8.4. Click here for solutions. Click here for answers. OTHER CONVERGENCE TESTS. 3 n. 2n 1! sn 3. 2 n n 2. 3n n 1. 1 n 1 5 n 1 n n 2


 Lee Baldwin
 1 years ago
 Views:
Transcription
1 SECTION OTHER CONVERGENCE TESTS OTHER CONVERGENCE TESTS A Click here for aswers. S Click here for solutios. 4 Test the series for covergece or divergece s s 3 l 5 8 Approximate the sum of the series to the idicated accuracy. 5. 2! 6. 2! ! 0 2 s 3 2 l Determie whether the series is absolutely coverget, coditioally coverget, or diverget s ! arcta ! !! ! si 2 2 cos6 s 8 3! 2 2 2! ! (five decimal places) 6 Copyright 203, Cegage Learig. All rights reserved.
2 2 SECTION OTHER CONVERGENCE TESTS ANSWERS E Click here for exercises. S Click here for solutios.. Coverget 2. Coverget 3. Diverget 4. Coverget 5. Coverget 6. Diverget 7. Coverget 8. Coverget 9. Diverget 0. Diverget. Diverget 2. Coverget 3. Coverget 4. Coverget Absolutely coverget 20. D ive rg e t 2. Absolutely coverget 22. Absolutely coverget 23. Absolutely coverget 24. Absolutely coverget 25. D ive rg e t 26. D ive rg e t 27. Absolutely coverget 28. D ive rg e t 29. Absolutely coverget 30. Absolutely coverget 3. Absolutely coverget 32. Absolutely coverget 33. D ive rg e t 34. Absolutely coverget 35. D ive rg e t 36. Absolutely coverget 37. Absolutely coverget 38. D ive rg e t Copyright 203, Cegage Learig. All rights reserved.
3 SECTION OTHER CONVERGENCE TESTS 3 SOLUTIONS E Click here for exercises. Copyright 203, Cegage Learig. All rights reserved.. ( ) 3 +4.b = 3 > 0 ad b+ <b for +4 all ; b =0so the series coverges by the Alteratig Series Test =0 ( ) b = 5 is decreasig ad positive for all, ad =0so the series 3 +2 coverges by the Alteratig Series Test. ( ) +. =so + ( ) + does ot exist ad the series diverges by the Test for Divergece. 4. ( ) 2.b = > 0 ad b+ <b for all, 2 ad =0, so the series coverges by the Alteratig 2 Series Test. 5. ( ) +3.b = +3 is positive ad decreasig, ad +3 =0, so the series coverges by the Alteratig Series Test. 6. ( ) = 5 so ( )+ does ot exist ad the series diverges 5 + by the Test for Divergece. ( ) 7. l.b = is positive ad decreasig for l =2 2, ad =0so the series coverges by the l Alteratig Series Test. 8. ( ) 2 +.b = > 0 for all b + <b ( +) 2 + < 2 + ( +) ( 2 + ) < [ ( +) 2 + ] < < 2 +, which is true for all. Also 2 + = / =0. Therefore the series +/2 coverges by the Alteratig Series Test. 9. ( ) =,so 2 ( ) does ot exist. Thus the series diverges 2 + by the Test for Divergece. 0. =( ) 2 4 +,so a = as. 2 Therefore, a 0(i fact the it does ot exist) ad the series ( ) 2 diverges by the Test for 4 + Divergece.. =( ) ,so a = as. Therefore, 0(i fact the it does ot exist) ad the series ( ) 2 2 diverges by the Test for Divergece. 