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# CALCULUS II MA 114

UK

GPA 3.54

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This 7 page Class Notes was uploaded by Kennith Herman on Friday October 23, 2015. The Class Notes belongs to MA 114 at University of Kentucky taught by Staff in Fall. Since its upload, it has received 23 views. For similar materials see /class/228154/ma-114-university-of-kentucky in Mathematics (M) at University of Kentucky.

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Date Created: 10/23/15

Calculus U M14114 007 009 Fall 2003 Russell Brown The second exam will be given on Tuesday evening 21 October in CB 122 The exam will cover the sections 71 76 and 78 79 Students should review the trigonometric identities from the rst exam Please know the entries 1 17 on the integral table in the back of Stewart H E0 00 r U 03 5 H Integration by parts Students should be able to recognize integrals which can be evaluated using integration by parts Trigonometric integrals Know the strategy for evaluating fsinkz cos lz dz when j and k are non negative integers Trigonometric substitution Use of the substitution z sinu to evaluate integrals of the form fzk a2 7 y dz Integration of rational functions by partial fractions You do not need to be able to use the reduction formula to integrate functions where the denominator is an irreducible quadratic raised to a power larger than one Rationalizing substitutions Substituting u 7 ax b Approximate integration Memorize Simpson7s rule and the trapezoid rule Be able to use the error estimates to estimate the accurracy of the integration You do not need to memorize the error estimates Evaluate improper integrals Be able to state and use the comparison theorem Sample exams Compute 5 of the 6 inde nite integrals below Write here the letter of the integral that is not to be graded If you do not specify the an integral which is not to be graded7 we will take the ve lowest scores a sin2zdz b sin3zdz C d d z22z2 z e zcos2z dz dz F 9 f dz Find the form of the partial fraction decomposition for the following rational functions DO NOT SOLVE FOR THE CONSTANTS a W z2 7 43x b x 13z 2z2 7 4 z 1 z2z2 12z2 2x 1 C The trapezoid rule Tn and Simpson7s rule Sn for approximating the integral 7 fz dz are Tn gov 2fz1 2fxn1 f s 7 gum 4fz1 2mg 4fltzsgt 4f W71 We The errors satisfy M20 7 13 12712 M40 7 15 E lt l Tl 180714 and EA 3 where Mk is a number which satis es lfkzl S Mk for all z with a S x S b D Use the trapezoid rule and Simpson7s rule with n 4 to approximate the integral 6 sin2z dz 3 Give your answers correctly rounded to four decimal places Find 71 so that the error in the trapezoid rule is at most 10 47 6 sin2z dz 7 Tn S 1074 3 A O V Find 71 so that the error in Simpson7s rule is at most 1047 6 sin2z dz 7 Sn 3 1074 3 A D V State the comparison theorem for improper integrals A C7 V Use the comparison theorem to determine if the following improper integrals converge U H i sin2xe dx 1 H 00 2sinx 11 1 s dx There are thirty two teams in the 2002 FIFA Copa del Mundo Thus it is interesting to know the value of the integral 00 3264 dx 0 Suppose that we know that 00 msle m dx A 0 Find a simple expression involving A which gives the value of the integral 00 msze m dx 0 It is known that A 8 222838654 177 922817 725562880 000 000 but I doubt this is very helpful Compute the following integrals if possible If an improper integral diverges say so 7r2 a sin3 x dx 0 b sin2 mdx 7r2 sinx C 0 1 cos2 x z 1 d zem dx 0 2 x d e x2 4 z 1 f x2 3 dz 1 d g 2 5 x h 1 1 d x 0 V1 7 x2 7r2 i tan x dx 0 1 lt1 0 1 2 dx 2 Which of the integrals a7 c7 h7 i and j are improper For each of these integrals7 give the point or points where it is improper 3 a Give the de nition of a rational function Give an example of a rational function b Give the de nition of a proper rational function Give examples of a proper rational function and an improper rational function 4 a Find the anti derivative l 7 x2 dx b Use your answer to compute 1 v1 7 x2 dz 0 c The integral in part b represents the area of a familiar region Use a geometric argument to give the area and check your answer to part b There were additional problems on the original exam however these problems are omitted from the review sheet because they examine material that was not covered in this course October 127 2003 Calculus U Fall 2003 MAZZ439UU7 009 Russell Brown The third exam will take place 730 930 on Tuesday7 18 November 2003 in CB122 The exam will cover 817 82 and 1017108 The topics to be examined are H Separable differential equations and applications E0 Arc length of graphs 9 Sequences Monotonic sequences 7 Convergent series Telescoping series7 geometric series 9 lntegral test and its use in estimating series 03 Comparison test and limit comparison test I Alternating series and the estimate for the error 00 Absolute convergence7 conditional convergence Ratio test p 1 0 Convergence of power series7 radius of convergence Review assignment Chapter 87 page 5417 1787 9a7 257 267 27 Chapter 107 page 6757676 1797 117 127 14760 Sample exam questions 1 Find the following limits 2n2 7 1 1 a L530 3712 987 n b l nin o n 1 2 Find the sum of each of the following series 1 1 1 1 1 a 7 mn n 0 1 WE lt 1 1n n1 3 Determine if each of the following series is absolutely convergent7 conditionally convergent or divergent Describe brie y how you test the series for convergence or divergence 00 1 a 2 3n 1 i0 1 b n1 00 C 2P3 n1 4 Let W 1 1 s Z and 5N n1 n5 n1 n5 Use the integral test to nd a value of N so that ls 7 le lt 10 5 5 Find the radius and interval of convergence for the following power series 2672 b 2 2n 1 c i n2 7 5 6 a State the alternating series estirnation theorern b Find a value of N so that 1 Find the limits of the following sequences 712 1 l a 21 212 1 n b 1 301n 1 C lirn Hoe 1 2n 2 9 9 Suppose that a sequence 117027 is de ned by 1 an147 and a1 1 an It can be shown that this sequence is bounded and increasing Assuming that the sequence is bounded and increasing7 explain why the sequence has a limit and compute the exact value of the limit lim an Hoe You may check your answer by nding an approximate value for the limit using your calculator Determine if each of the following series converges and7 for the convergent series7 compute the sum of the series 00 27L a n0 Z 2 n4 iltgini2gt n a State the integral test for convergence of a series b What N is needed so that 00 i 7 i lt 10 57 n1 5 n1 5 Do not evaluate the partial sum For which values of z does the series 00 2 7 3 H H COHVSI gS November 117 2003

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