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may 2009 - FACULTY OF ARTS AND SCIENCE We 0888 University...

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Unformatted text preview: FACULTY OF ARTS AND SCIENCE We: 0888 University of Toronto FINAL EXAMINATIONS, APRIL/MAY 2009 MAT 135Y1Y Calculus I Ifl‘j Duration -— 3 hours W NAME ’ 91M ( Please PRINT full name, _ and UNDERLINE surname): STUDENT NO: SIGNATURE OF STUDENT (in INK 01' BALL-POINT PEN): This Exam has 2 Parts: PART A: 8 questions (55 marks). PART B: 18 multiple choice questions (45 marks). Indicate your answer to each multiple-choice question in PART B by completely filling in the appropriate circle in the ANSWER BOX on this front page. (Use a dark pencil!) FOR MARKERS ONLY « -- ANSWER BOX FOR PART B NOTE: 1. Before you start, check that this Exam has 20 pages. 2. No aids allowed. NO CALCULATORS! 3. DO NOT TEAR OUT THIS PAGE OR ANY OTHER PAGE. M®l COMPUTER CARDS AND ANSWER BOOKS WILL NOT BE USED. NO SCRAP PAPER! 9°51???pr Page 1 of 20 Code: 0888 PART A [55 marks] Answer all questions in PART A in Spaces provided. Show all your work for PART A. Any answer in PART A Without proper justification may receive very little or no credit. Use the back of each page for rough work. Marks for each question in PART A are indicated by [ ]. DO NOT TEAR OUT ANY PAGES. 1. Find jmcosxdm. [5/ 2. Find / tan82 a: Sec4 xdsc. Page 2 of 20 Code: 0888 1 3. F' d de. 1n ijVm2~4 [7/ Page 3 of 20 Code: 0888 4. Find the arc length of the curve 3/ = 1n(sec cc) , 0 S a: < 1’. 4 . [7/ Page 4 of 20 Code: 0888 d 5. Find the solution of the differential equation 8-:- = 2:239:32 + 1)e'3y that satisfies the condi— tion 31(1) =1n2 . [7] Page 5 of 20 [8] Page 6 of 20 COde: 0888 Code: 0888 nxn m . Remember to fully Justlfy your 00 7. Find the interval of convergence of the series 2 n=4 answer. [8] Page 7 of 20 e Code: 0888 8. NOTE: This is a hard problem and will be marked extremely strictly. Very little or no credit will be given unless your solution is completely correct. Find [\lx— \/$2-25 dzc. Hint: Investigate V3; + 5 —— Va: —- 5. Page 8 of 20 PART B [45 marks] Code: 0888 18 multiple choice questions PLEASE READ CAREFULLY: Each of the following multiple—choice questions has exactly one correct answer. Indicate your answer to each question by completely filling in the appropriate circle in the ANSWER BOX on the front page. Use a dark pencil. MARKING SCHEME: 2% marks for a correct answer, 0 for no answer, a wrong answer or giving more than one answer. You are not required to justify your answers in PART 13. NOTE: If there is any discrepancy between the circles you darken on these inside pages and those you darken on the front page, the circles you darken on the front page will be regarded as your final answers. Note that only the circles you darken will count. For Part B, your computations and answers (other than the circles you darken) will NOT count. WARNING: If you darken the circles on these inside pages but do not darken the circles on the front page, you will still get credit for your correct answers, but there will be a PENALTY of minus 4 marks. YOU MUST NOT TEAR OUT ANY PAGES OF THIS EXAM. 1. Find the value of lirn w 37—)0 a: 1 ® ”:2: undefined © 0 1 © 2; 1 ® “‘3‘ 2. Let f(:v) = 3:4 - 2x3 —- 3691:2 +533 -4 , for all x. Then the graph of f has a point of inflection at ® cc=~2 and m;3 only. so a 3 only. © :1: z: -2 and :1: = ~3 only. ® a: 2 -2 only. ® w=2andm=3only Page 9 of 20 Code: 0888 INDICATE YOUR ANSWERS ON THE FRONT PAGE. Penalty for not doing so is MINUS 4 marks. 3. The length of a rectangle is increasing at 5 ft/sec while its width is decreasing at 4 ft: / sec. At what rate will the area of the rectangle be changing when the length of the rectangle is 500 ft and the Width is 300 ft? increasing at 400 sq ft/sec. decreasing at 500 sq ft/ sec. increasing at 450 sq ft / sec. increasing at 500 sq ft / sec. @©@@® decreasing at 400 sq ft / sec. 4:133 + 33:2 —- 103: + 1 4- The graph 0f :9 = has two vertical aysmptotes and one slant (Le. oblique) asymptote. The slgfit+a§§mptote is the line ® 3/ = 4x + g: y = 4x -— 2 © y = 43: @ y = 4:3 + 3 ® 3/ 2 4:1: — 5 Page 10 of 20 Code: 0888 INDICATE YOUR ANSWERS ON THE FRONT PAGE. Penalty for not doing so is MINUS 4 marks. 