2. ( ) +4. b = > 0 for all. Let +4 x f (x) = x +4. The f 4 x (x) = 2 x (x +4) 2 < 0 if x>4, so{b } is decreasig after =4. +4 = =0. So the series +4/ coverges by the Alteratig Series Test. 3. ( ) + 2. b = > 0 ad b b which is 2 + certaily true. (/2 )=0by l Hospital s Rule, so the series coverges by the Alteratig Series Test l decreases ad 3 l =0,sobythe Alteratig Series Test the series coverges. ( ) 5. (2 )!. b 5 = (2 5 )! = 362,880 < , so ( ) 4 (2 )! ( ) (2 )! b 4 = (2 4)! = ad 40,320 s 3 = , so, correct to four 720 ( ) decimal places, (2)! =0
4 4 SECTION OTHER CONVERGENCE TESTS Copyright 203, Cegage Learig. All rights reserved. 7. b 6 = 2 6 6! = < 0.000, so 46,080 ( ) 5 ( ) ! 2! =0 =0 8. b 8 =/ < ad s 7 = ,625 46, , so correct to five decimal places, = ( ) /2 is a coverget pseries (p = 3 2 > ), so the give series is absolutely coverget. 20. Usig the Ratio Test, + = ( 3)+ / ( +) 3 ( 3) / 3 ( ) 3 =3 =3> + so the series diverges. 2. Usig the Ratio Test, a+ = ( 3)+ / ( +)! ( 3) /! = 3 =0<,so + the series is absolutely coverget. 22. a+ =! ( +)! = =0<,so + ( ) the series is absolutely coverget by the Ratio! 23. Test. 2 + < ad coverges (p =2> ), so 2 2 coverges absolutely by the Compariso Test a+ = /[(2 +)!] /[(2 )!] = (2 +)2 =0 so by the Ratio Test the series is absolutely coverget = 2 3,so ( ) Test for Divergece. 26. ( ) does ot exist, so diverges by the Test for Divergece. 2 diverges by the 3 4 ( ) a+ = 2 /[ + ( +)3 +2] 2 /(3 + ) = = 2 3 < so the series coverges absolutely by the Ratio Test. 28. a+ = 5 /[ ( +2) ] 5 /[ ( +) ] = 5 ( ) = > 29. a+ = ( +2)5 + /[ ( +)3 2(+) ] ( +)5 /(3 2 ) 5 ( +2) = 9( +) 2 = 5 9 < so the series coverges absolutely by the Ratio Test. so si 2 2 ad 2 si 2 2 coverges (pseries, p =2> ), 2 coverges absolutely by the Compariso Test. arcta < π/2 3 ad π/2 coverges (p =3> ), so 3 3 ( ) arcta coverges absolutely by the 3 Compariso Test. cos π 6, so sice coverges (p = 3 2 > ), the give series coverges absolutely by the Compariso Test. 33. a+ = ( +)!/0 + + =!/0 0 = 34. a+ [ = 8 ( +) 3 ]/ [( +)!] (8 3 )/ (!) = 8 ( +) =0< so the series coverges absolutely by the Ratio Test. 35. a+ = ( +) + / /5 2+3 ( ) + = ( +)= 25
5 SECTION OTHER CONVERGENCE TESTS = ( 2)+ ( +) 2 /[( +3)!] ( 2) 2 /[( +2)!] 2( +) 2 = 2 ( +3) =0< so the series coverges absolutely by the Ratio Test. 37. a+ = ( +3)! /[ ( +)!0 +] ( +2)!/(!0 ) = = 0 < so the series coverges absolutely by the Ratio Test. 4 7 (3 2) (3 +) 38. a+ = (2 +)(2 +3) 4 7 (3 2) (2 +) 3 + = 2 +3 = 3 2 > Copyright 203, Cegage Learig. All rights reserved.
