MATH 2300 review problems for Exam 2
|
|
- Madlyn Wade
- 6 years ago
- Views:
Transcription
1 MATH 2300 review problems for Exam 2. A metal plate of costat desity ρ (i gm/cm 2 ) has a shape bouded by the curve y = x, the x-axis, ad the lie x =. Fid the mass of the plate. Iclude uits. Fid the ceter of mass of the plate. Iclude uits. 2. (Exercise 7 from Sectio 6.6 i Stewart s Calculus Cocepts ad Cotexts) Suppose that 2J of work is eeded to stretch a sprig from its atural legth of 30 cm to a legth of 42 cm. How much work is eeded to stretch the sprig from 35 cm to 40 cm? How far beyod its atural legth will a force of 30 N keep the sprig stretched? 3. (Exercise from Sectio 6.6 i Stewart s Calculus Cocepts ad Cotexts) Show how to approximate the required work by a Riema sum. The express the work as a itegral ad evaluate it: a heavy rope, 50 ft log, weighs 0.5 lb/ft ad hags over the edge of a buildig 20 ft high. How much work is doe i pullig the rope to the top of the buildig? How much work is doe i pullig half the rope to the top of the buildig? 4. (Exercise 5 from Sectio 6.6 i Stewart s Calculus Cocepts ad Cotexts) Show how to approximate the required work by a Riema sum. The express the work as a itegral ad evaluate it: a leaky 0-kg bucket is lifted from the groud to a height of 2m at a costat speed with a rope that weighs 0.8 kg/m. Iitially the bucket cotais 36 kg of water, but the water leads at a costat rate ad fiishes draiig just as the bucket reaches the 2-m level. How much work is doe? 5. Complete Exercise 9 from Sectio 6.6 o pg. 473 i Stewart s Calculus Cocepts ad Cotexts 6. Fid the limit of all the sequeces i the sequece activity: 7. Does {a }, where a =, coverge? If so, what does it coverge to? 8. Decide whether each of the followig sequeces coverges. If a series coverges, what does it coverge to? If ot, why ot? The sequece whose -th term is a =. The sequece whose -th term is b = +. The sequece whose -th term is c = cos(π). The sequece {d }, where d = 2 ad d = 2d for >. 9. Fid the sum of the series. For what values of the variable does the series coverge to this sum? + x 2 + x2 4 + x3 8 +
2 y y 2 + y 3 y z + z2 3 + z For each of the followig series, determie whether or ot they coverge. If they coverge, determie what they coverge to. ( ) = ( π) 2 =3 e (.07) + = For each of the followig series, determie if it coverges absolutely, coverges coditioally, or diverges. Completely justify your aswers, icludig all details. (f) (g) (h) (i) =2 =2 =2 ( ) + 5 (l ) 2 (l ) ( 3)! ( 4 2 ( 3) + + ) 2 ( ) arcta
3 (j) (k) (l) si 2 ( +! ) 2. Cosider the series ( )! Cofirm usig the Alterate Series Test that the series coverges. How may terms must be added to estimate the sum to withi.000? Estimate the sum to withi How may terms of ( ) should be added to estimate the sum to withi.0? No calculators. 4. Check whether the followig series coverge or diverge. I each case, give the aswer for covergece, ad ame the test you would use. If you use a compariso test, ame the series b you would compare to. (f) (g) (h) (i) ( 2 + ) diverges use itegral or limit compariso test, comparig to ( 2 + ) 2 coverges term-size compariso test with 3 ( + ) diverges divergece test (th term test) (that is, the the term does ot go to zero) diverges limit compariso test or th term test ( ) si 2 (hit: cosider 2! coverges ratio test (2)! ( + 3)! diverges ratio test! ( + 2)! coverges simplify to compariso test.! coverges ratio test ) coverges limit compariso test 2 (+)(+2) ad the use term-size compariso test or limit
4 5. Cosider the series l. Are the followig statemets true or false? Fully justify your aswer. The series coverges by limit compariso with the series The series coverges by the ratio test. False The series coverges by the itegral test. False 6. Cosider the series aswer.. False ( ) l. Are the followig statemets true or false? Fully justify your The series coverges by limit compariso with the series The series coverges by the ratio test.false The series coverges by the itegral test. False The series coverges by the alteratig series test.true The series coverges absolutely. False.False 7. Suppose the series a is absolutely coverget. Are the followig true or false? Explai. a is coverget.true, because if a series coverges absolutely, it must coverge The sequece a is coverget. lim = 0. ( ) a is coverget.true, i fact it coverges absolutely. true, because a coverges, so by the Divergece Test, The sequece a coverges to. False, the sequece coverges to 0 by the Divergece Test. a is coditioally coverget. False (f) a coverges. True, this ca be show usig the term-size compariso test. 8. Does the followig series coverge or diverge? 3 2 You must justify your aswer to receive credit. Yes, by the ratio test, because, if a deotes the th term of this series, the the limit of a + /a as is zero (which is less tha ). 9. A ball is dropped from a height of 0 feet ad bouces. Assume that there is o air resistace. Each bouce is 3 4 of the height of the bouce before. () Fid a expressio for the height to which the ball rises after it hits the floor for the th time. (2) Fid a expressio for the total vertical distace the ball has traveled whe it hits the floor for the th time. (3) Usig without proof the fact that a ball dropped from a height of h feet reaches the groud i h/4 secods: Will the ball bouce forever? If ot, how log it will take for the ball to come to rest?!