5. The sum of two positive numbers is 16. What is the smallest possible value of the sum of their squares? @ 132 126 © 134 ® 128 ® 130 6. Find the area of the region enclosed between the curves y 2 1 + 2:1; and y = 1+ :1: + V593 69% :- ©i @2- @2 Page 11 of 20 Code: 0888 INDICATE YOUR ANSWERS ON THE FRONT PAGE. Penalty for not doing so is MINUS 4 marks. 7. Let R be the region enclosed between the curves y = a: and y =2 x2 . Find the volume of the solid generated by revolving R about the a: -- axis . 57r @ 1‘5 71' .3 8. Find the average value of the function f (as) =2 3562 + 83:3 on [1, 3] . ® 190 80 © 93 ® 65 ® 127 . dR dW 9. ConSIder the predator—prey system ”cl? = 5R -— 4RW, ~22;— = -3W + 2RW. When the system is in equilibrium with W # 0 , R 73 0, then RW = 15 @ 5‘ 5 5 7 © ‘2: 7 © 2 5 ® 5 Page 12 of 20 Code: 0888 INDICATE YOUR ANSWERS ON THE FRONT PAGE. Penalty for not doing so is MINUS 4 marks. 10. Suppose that a population grows according to a logistic model, i.e. the growth is modeled dP by the differential equation ~22},— = kP(1 —- E?) . Suppose that the carrying capacity K is 10, 000 . Suppose further that the initial population is 2, 000 and that it grows to 4, 000 after one year. What will be the population after another year (Le. 2 years from the beginning)? @ 6,800 6, 400 @ 7,000 @ 6, 600 - ® 6, 500 Page 13 of 20 Code: 0888 INDICATE YOUR ANSWERS ON THE FRONT PAGE. Penalty for not doing so is MINUS 4 marks. 11. Consider the following three series: CO 1 I. 7; (n + 2)\/ln(n + 2) Decide which of the Series converge (or converages). @ II only ' II and III only © I, and II and III ® III only (15) I and III only Page 14 of 20 Code: 0888 INDICATE YOUR ANSWERS ON THE FRONT PAGE. Penalty for not doing so is MINUS 4 marks. 12. Consider the following series: n-O n4+n2+25 °° n (1+n)“ ;(_1)+1 n! Which one of the following statements is correct? I and II both converge conditionally. I converges absolutely and II converges conditionally. I converges conditionally and II diverges. I converges conditionally and II converges absolutely. @©©®@ I and II both converge absolutely. Page 15 of 20 Code: 0888 INDICATE YOUR ANSWERS ON THE FRONT PAGE. Penalty for not doing so is MINUS 4 marks. 13. Find the coefficient of :34 of the Maclaurin series for f(x) = sin2 a: . 1 @723 Q l 4:: @©@ 1 on! 22.2. 14. Let an 2 (1 + ’n n ) . Then the sequence {can} converges to 1. converges to 2 . diverges. converges to 1112. converges to e2 . @©@®® Page 16 of 20 Code: 0888 INDICATE YOUR ANSWERS ON THE FRONT PAGE. Penalty for not doing so is MINUS 4 marks. my 15. If y is a differentiable function of m such that (1 + :63)y2 + 4 f V 53:2 + t2 dt = 112 , find 2:1: the value of ill-:- at the point Where y = 2 (1.1.93 :3 = 2). 27 @"5‘5’ not determinable due to insufficient information 18 ’35 23 "Z4" 28 ”3'9? ®©© Page 17 of 20 INDICATE YOUR ANSWERS ON THE FRONT PAGE. Penalty for not doing so is MINUS 4 marks. 16. Find the value of Adex @ 1% 3; © *3 © 32: ® '2: Page 18 of 20 Code: 0888 Code: 0888 INDICATE YOUR ANSWERS ON THE FRONT PAGE. Penalty for not doing so is MINUS 4 marks. °° 1 + 8x — 2:02 1 , Th ‘ ' t 1 ————— d 7 e 1mproper 1n egra /1 8m4 + 21:3 + 43:2 + :1: ac 6 ® converges to In (E) . 5 converges to In (5) . 0 diverges. converges to In (—3) . @©@ 5 converges to 1n (1) . Page 19 of 20 we». Code: 0888 INDICATE YOUR ANSWERS ON THE FRONT PAGE. Penalty for not doing so is MINUS 4 marks. 18. Let f(”)(a) denote the value of the nth derivative of f at a. If fins) 2 23—+8:2 , then #95) (0) == @ (950294 (950295 © (950297 @ (950293 ® (950296 Page 20 of 20 ...
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