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MATH22212-04 - FACULTY OF SCIENCE — FINAL EXAMINATION...

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Unformatted text preview: FACULTY OF SCIENCE — FINAL EXAMINATION MATHEMATICS 222 —— CALCULUS 3 Examiner: Professor S. W. Drury ' Date: Friday, December 17, 2004 Associate examiner: ProfessorW. Jonsson Time: 9:00 am. —~ 12:00 noon STUDENT NAME: STUDENT NUMBER: VERSION 1 Instructions 1. 10. 11. Identify yourself by writing your name and student number on both this paper and the computer readable sheet. The consequences of writing illegibly should be obvious. This examination paper may n_ot be removed from the examination room by the candidate. All work, includ— ing this paper, the computer readable sheet and any answers or rough work in examination booklets must be handed in. ' . Answers to part I (questions 1 thru 10) are to be entered onto the computer readable sheet provided using a soft lead pencil. This exam is Version 1. It is important that the version number for the question paper be entered into the column next to the student number on the computer readable sheet. Enter your student number and fill in the appropriate disk below each digit. Fill in the check bits as indicated. Enter your name, the course number (MATH 222) and sign the card. . The answers to part 11, (questions 11 thru 16) are to be written on the blank part of the question paper below the question. . Questions 1 thru 10 are worth 3 points each, questions 11 thru 16 are worth 12 points each. The total number of points available is 102. As a rough guide, at least initially, you should not spend more than 5 minutes on a question in part I nor more than 20 minutes on a question in part II. - . This examination booklet consists of this cover, pages 2 through 9 containing questions and two further continuation pages. . You are expected to show all your work. Write your solution in the space provided below the question. 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! You are advised to spend the first minute or two scanning the problems. (Please inform the invigilator if you find that your booklet is defective.) 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. This is a closed book exam. No notes, calculators, mobile phones, PDA's etc. are allowed. PLEASE DO NOT WRITE INSIDE THIS BOX choice I30 and gr total [102 CalculusB MATH 222 Final Exam — December 17, 2004 .. Version 1 Part I — Answer on the computer readable sheet 1. (3 points) When Taylor’s Theorem is applied to (1 + 42:)? at m = 0 one obtains (1 +4a:)‘i‘ = 1 +2x+R(x) where the precise form of the Lagrange Remainder R(:c) is given by (A) —4(1+4w)—%u2, (B) —2(1+4u)"%x.2, (C) —-(1+4u)_%m2, (D) —(1+4:t:)_%sc2, (B) —4(1+4u)-%u2. where u is an unknown number between 0 and m. 2. (3 points) The Maclaurin series of 1n(1 + 2x + 32) up to the term in :53 is (A) 2m + $2 + £733 ' (B) 23 “ '32 + #73 (C) 2:1: +2:2 + §m3 (D) 21: -- m2 + 325-33 (E) 25:: +w2 — %w3 3. (3 points) The power series W 211. 271. a; (n + 3) :1: has radius of convergence (m1, m)fi. (01% (mg, m)w 4. (3 points) The surface 2 = 3:2 + 2m; + 43,2 has a horizontal tangent plane at M)mm=mm, ®)mm2cwu m)[email protected] (D) (may) = (_%10)’ (E) (:3: y) 2 ("1! 1)‘ _ 5. (3 points) Let g(a:, y) = f (:1:2 -— 2953;) with f a differentiable function of one variable, then necessarily __ 39 69 __ _ 59 _ 69' _ _ 69. _89 u (A) (3: m5; +$-6_w — 0, a (B)a(a: way 3:653 w— 0, a (C)a(z wan: +5563; — 0, (D) (a: 2;) am a: By , (E) (a: + 3;) 6:5 :1: 6y 0. 6. (3 points) The rate of increase of the function 2:2 + yz per unit length at the point (2, 2, 2) in the direction of the point (1, 4, 4) is (A) 0. (3) J2: (C) 3 (D) El; (E) 2- Answer on computer readable card 2 Exam continued overleaf. . . Calculus3 MATH 222 Final Exam— December 17, 2004 ' Version 1 r , ' 7. (3 points) Consider the Taylor expansion of g(t) = / 3a: sin(m)d:r about t = 0. Rounded to six decimal 0 places, the value of g(0.1) is (A) 0.900999, (B) 0.000998, (C) 0.001901, (D) 0.001002, (E) 0.000995. 