4.3. The Integral and Comparison Tests
4.3. THE INTEGRAL AND COMPARISON TESTS 9 4.3. The Itegral ad Compariso Tests 4.3.. The Itegral Test. Suppose f is a cotiuous, positive, decreasig fuctio o [, ), ad let a = f(). The the covergece or divergece
More informationSAMPLE QUESTIONS FOR FINAL EXAM. (1) (2) (3) (4) Find the following using the definition of the Riemann integral: (2x + 1)dx
SAMPLE QUESTIONS FOR FINAL EXAM REAL ANALYSIS I FALL 006 3 4 Fid the followig usig the defiitio of the Riema itegral: a 0 x + dx 3 Cosider the partitio P x 0 3, x 3 +, x 3 +,......, x 3 3 + 3 of the iterval
More informationApproximating the Sum of a Convergent Series
Approximatig the Sum of a Coverget Series Larry Riddle Ages Scott College Decatur, GA 30030 lriddle@agesscott.edu The BC Calculus Course Descriptio metios how techology ca be used to explore covergece
More informationTheorems About Power Series
Physics 6A Witer 20 Theorems About Power Series Cosider a power series, f(x) = a x, () where the a are real coefficiets ad x is a real variable. There exists a real oegative umber R, called the radius
More informationSection 11.3: The Integral Test
Sectio.3: The Itegral Test Most of the series we have looked at have either diverged or have coverged ad we have bee able to fid what they coverge to. I geeral however, the problem is much more difficult
More information4 n. n 1. You shold think of the Ratio Test as a generalization of the Geometric Series Test. For example, if a n ar n is a geometric sequence then
SECTION 2.6 THE RATIO TEST 79 2.6. THE RATIO TEST We ow kow how to hadle series which we ca itegrate (the Itegral Test), ad series which are similar to geometric or pseries (the Compariso Test), but of
More informationSequences, Series and Convergence with the TI 92. Roger G. Brown Monash University
Sequeces, Series ad Covergece with the TI 92. Roger G. Brow Moash Uiversity email: rgbrow@deaki.edu.au Itroductio. Studets erollig i calculus at Moash Uiversity, like may other calculus courses, are itroduced
More informationTaylor Series and Polynomials
Taylor Series ad Polyomials Motivatios The purpose of Taylor series is to approimate a fuctio with a polyomial; ot oly we wat to be able to approimate, but we also wat to kow how good the approimatio is.
More informationINFINITE SERIES KEITH CONRAD
INFINITE SERIES KEITH CONRAD. Itroductio The two basic cocepts of calculus, differetiatio ad itegratio, are defied i terms of limits (Newto quotiets ad Riema sums). I additio to these is a third fudametal
More informationBuilding Blocks Problem Related to Harmonic Series
TMME, vol3, o, p.76 Buildig Blocks Problem Related to Harmoic Series Yutaka Nishiyama Osaka Uiversity of Ecoomics, Japa Abstract: I this discussio I give a eplaatio of the divergece ad covergece of ifiite
More informationInfinite Sequences and Series
CHAPTER 4 Ifiite Sequeces ad Series 4.1. Sequeces A sequece is a ifiite ordered list of umbers, for example the sequece of odd positive itegers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29...
More informationa 4 = 4 2 4 = 12. 2. Which of the following sequences converge to zero? n 2 (a) n 2 (b) 2 n x 2 x 2 + 1 = lim x n 2 + 1 = lim x
0 INFINITE SERIES 0. Sequeces Preiary Questios. What is a 4 for the sequece a? solutio Substitutig 4 i the expressio for a gives a 4 4 4.. Which of the followig sequeces coverge to zero? a b + solutio
More informationOur aim is to show that under reasonable assumptions a given 2πperiodic function f can be represented as convergent series
8 Fourier Series Our aim is to show that uder reasoable assumptios a give periodic fuctio f ca be represeted as coverget series f(x) = a + (a cos x + b si x). (8.) By defiitio, the covergece of the series
More informationAP Calculus AB 2006 Scoring Guidelines Form B
AP Calculus AB 6 Scorig Guidelies Form B The College Board: Coectig Studets to College Success The College Board is a otforprofit membership associatio whose missio is to coect studets to college success
More informationSequences II. Chapter 3. 3.1 Convergent Sequences
Chapter 3 Sequeces II 3. Coverget Sequeces Plot a graph of the sequece a ) = 2, 3 2, 4 3, 5 + 4,...,,... To what limit do you thik this sequece teds? What ca you say about the sequece a )? For ǫ = 0.,
