2011 for the course MAT 135 taught by Professor Lam during the Spring '08 term at University of Toronto- Toronto.

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mat135y-a09 - % UNIVERSITY OF TORONTO ' J35 Faculty of Arts...

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