8. (3 points) The volume of the region of three dimensional space given by 0 S 2 S 4 — :52 —— y2 is (A) 123, (B) 4n, (C) SW, (D) 3331,03) 12w. 9. (3 points) The volume ofa circular cone of radius r and height h is énrzh. Suppose that the height of the cone is increasing at a rate of 3 cm/sec and that the radius is decreasing at a rate of 2 cm/sec. Then at an instant when the height is 30 cm and the volume is 1000a cubic cm, the volume is (A) increasing at a rate of 10071" cubic cmfsec. (B) increasing at a rate of 3001r cubic cm/sec. (C) decreasing at a rate of 10071- cubic CID/sec. (D) decreasing at a rate of 300w cubic cm/sec. (E) increasing at a rate of 500w cubic cm/sec. 10. (3 points) A region R of three dimensional space is given by the constraints Z2 2 3:2 +312 and 3:2 +y2+z2 5 4. The integral ff My, nuance R transforms into spherical coordinates as 7r/2 2 211’ (A) f f f(x,y,z)p2sin(¢)d6dpd¢ (6:0 p=0 9:0 arr/2 4 2n (B) f f f(w,y,z)p2(sin(¢))2d0dpdqs ¢=0 11:!) 9:0 «[2 2 2n 2 ' 2 (C) [H fp=nf6=onmap (sm(¢)) dsdpdqs 1r/4 2 2n (D) f j f(w,y,z)p2sin(¢)d6dpd¢ ago p=0 9:0 7r/4 4 2n (E) j ] nay,z)p2(sin(¢))2d9dpd¢ .520 p=0 9:0 where in the answers, it is to be assumed that f (2:, y, 2:) has been correctly rewritten in terms of p, to and 6. Answer on computer readable card 3 Exam continued overleaf. . . Calculu53 MATH 222 Final Exam — December 17, 2004 _ Version 1 Part II — Answer in the space provided beiow the question 11. (12 points) Find and classify the critical points of the function f (9:, y) = 2:53 + 79:2 — 22:11 + y2 in the whole (9:, y)-Plane- Continue solution opposite 4 Exam continued overleaf. . . then on page - Calculus3 MATH 222 Final Exam — December 17, 2004 Version 1 3 y - 12. (12 points) Suppose that the integral / f f (:3, y) dm dy is expressed as an area integral ff f (2:, y) dA yzfl an=1," —2y R . over a region of R of the plane. (i) Describe the region R and draw a diagram showing R. (ii) Express this same integral in terms of one or more iterated integrals in which the order of integration has been reversed. Continue solution op osite then on page E 5 Exam contmued overleaf. . . CalculusB MATH 222 Final Exam w December 17, 2004 ' Version 1 13. (12 points) ' Suppose that z = z(:r:, y) is determined implicitly in terms of a: and y by the equation 23.7: + zy2 + 3mg = 5 62z 3w6y(1'1)' in such a way that 2(1, 1) = 1. Find 3—:(1, 1), 3—:(1, 1) and Continue solution opposite 6 Exam continued overleaf. . . then on page CalculusS MATH 222 Final Exam - December 17, 2004 I Version I 14. (12 points) Use cylindxical coordinates to evaluate ff/Zyzdmdydz R . where R is the region of 3-space given by the two inequalities y 2 0 and 0 g z <_Z 1 — m2 -— ya. Continue solution opposite 7 Exam continued overleaf. . . then on page Calculus3 MATH 222 Final Exam - December 17, 2004 Version I 15. (12 points) Use the method of Lagrange multipliers to find the maximum value of the function f (a: ,y, z): 9343;; 22 ontheportionoftheplaneSz+3y+4z _. 18h1thequadrantw_ > 0, y_ > 0, z > 0. Continue solution 0 posite 8 Exam continued overleaf. . . then on page Calculus3 MATH 222 Final Exam— December 17, 2004 Version 1 16. (i) (6 points) Find the arclength of the helix t |—} (315, 4cos(t), —4 sin(t)) between the points (0,4, 0) and (371-, —4, 0). ‘ (ii) (6 points) Find the surface area of the portion of the surface z = my lying above the disk :32 + y2 S 1 in the (ac, y)-plane. - - Continue solution opposite 9 Exam continued overleaf. . . then on page CalculusB MATH 222 Final Exam - December 17, 2004 Version 1 CONTINUATION PAGE FOR PROBLEM NUMBER You must refer to this continuation page on the page where the problem is printed! Continue solution opposite |:‘ ' 10 Exam continued overleaf. . . then on page Calculus3 MATH 222 Final Exam -— December 17, 2004 Version 1 CONTINUATION PAGE FOR PROBLEM NUMBER You must refer to this continuation page on the page where the problem is printed! Continue solution opposite I 1 then on the back ...
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