5 Wat more practice? Here s some more! 20. I theory, drugs that decay expoetially always leave a residue i the body. However i practice, oce the drug has bee i the body for 5 half-lives, it is regarded as beig elimiated. If a patiet takes a tablet of the same drug every 5 half-lives forever, what is the upper limit to the amout of drug that ca be i the body? 2. Let {f } be the sequece defied recursively by f = 5 ad f = f Check that the sequece g whose -th term is g = satisfies this recurrece relatio, ad that g = 5. (This tells us g = f for all.) Use the result of part to fid f 20 quickly. 22. Fid the values of a for which the series coverges/diverges: (f) (g) (h) (i) ( 2a ( a 2 ( 2 ) ) a ) (l a) (l ) a ( + a ) ( + a) l a a l 23. Determie if these sequeces coverge absolutely, coverge coditioally or diverge. cos 2 coverges absolutely, take the absolute value ad compare to usig the term-size 2 compariso test
6 ( ) 2 coverges coditioally. Take the absolute value, the use limit compariso, + comparig to to show it does ot coverge absolutely. Use alteratig series test to show origial series coverges (ad thus coverges coditioally) For each of the followig statemets, determie if it is true Always, Sometimes or Never. 24. If a sequece a coverges, the the sequece ( ) a also coverges. 25. If a sequece ( ) a coverges to 0, the the sequece a also coverges to The average value of a fuctio is egative. 27. The geometric series 28. The geometric series 29. If 5r coverges to c 5c coverges to 5 4 a coverges, the 30. If a series 3. If the series 32. If the series a coverges. 5 r a is coverget, the lim a = 0 ( ) a is coditioally coverget, the the sequece a coverges. ( ) a is coverget, the the series 33. If lim a + a =, the a coverges coditioally. 34. If the series a diverges, the the series a is coverget. a diverges.
MATH 2300 review problems for Exam 2
MATH 2300 review problems for Exam 2. A metal plate of costat desity ρ (i gm/cm 2 ) has a shape bouded by the curve y = x, the x-axis, ad the lie x =. Fid the mass of the plate. Iclude uits. (b) Fid the
More informationMATH 2300 review problems for Exam 2
MATH 2300 review problems for Exam 2. A metal plate of costat desity ρ (i gm/cm 2 ) has a shape bouded by the curve y = x, the x-axis, ad the lie x =. (a) Fid the mass of the plate. Iclude uits. Mass =
More informationMIDTERM 2 CALCULUS 2. Monday, October 22, 5:15 PM to 6:45 PM. Name PRACTICE EXAM
MIDTERM 2 CALCULUS 2 MATH 23 FALL 218 Moday, October 22, 5:15 PM to 6:45 PM. Name PRACTICE EXAM Please aswer all of the questios, ad show your work. You must explai your aswers to get credit. You will
More informationAlternating Series. 1 n 0 2 n n THEOREM 9.14 Alternating Series Test Let a n > 0. The alternating series. 1 n a n.
0_0905.qxd //0 :7 PM Page SECTION 9.5 Alteratig Series Sectio 9.5 Alteratig Series Use the Alteratig Series Test to determie whether a ifiite series coverges. Use the Alteratig Series Remaider to approximate
More informationCarleton College, Winter 2017 Math 121, Practice Final Prof. Jones. Note: the exam will have a section of true-false questions, like the one below.
Carleto College, Witer 207 Math 2, Practice Fial Prof. Joes Note: the exam will have a sectio of true-false questios, like the oe below.. True or False. Briefly explai your aswer. A icorrectly justified
More informationMath 132, Fall 2009 Exam 2: Solutions
Math 3, Fall 009 Exam : Solutios () a) ( poits) Determie for which positive real umbers p, is the followig improper itegral coverget, ad for which it is diverget. Evaluate the itegral for each value of
More informationSCORE. Exam 2. MA 114 Exam 2 Fall 2016
Exam 2 Name: Sectio ad/or TA: Do ot remove this aswer page you will retur the whole exam. You will be allowed two hours to complete this test. No books or otes may be used. You may use a graphig calculator
More informationMath 116 Second Midterm November 13, 2017
Math 6 Secod Midterm November 3, 7 EXAM SOLUTIONS. Do ot ope this exam util you are told to do so.. Do ot write your ame aywhere o this exam. 3. This exam has pages icludig this cover. There are problems.
