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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', /$Zsin(ra)dz. /%(lna:)”d.t); - log rules (another particular weakness), limit rules, derivative rules: L’HR for limits of indeterminate forms. MATH 138 FINAL EXAIVIINATION INFORMATION VVIN'I‘ER 2009 o For problem 1: - how to recognize which of I of P, PFD, or substitution (including trigonometric substitutions) is suitable for a given integral;
— deﬁnitions of improper integrals of Type 1 and Type 2, and one—sided limits for Type 2:
— LSR. LPR, LQR, LCR for sequence limits. 0 For problems 2-3: how to set up Riemann integrals for area, volume of revolution, and arc length ; dL — ﬁnding arc length using speed = time of rate of change of distance = a = = ||x’ for vector curves.
so L =/ ||v||dt for a. S t S b;
~ how to sketch curves of the form x = (a coswt,bsinwt), x = (at.", bt'"), x = a(cos3 t,sin3 t) (see pages A18 - A23 of your text for an excellent review of conics, a ”must know"); how to solve separable and linear DES. check for equilibrium solutions, solve for constants using given data.
analyze and interpret direction ﬁelds; you do NOT need to know any speciﬁc applications - models will be
given. 0 For problems 4-5: how and when to apply 11““ TT, CT, LCT. IT, RT. AST; types of series for which RT fails: meaning of absolute versus conditional convergence;
so ﬁnding the interval of convergence for a power series 2 (2,,(2: — 0)";
using the error bounds from the Corollary of AST or IT; using known series to ﬁnd other Maclaurin series. Note: When an exam question asks you to use a ‘kn0wn’ series, you may use any of: 1. oo
“‘2 °° un
u_ _ _ i
.e —1+u+-2—!+---~n;0n! f01uelR
3 5 0° 2n+i
u u u,
.sinu:u—-—+——~--= —1"—f0 ER
3! 5! 7;} )(2n+1)! ”‘
U2 u“ °° 112"
. " =1—— ——---= —-1" f ‘
cosu 2[-1-4! "2:25 ) (2”)! or uElR oo
_1,_ _ __
(1+u)p=1+ZM—)(p—(n—l))u" for |ul<1 audpelR. I
You may also use that the ‘p—series‘ E n—p converges for p > 1 and diverges for p 5 1. n=1 MA’I‘H 138 FINAL EXAMINATION INFORMATION \VINTER 2009 o For problems 6: - how to ﬁnd PN(ar) for a given f($) 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
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
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
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
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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.
- Winter '07