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Unformatted text preview: MATH 138 FINAL EXAMINATION INFORMATION VVIN'I‘ER 2009 1. Ln 7. The final examination in MATH 138 will be held TUESDAY, APRIL 14, 2009, 9:00 - 11:30 a.m. in the PAC (except SEC 005) (SEC 005 will write the exam in RCH 103, 105, 110, 112) An open tutorial on Assignment. 10 will be held 4:30 - 6:30 pm. on April 6 in RCH 10]. Prior to the final examination, tutorial assistance will be available in MC 4066 on Sunday. April 12th, from 12:00 noon to 5:00 pm. and Monday. April 13th, from 9:00 am. to 5:00 pm. There will be an open review session on Monday, April 13th, from 3:00 pm. to 5:00 pm. in DC 1350, with a full discussion of solutions for Sample Exam problems and related questions. In addition, individual instructors may hold review sessions for their own sections - check with your instructor. . A sample exam (from Winter 2008) is posted on this website. The math society also has some previous exams from which sOme problems are usefuli . Your final grade will be caluclated as follows: the 2 Maple Labs plus the best 8 of your 9 assignments (10%). plus your two tests (30%, with the best test weighted 20%. and the other weighted 10%). plus your final exani (60%) equals your final grade (100%). NOTE: If you have missed a test and have not given your instructor a valid medical form. a grade of 0 will be used. . Your final examination is structured as follows: 0 Problem 1 deals with basic integration techniques, improper integrals, and sequence limits using the limit theorems. (~15-20 marks) 0 Problems 2 and 3 involve applications (areas, volumes, curve sketches and arc length for vector functions x = F(t) = (x(t). y(t)) in R2, separable and linear DES, direction field analysis. DE models). (~25-30 marks) 0 Problems 4 and 5 are on series convergence, power series, and operations. (~25 marks) 0 Problem 6 is on Taylor polynomials, the Remainder Theorem, Taylor’s Inequality. (~10 marks) 0 Problems 7 and 8 are theoretical, involving definitions. theorems. and proofs of theorems, and applying them to short-answer true/false questions. (~20 marks) What you need to know: 0 From Calculus 1, Math 137: - properties and graphs of basic functions (mp. 63.11] I. sin 1‘, cos x. tan :5. sec 1:, arcsinzt. arctan I how to find derivatives and antiderivatives of all basic functions. A Particular weaknesses we‘ve noted 1 1 1 are antiderivatives of the form / —d:z: (erg. / —2d:c. / —d;t). and of inverse functions (eg. :5? :1: r2/3 /1n:rd:c. /arctana:d1x farcsin cedar), as well as trigonometric substitutions. ,. rc - the method of substitution for integrals (eg, / ax2+b (11', /$Zsin(ra)dz. /%(lna:)”d.t); - log rules (another particular weakness), limit rules, derivative rules: L’HR for limits of indeterminate forms. MATH 138 FINAL EXAIVIINATION INFORMATION VVIN'I‘ER 2009 o For problem 1: - how to recognize which of I of P, PFD, or substitution (including trigonometric substitutions) is suitable for a given integral; — definitions of improper integrals of Type 1 and Type 2, and one—sided limits for Type 2: — LSR. LPR, LQR, LCR for sequence limits. 0 For problems 2-3: how to set up Riemann integrals for area, volume of revolution, and arc length ; dL — finding arc length using speed = time of rate of change of distance = a = = ||x’ for vector curves. b ; so L =/ ||v||dt for a. S t S b; 0 ~ how to sketch curves of the form x = (a coswt,bsinwt), x = (at.", bt'"), x = a(cos3 t,sin3 t) (see pages A18 - A23 of your text for an excellent review of conics, a ”must know"); how to solve separable and linear DES. check for equilibrium solutions, solve for constants using given data. analyze and interpret direction fields; you do NOT need to know any specific applications - models will be given. 0 For problems 4-5: how and when to apply 11““ TT, CT, LCT. IT, RT. AST; types of series for which RT fails: meaning of absolute versus conditional convergence; so finding the interval of convergence for a power series 2 (2,,(2: — 0)"; n=0 using the error bounds from the Corollary of AST or IT; using known series to find other Maclaurin series. Note: When an exam question asks you to use a ‘kn0wn’ series, you may use any of: 1. oo 1_u=1+u+u2+-~-=Zu"f0r |u|<1 n=0 “‘2 °° un u_ _ _ i .e —1+u+-2—!+---~n;0n! f01uelR 3 5 0° 2n+i u u u, .sinu:u—-—+——~--= —1"—f0 ER 3! 5! 7;} )(2n+1)! ”‘ U2 u“ °° 112" . " =1—— ——---= —-1" f ‘ cosu 2[-1-4! "2:25 ) (2”)! or uElR oo _1,_ _ __ (1+u)p=1+ZM—)(p—(n—l))u" for |ul<1 audpelR. I n. n=l so i 1 You may also use that the ‘p—series‘ E n—p converges for p > 1 and diverges for p 5 1. n=1 MA’I‘H 138 FINAL EXAMINATION INFORMATION \VINTER 2009 o For problems 6: - how to find PN(ar) for a given f($) and centre :c = 0, using f(a). f’(a)...., or a known series: — statements of TRT and Taylor’s Inequality, and how to bound the error in approximating f by PN(:r) or b b / f by / PN($)drr; using substitutions in these inequalities. a a o For problems 7-8: - precise statements of all definitions and theorems used in the course; — how to prove lim f (as) = L or lim 3:" = L using the formal definitions of limit: THO n—roo — how to prove the following five theorems: (i) If u’ and v’ are continuous on [a. b], then b b / u(1:)v'(:r)d1' = [u(.1:)v(:r)]2 —/ v(u:)u’(1:)drc. (I by P, page 5 of Course Notes) (ii) If lim 33,, = p and lim yn = q, then lim (3,, + y") = p + q. (LSFL page 69 of CourseNotes) fl—‘m n—ooo n—ioo -x (iii) If f (1:) is continuous, positive, and decreasing on [1a, 00], and / f converges. then the series 1 00 2 an with a7] = f (11) also converges. (part of IT ~ pages 85-86 of Course Notes) 11:] co 00 iv If bum" converges, then ()n‘m" converges absolute] for all a: such that 1' < $0 0 Y n=0 "=0 (page 98 of Course Notes) (v) If f(1:) has bounded derivatives of all orders on an interval 1 containing :1: = a. then °° ('1) lim |f(a:) — PN(a:) = O, i.e.. = 2 f (a) (:r — a)". (CTSf on page 113 of Course Notes) N—wn n: n! 8. In studying for the exam, we recommend that you 0 make a list of the statements of all definitions and theorems and practice them with a friend (or make it a game with a group of friends!); 0 learn the theorem proofs by writing out the theorems, thinking about the reasoning behind each step, and then trying to write them again without looking; 0 go over your assignments. especially to see where you went wrong: retry those problems: 0 practice problems you haven‘t tried before from the assignments; 0 go over Test 1 and Test 2 and ensure you understand the solutions, and have corrected any mistakes. 0 when you feel confident you’ve reviewed everything, do the Sample Exam. Note: Just reading over solved problems is a passive activity. You will learn best by doing — math is a participatory sport! ...
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This note was uploaded on 09/04/2009 for the course MATH 138 taught by Professor Anoymous during the Winter '07 term at Waterloo.

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