More informationSOLUTIONS TO EXAM 3. Solution: Note that this defines two convergent geometric series with respective radii r 1 = 2/5 < 1 and r 2 = 1/5 < 1.
SOLUTIONS TO EXAM 3 Problem Fid the sum of the followig series 2 + ( ) 5 5 2 5 3 25 2 2 This series diverges Solutio: Note that this defies two coverget geometric series with respective radii r 2/5 < ad
More informationSCORE. Exam 2. MA 114 Exam 2 Fall 2017
Exam Name: Sectio ad/or TA: Do ot remove this aswer page you will retur the whole exam. You will be allowed two hours to complete this test. No books or otes may be used. You may use a graphig calculator
More informationMTH 142 Exam 3 Spr 2011 Practice Problem Solutions 1
MTH 42 Exam 3 Spr 20 Practice Problem Solutios No calculators will be permitted at the exam. 3. A pig-pog ball is lauched straight up, rises to a height of 5 feet, the falls back to the lauch poit ad bouces
More informationPlease do NOT write in this box. Multiple Choice. Total
Istructor: Math 0560, Worksheet Alteratig Series Jauary, 3000 For realistic exam practice solve these problems without lookig at your book ad without usig a calculator. Multiple choice questios should
More informationMath 116 Practice for Exam 3
Math 6 Practice for Exam Geerated October 0, 207 Name: SOLUTIONS Istructor: Sectio Number:. This exam has 7 questios. Note that the problems are ot of equal difficulty, so you may wat to skip over ad retur
More informationSCORE. Exam 2. MA 114 Exam 2 Fall 2016
MA 4 Exam Fall 06 Exam Name: Sectio ad/or TA: Do ot remove this aswer page you will retur the whole exam. You will be allowed two hours to complete this test. No books or otes may be used. You may use
More informationRead carefully the instructions on the answer book and make sure that the particulars required are entered on each answer book.
THE UNIVERSITY OF WARWICK FIRST YEAR EXAMINATION: Jauary 2009 Aalysis I Time Allowed:.5 hours Read carefully the istructios o the aswer book ad make sure that the particulars required are etered o each
More informationINFINITE SERIES PROBLEMS-SOLUTIONS. 3 n and 1. converges by the Comparison Test. and. ( 8 ) 2 n. 4 n + 2. n n = 4 lim 1
MAC 3 ) Note that 6 3 + INFINITE SERIES PROBLEMS-SOLUTIONS 6 3 +
More information4x 2. (n+1) x 3 n+1. = lim. 4x 2 n+1 n3 n. n 4x 2 = lim = 3
Exam Problems (x. Give the series (, fid the values of x for which this power series coverges. Also =0 state clearly what the radius of covergece is. We start by settig up the Ratio Test: x ( x x ( x x
More informationMath 113 Exam 4 Practice
Math Exam 4 Practice Exam 4 will cover.-.. This sheet has three sectios. The first sectio will remid you about techiques ad formulas that you should kow. The secod gives a umber of practice questios for
More informationSolutions to quizzes Math Spring 2007
to quizzes Math 4- Sprig 7 Name: Sectio:. Quiz a) x + x dx b) l x dx a) x + dx x x / + x / dx (/3)x 3/ + x / + c. b) Set u l x, dv dx. The du /x ad v x. By Itegratio by Parts, x(/x)dx x l x x + c. l x
More informationINFINITE SEQUENCES AND SERIES
11 INFINITE SEQUENCES AND SERIES INFINITE SEQUENCES AND SERIES 11.4 The Compariso Tests I this sectio, we will lear: How to fid the value of a series by comparig it with a kow series. COMPARISON TESTS
More informationMIDTERM 3 CALCULUS 2. Monday, December 3, :15 PM to 6:45 PM. Name PRACTICE EXAM SOLUTIONS
MIDTERM 3 CALCULUS MATH 300 FALL 08 Moday, December 3, 08 5:5 PM to 6:45 PM Name PRACTICE EXAM S Please aswer all of the questios, ad show your work. You must explai your aswers to get credit. You will
More informationSec 8.4. Alternating Series Test. A. Before Class Video Examples. Example 1: Determine whether the following series is convergent or divergent.
Sec 8.4 Alteratig Series Test A. Before Class Video Examples Example 1: Determie whether the followig series is coverget or diverget. a) ( 1)+1 =1 b) ( 1) 2 1 =1 Example 2: Determie whether the followig
More informationn=1 a n is the sequence (s n ) n 1 n=1 a n converges to s. We write a n = s, n=1 n=1 a n
Series. Defiitios ad first properties A series is a ifiite sum a + a + a +..., deoted i short by a. The sequece of partial sums of the series a is the sequece s ) defied by s = a k = a +... + a,. k= Defiitio
More informationREVIEW 1, MATH n=1 is convergent. (b) Determine whether a n is convergent.
REVIEW, MATH 00. Let a = +. a) Determie whether the sequece a ) is coverget. b) Determie whether a is coverget.. Determie whether the series is coverget or diverget. If it is coverget, fid its sum. a)
More informationInfinite Sequence and Series
Chapter 7 Ifiite Sequece ad Series 7. Sequeces A sequece ca be thought of as a list of umbers writte i a defiite order: a,a 2,a 3,a 4,...,a,... The umber a is called the first term, a 2 is the secod term,
More informationJANE PROFESSOR WW Prob Lib1 Summer 2000
JANE PROFESSOR WW Prob Lib Summer 000 Sample WeBWorK problems. WeBWorK assigmet Series6CompTests due /6/06 at :00 AM..( pt) Test each of the followig series for covergece by either the Compariso Test or
More informationReview for Test 3 Math 1552, Integral Calculus Sections 8.8,
Review for Test 3 Math 55, Itegral Calculus Sectios 8.8, 0.-0.5. Termiology review: complete the followig statemets. (a) A geometric series has the geeral form k=0 rk.theseriescovergeswhe r is less tha
More informationNATIONAL UNIVERSITY OF SINGAPORE FACULTY OF SCIENCE SEMESTER 1 EXAMINATION ADVANCED CALCULUS II. November 2003 Time allowed :
NATIONAL UNIVERSITY OF SINGAPORE FACULTY OF SCIENCE SEMESTER EXAMINATION 003-004 MA08 ADVANCED CALCULUS II November 003 Time allowed : hours INSTRUCTIONS TO CANDIDATES This examiatio paper cosists of TWO
More informationBC: Q401.CH9A Convergent and Divergent Series (LESSON 1)
BC: Q40.CH9A Coverget ad Diverget Series (LESSON ) INTRODUCTION Sequece Notatio: a, a 3, a,, a, Defiitio: A sequece is a fuctio f whose domai is the set of positive itegers. Defiitio: A ifiite series (or
More informationMath 106 Fall 2014 Exam 3.2 December 10, 2014
Math 06 Fall 04 Exam 3 December 0, 04 Determie if the series is coverget or diverget by makig a compariso (DCT or LCT) with a suitable b Fill i the blaks with your aswer For Coverget or Diverget write
More informationIn this section, we show how to use the integral test to decide whether a series
Itegral Test Itegral Test Example Itegral Test Example p-series Compariso Test Example Example 2 Example 3 Example 4 Example 5 Exa Itegral Test I this sectio, we show how to use the itegral test to decide
More informationCHAPTER 10 INFINITE SEQUENCES AND SERIES
CHAPTER 10 INFINITE SEQUENCES AND SERIES 10.1 Sequeces 10.2 Ifiite Series 10.3 The Itegral Tests 10.4 Compariso Tests 10.5 The Ratio ad Root Tests 10.6 Alteratig Series: Absolute ad Coditioal Covergece
More informationPLEASE MARK YOUR ANSWERS WITH AN X, not a circle! 1. (a) (b) (c) (d) (e) 3. (a) (b) (c) (d) (e) 5. (a) (b) (c) (d) (e) 7. (a) (b) (c) (d) (e)
Math 0560, Exam 3 November 6, 07 The Hoor Code is i effect for this examiatio. All work is to be your ow. No calculators. The exam lasts for hour ad 5 mi. Be sure that your ame is o every page i case pages
More informationSeries III. Chapter Alternating Series
Chapter 9 Series III With the exceptio of the Null Sequece Test, all the tests for series covergece ad divergece that we have cosidered so far have dealt oly with series of oegative terms. Series with
More informationMath 116 Practice for Exam 3
Math 6 Practice for Eam 3 Geerated April 4, 26 Name: SOLUTIONS Istructor: Sectio Number:. This eam has questios. Note that the problems are ot of equal difficulty, so you may wat to skip over ad retur
More informationPractice Test Problems for Test IV, with Solutions
Practice Test Problems for Test IV, with Solutios Dr. Holmes May, 2008 The exam will cover sectios 8.2 (revisited) to 8.8. The Taylor remaider formula from 8.9 will ot be o this test. The fact that sums,
More informationSection 11.8: Power Series
Sectio 11.8: Power Series 1. Power Series I this sectio, we cosider geeralizig the cocept of a series. Recall that a series is a ifiite sum of umbers a. We ca talk about whether or ot it coverges ad i
More informationTesting for Convergence
9.5 Testig for Covergece Remember: The Ratio Test: lim + If a is a series with positive terms ad the: The series coverges if L . The test is icoclusive if L =. a a = L This
More informationFall 2018 Exam 2 PIN: 17 INSTRUCTIONS
MARK BOX problem poits 0 0 HAND IN PART 0 3 0 NAME: Solutios 4 0 0 PIN: 6-3x % 00 INSTRUCTIONS This exam comes i two parts. () HAND IN PART. Had i oly this part. () STATEMENT OF MULTIPLE CHOICE PROBLEMS.
More informationMa 530 Infinite Series I
Ma 50 Ifiite Series I Please ote that i additio to the material below this lecture icorporated material from the Visual Calculus web site. The material o sequeces is at Visual Sequeces. (To use this li
More informationMA131 - Analysis 1. Workbook 9 Series III
MA3 - Aalysis Workbook 9 Series III Autum 004 Cotets 4.4 Series with Positive ad Negative Terms.............. 4.5 Alteratig Series.......................... 4.6 Geeral Series.............................
More informationMath 116 Final Exam December 12, 2014
Math 6 Fial Exam December 2, 24 Name: EXAM SOLUTIONS Istructor: Sectio:. Do ot ope this exam util you are told to do so. 2. This exam has 4 pages icludig this cover. There are 2 problems. Note that the
More informationQuiz No. 1. ln n n. 1. Define: an infinite sequence A function whose domain is N 2. Define: a convergent sequence A sequence that has a limit
Quiz No.. Defie: a ifiite sequece A fuctio whose domai is N 2. Defie: a coverget sequece A sequece that has a limit 3. Is this sequece coverget? Why or why ot? l Yes, it is coverget sice L=0 by LHR. INFINITE
More informationChapter 10: Power Series
Chapter : Power Series 57 Chapter Overview: Power Series The reaso series are part of a Calculus course is that there are fuctios which caot be itegrated. All power series, though, ca be itegrated because
More informationCalculus 2 Test File Fall 2013
Calculus Test File Fall 013 Test #1 1.) Without usig your calculator, fid the eact area betwee the curves f() = 4 - ad g() = si(), -1 < < 1..) Cosider the followig solid. Triagle ABC is perpedicular to
More informationChapter 6: Numerical Series
Chapter 6: Numerical Series 327 Chapter 6 Overview: Sequeces ad Numerical Series I most texts, the topic of sequeces ad series appears, at first, to be a side topic. There are almost o derivatives or itegrals
More informationCalculus with Analytic Geometry 2
Calculus with Aalytic Geometry Fial Eam Study Guide ad Sample Problems Solutios The date for the fial eam is December, 7, 4-6:3p.m. BU Note. The fial eam will cosist of eercises, ad some theoretical questios,
More informationChapter 6 Overview: Sequences and Numerical Series. For the purposes of AP, this topic is broken into four basic subtopics:
Chapter 6 Overview: Sequeces ad Numerical Series I most texts, the topic of sequeces ad series appears, at first, to be a side topic. There are almost o derivatives or itegrals (which is what most studets
More informationAP Calculus Chapter 9: Infinite Series
AP Calculus Chapter 9: Ifiite Series 9. Sequeces a, a 2, a 3, a 4, a 5,... Sequece: A fuctio whose domai is the set of positive itegers = 2 3 4 a = a a 2 a 3 a 4 terms of the sequece Begi with the patter
More informationMath 5C Discussion Problems 2 Selected Solutions
Math 5 iscussio Problems 2 elected olutios Path Idepedece. Let be the striaght-lie path i 2 from the origi to (3, ). efie f(x, y) = xye xy. (a) Evaluate f dr. olutio. (b) Evaluate olutio. (c) Evaluate
More informationChapter 7: Numerical Series
Chapter 7: Numerical Series Chapter 7 Overview: Sequeces ad Numerical Series I most texts, the topic of sequeces ad series appears, at first, to be a side topic. There are almost o derivatives or itegrals
More informationf(x) dx as we do. 2x dx x also diverges. Solution: We compute 2x dx lim
Math 3, Sectio 2. (25 poits) Why we defie f(x) dx as we do. (a) Show that the improper itegral diverges. Hece the improper itegral x 2 + x 2 + b also diverges. Solutio: We compute x 2 + = lim b x 2 + =
More informationAns: a n = 3 + ( 1) n Determine whether the sequence converges or diverges. If it converges, find the limit.
. Fid a formula for the term a of the give sequece: {, 3, 9, 7, 8 },... As: a = 3 b. { 4, 9, 36, 45 },... As: a = ( ) ( + ) c. {5,, 5,, 5,, 5,,... } As: a = 3 + ( ) +. Determie whether the sequece coverges
More informationPart I: Covers Sequence through Series Comparison Tests
Part I: Covers Sequece through Series Compariso Tests. Give a example of each of the followig: (a) A geometric sequece: (b) A alteratig sequece: (c) A sequece that is bouded, but ot coverget: (d) A sequece
More informationMath 163 REVIEW EXAM 3: SOLUTIONS
Math 63 REVIEW EXAM 3: SOLUTIONS These otes do ot iclude solutios to the Cocept Check o p8. They also do t cotai complete solutios to the True-False problems o those pages. Please go over these problems
More information= lim. = lim. 3 dx = lim. [1 1 b 3 ]=1. 3. Determine if the following series converge or diverge. Justify your answers completely.
MTH Lecture 2: Solutios to Practice Problems for Exam December 6, 999 (Vice Melfi) ***NOTE: I ve proofread these solutios several times, but you should still be wary for typographical (or worse) errors..
More informationMath 106 Fall 2014 Exam 3.1 December 10, 2014
Math 06 Fall 0 Exam 3 December 0, 0 Determie if the series is coverget or diverget by makig a compariso DCT or LCT) with a suitable b Fill i the blaks with your aswer For Coverget or Diverget write Coverget
More informationMath 113 Exam 3 Practice
Math Exam Practice Exam 4 will cover.-., 0. ad 0.. Note that eve though. was tested i exam, questios from that sectios may also be o this exam. For practice problems o., refer to the last review. This
More information5 Sequences and Series
Bria E. Veitch 5 Sequeces ad Series 5. Sequeces A sequece is a list of umbers i a defiite order. a is the first term a 2 is the secod term a is the -th term The sequece {a, a 2, a 3,..., a,..., } is a
More informationMTH 246 TEST 3 April 4, 2014
MTH 26 TEST April, 20 (PLEASE PRINT YOUR NAME!!) Name:. (6 poits each) Evaluate lim! a for the give sequece fa g. (a) a = 2 2 5 2 5 (b) a = 2 7 2. (6 poits) Fid the sum of the telescopig series p p 2.
More informationMath 5C Discussion Problems 2
Math iscussio Problems Path Idepedece. Let be the striaght-lie path i R from the origi to (3, ). efie f(x, y) = xye xy. (a) Evaluate f dr. (b) Evaluate ((, 0) + f) dr. (c) Evaluate ((y, 0) + f) dr.. Let
More information(A sequence also can be thought of as the list of function values attained for a function f :ℵ X, where f (n) = x n for n 1.) x 1 x N +k x N +4 x 3
MATH 337 Sequeces Dr. Neal, WKU Let X be a metric space with distace fuctio d. We shall defie the geeral cocept of sequece ad limit i a metric space, the apply the results i particular to some special
More informationInfinite Sequences and Series
Chapter 6 Ifiite Sequeces ad Series 6.1 Ifiite Sequeces 6.1.1 Elemetary Cocepts Simply speakig, a sequece is a ordered list of umbers writte: {a 1, a 2, a 3,...a, a +1,...} where the elemets a i represet
More informationZ ß cos x + si x R du We start with the substitutio u = si(x), so du = cos(x). The itegral becomes but +u we should chage the limits to go with the ew
Problem ( poits) Evaluate the itegrals Z p x 9 x We ca draw a right triagle labeled this way x p x 9 From this we ca read off x = sec, so = sec ta, ad p x 9 = R ta. Puttig those pieces ito the itegralrwe
More informationSUMMARY OF SEQUENCES AND SERIES
SUMMARY OF SEQUENCES AND SERIES Importat Defiitios, Results ad Theorems for Sequeces ad Series Defiitio. A sequece {a } has a limit L ad we write lim a = L if for every ɛ > 0, there is a correspodig iteger
More informationMATH 31B: MIDTERM 2 REVIEW
MATH 3B: MIDTERM REVIEW JOE HUGHES. Evaluate x (x ) (x 3).. Partial Fractios Solutio: The umerator has degree less tha the deomiator, so we ca use partial fractios. Write x (x ) (x 3) = A x + A (x ) +
More informationMath 152 Exam 3, Fall 2005
c IIT Dept. Applied Mathematics, December, 005 PRINT Last ame: KEY First ame: Sigature: Studet ID: Math 5 Exam 3, Fall 005 Istructios. For the multiple choice problems, there is o partial credit. For the
More informationMath 116 Second Exam
Math 6 Secod Exam November, 6 Name: Exam Solutios Istructor: Sectio:. Do ot ope this exam util you are told to do so.. This exam has 9 pages icludig this cover. There are 8 problems. Note that the problems
More informationMath 113 (Calculus 2) Section 12 Exam 4
Name: Row: Math Calculus ) Sectio Exam 4 8 0 November 00 Istructios:. Work o scratch paper will ot be graded.. For questio ad questios 0 through 5, show all your work i the space provided. Full credit
More informationMATH 129 FINAL EXAM REVIEW PACKET (Revised Spring 2008)
MATH 9 FINAL EXAM REVIEW PACKET (Revised Sprig 8) The followig questios ca be used as a review for Math 9. These questios are ot actual samples of questios that will appear o the fial exam, but they will
More information10.2 Infinite Series Contemporary Calculus 1
10. Ifiite Series Cotemporary Calculus 1 10. INFINITE SERIES Our goal i this sectio is to add together the umbers i a sequece. Sice it would take a very log time to add together the ifiite umber of umbers,
More informationFINALTERM EXAMINATION Fall 9 Calculus & Aalytical Geometry-I Questio No: ( Mars: ) - Please choose oe Let f ( x) is a fuctio such that as x approaches a real umber a, either from left or right-had-side,
More informationMAT1026 Calculus II Basic Convergence Tests for Series
MAT026 Calculus II Basic Covergece Tests for Series Egi MERMUT 202.03.08 Dokuz Eylül Uiversity Faculty of Sciece Departmet of Mathematics İzmir/TURKEY Cotets Mootoe Covergece Theorem 2 2 Series of Real
More informationMathematics 116 HWK 21 Solutions 8.2 p580
Mathematics 6 HWK Solutios 8. p580 A abbreviatio: iff is a abbreviatio for if ad oly if. Geometric Series: Several of these problems use what we worked out i class cocerig the geometric series, which I
More informationCalculus II exam 1 6/18/07 All problems are worth 10 points unless otherwise noted. Show all analytic work.
9.-0 Calculus II exam 6/8/07 All problems are worth 0 poits uless otherwise oted. Show all aalytic work.. (5 poits) Prove that the area eclosed i the circle. f( x) = x +, 0 x. Use the approximate the area
More information7.) Consider the region bounded by y = x 2, y = x - 1, x = -1 and x = 1. Find the volume of the solid produced by revolving the region around x = 3.
Calculus Eam File Fall 07 Test #.) Fid the eact area betwee the curves f() = 8 - ad g() = +. For # - 5, cosider the regio bouded by the curves y =, y = 3 + 4. Produce a solid by revolvig the regio aroud
More informationProblem Cosider the curve give parametrically as x = si t ad y = + cos t for» t» ß: (a) Describe the path this traverses: Where does it start (whe t =
Mathematics Summer Wilso Fial Exam August 8, ANSWERS Problem 1 (a) Fid the solutio to y +x y = e x x that satisfies y() = 5 : This is already i the form we used for a first order liear differetial equatio,
More informationMath 113 Exam 3 Practice
Math Exam Practice Exam will cover.-.9. This sheet has three sectios. The first sectio will remid you about techiques ad formulas that you should kow. The secod gives a umber of practice questios for you
More information10.5 Positive Term Series: Comparison Tests Contemporary Calculus 1
0. Positive Term Series: Compariso Tests Cotemporary Calculus 0. POSITIVE TERM SERIES: COMPARISON TESTS This sectio discusses how to determie whether some series coverge or diverge by comparig them with
More informationMH1101 AY1617 Sem 2. Question 1. NOT TESTED THIS TIME
MH AY67 Sem Questio. NOT TESTED THIS TIME ( marks Let R be the regio bouded by the curve y 4x x 3 ad the x axis i the first quadrat (see figure below. Usig the cylidrical shell method, fid the volume of
More informationSection 8.5 Alternating Series and Absolute Convergence
Sectio 85 Alteratig Series ad Absolute Covergece AP E XC II Versio 20 Authors Gregory Hartma, PhD Departmet of Applied Mathema cs Virgiia Military Is tute Bria Heiold, PhD Departmet of Mathema cs ad Computer
More informationThe Ratio Test. THEOREM 9.17 Ratio Test Let a n be a series with nonzero terms. 1. a. n converges absolutely if lim. n 1
460_0906.qxd //04 :8 PM Page 69 SECTION 9.6 The Ratio ad Root Tests 69 Sectio 9.6 EXPLORATION Writig a Series Oe of the followig coditios guaratees that a series will diverge, two coditios guaratee that
More informationMATH 2411 Spring 2011 Practice Exam #1 Tuesday, March 1 st Sections: Sections ; 6.8; Instructions:
MATH 411 Sprig 011 Practice Exam #1 Tuesday, March 1 st Sectios: Sectios 6.1-6.6; 6.8; 7.1-7.4 Name: Score: = 100 Istructios: 1. You will have a total of 1 hour ad 50 miutes to complete this exam.. A No-Graphig
More informationSeries Review. a i converges if lim. i=1. a i. lim S n = lim i=1. 2 k(k + 2) converges. k=1. k=1
Defiitio: We say that the series S = Series Review i= a i is the sum of the first terms. i= a i coverges if lim S exists ad is fiite, where The above is the defiitio of covergece for series. order to see
More information11.6 Absolute Convergence and the Ratio and Root Tests
.6 Absolute Covergece ad the Ratio ad Root Tests The most commo way to test for covergece is to igore ay positive or egative sigs i a series, ad simply test the correspodig series of positive terms. Does
More information9.3 The INTEGRAL TEST; p-series
Lecture 9.3 & 9.4 Math 0B Nguye of 6 Istructor s Versio 9.3 The INTEGRAL TEST; p-series I this ad the followig sectio, you will study several covergece tests that apply to series with positive terms. Note
More informationMath 113, Calculus II Winter 2007 Final Exam Solutions
Math, Calculus II Witer 7 Fial Exam Solutios (5 poits) Use the limit defiitio of the defiite itegral ad the sum formulas to compute x x + dx The check your aswer usig the Evaluatio Theorem Solutio: I this
More informationINFINITE SEQUENCES AND SERIES
INFINITE SEQUENCES AND SERIES INFINITE SEQUENCES AND SERIES I geeral, it is difficult to fid the exact sum of a series. We were able to accomplish this for geometric series ad the series /[(+)]. This is
More informationSection 5.5. Infinite Series: The Ratio Test
Differece Equatios to Differetial Equatios Sectio 5.5 Ifiite Series: The Ratio Test I the last sectio we saw that we could demostrate the covergece of a series a, where a 0 for all, by showig that a approaches
More information5.6 Absolute Convergence and The Ratio and Root Tests
5.6 Absolute Covergece ad The Ratio ad Root Tests Bria E. Veitch 5.6 Absolute Covergece ad The Ratio ad Root Tests Recall from our previous sectio that diverged but ( ) coverged. Both of these sequeces
More informationConvergence: nth-term Test, Comparing Non-negative Series, Ratio Test
Covergece: th-term Test, Comparig No-egative Series, Ratio Test Power Series ad Covergece We have writte statemets like: l + x = x x2 + x3 2 3 + x + But we have ot talked i depth about what values of x
More information62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 +
62. Power series Defiitio 16. (Power series) Give a sequece {c }, the series c x = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + is called a power series i the variable x. The umbers c are called the coefficiets of
More informationMATH 166 TEST 3 REVIEW SHEET
MATH 66 TEST REVIEW SHEET Geeral Commets ad Advice: The studet should regard this review sheet oly as a sample of potetial test problems, ad ot a ed-all-be-all guide to its cotet. Aythig ad everythig which
More informationMath 116 Final Exam December 19, 2016
Math 6 Fial Exam December 9, 06 UMID: EXAM SOLUTIONS Iitials: Istructor: Sectio:. Do ot ope this exam util you are told to do so.. Do ot write your ame aywhere o this exam. 3. This exam has 3 pages icludig
More informationMATH 129 FINAL EXAM REVIEW PACKET (Spring 2014)
MATH 9 FINAL EXAM REVIEW PACKET (Sprig 4) The followig questios ca be used as a review for Math 9. These questios are ot actual samples of questios that will appear o the fial eam, but the will provide
More informationThe Interval of Convergence for a Power Series Examples
The Iterval of Covergece for a Power Series Examples To review the process: How to Test a Power Series for Covergece. Fid the iterval where the series coverges absolutely. We have to use the Ratio or Root
More information6.3 Testing Series With Positive Terms
6.3. TESTING SERIES WITH POSITIVE TERMS 307 6.3 Testig Series With Positive Terms 6.3. Review of what is kow up to ow I theory, testig a series a i for covergece amouts to fidig the i= sequece of partial
More informationis also known as the general term of the sequence
Lesso : Sequeces ad Series Outlie Objectives: I ca determie whether a sequece has a patter. I ca determie whether a sequece ca be geeralized to fid a formula for the geeral term i the sequece. I ca determie
More informationMATH Exam 1 Solutions February 24, 2016
MATH 7.57 Exam Solutios February, 6. Evaluate (A) l(6) (B) l(7) (C) l(8) (D) l(9) (E) l() 6x x 3 + dx. Solutio: D We perform a substitutio. Let u = x 3 +, so du = 3x dx. Therefore, 6x u() x 3 + dx = [
More informationTaylor Series (BC Only)
Studet Study Sessio Taylor Series (BC Oly) Taylor series provide a way to fid a polyomial look-alike to a o-polyomial fuctio. This is doe by a specific formula show below (which should be memorized): Taylor
More information