More informationIn nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008
I ite Sequeces Dr. Philippe B. Laval Keesaw State Uiversity October 9, 2008 Abstract This had out is a itroductio to i ite sequeces. mai de itios ad presets some elemetary results. It gives the I ite Sequeces
More informationLecture 4: Cauchy sequences, BolzanoWeierstrass, and the Squeeze theorem
Lecture 4: Cauchy sequeces, BolzaoWeierstrass, ad the Squeeze theorem The purpose of this lecture is more modest tha the previous oes. It is to state certai coditios uder which we are guarateed that limits
More informationTrigonometric Form of a Complex Number. The Complex Plane. axis. ( 2, 1) or 2 i FIGURE 6.44. The absolute value of the complex number z a bi is
0_0605.qxd /5/05 0:45 AM Page 470 470 Chapter 6 Additioal Topics i Trigoometry 6.5 Trigoometric Form of a Complex Number What you should lear Plot complex umbers i the complex plae ad fid absolute values
More informationSequences and Series
CHAPTER 9 Sequeces ad Series 9.. Covergece: Defiitio ad Examples Sequeces The purpose of this chapter is to itroduce a particular way of geeratig algorithms for fidig the values of fuctios defied by their
More informationAP Calculus BC 2003 Scoring Guidelines Form B
AP Calculus BC Scorig Guidelies Form B The materials icluded i these files are iteded for use by AP teachers for course ad exam preparatio; permissio for ay other use must be sought from the Advaced Placemet
More informationSeries Convergence Tests Math 122 Calculus III D Joyce, Fall 2012
Some series converge, some diverge. Series Convergence Tests Math 22 Calculus III D Joyce, Fall 202 Geometric series. We ve already looked at these. We know when a geometric series converges and what it
More informationTAYLOR SERIES, POWER SERIES
TAYLOR SERIES, POWER SERIES The followig represets a (icomplete) collectio of thigs that we covered o the subject of Taylor series ad power series. Warig. Be prepared to prove ay of these thigs durig the
More informationCopyright 2013 wolfssl Inc. All rights reserved. 2
  Copyright 2013 wolfssl Inc. All rights reserved. 2 Copyright 2013 wolfssl Inc. All rights reserved. 2 Copyright 2013 wolfssl Inc. All rights reserved. 3 Copyright 2013 wolfssl Inc. All rights reserved.
More informationModified Line Search Method for Global Optimization
Modified Lie Search Method for Global Optimizatio Cria Grosa ad Ajith Abraham Ceter of Excellece for Quatifiable Quality of Service Norwegia Uiversity of Sciece ad Techology Trodheim, Norway {cria, ajith}@q2s.tu.o
More informationExample 2 Find the square root of 0. The only square root of 0 is 0 (since 0 is not positive or negative, so those choices don t exist here).
BEGINNING ALGEBRA Roots ad Radicals (revised summer, 00 Olso) Packet to Supplemet the Curret Textbook  Part Review of Square Roots & Irratioals (This portio ca be ay time before Part ad should mostly
More informationLesson 15 ANOVA (analysis of variance)
Outlie Variability betwee group variability withi group variability total variability Fratio Computatio sums of squares (betwee/withi/total degrees of freedom (betwee/withi/total mea square (betwee/withi
More informationMath 113 HW #11 Solutions
Math 3 HW # Solutios 5. 4. (a) Estimate the area uder the graph of f(x) = x from x = to x = 4 usig four approximatig rectagles ad right edpoits. Sketch the graph ad the rectagles. Is your estimate a uderestimate
More informationA power series about x = a is the series of the form
POWER SERIES AND THE USES OF POWER SERIES Elizabeth Wood Now we are finally going to start working with a topic that uses all of the information from the previous topics. The topic that we are going to
More informationNATIONAL SENIOR CERTIFICATE GRADE 12
NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P NOVEMBER 0 MARKS: 50 TIME: 3 hours This questio paper cosists of 8 pages, diagram sheet ad iformatio sheet. Please tur over Mathematics/P DBE/November 0
More informationTo discuss this topic fully, let us define some terms used in this and the following sets of supplemental notes.
INFINITE SERIES SERIES AND PARTIAL SUMS What if we wanted to sum up the terms of this sequence, how many terms would I have to use? 1, 2, 3,... 10,...? Well, we could start creating sums of a finite number
More informationRepeating Decimals are decimal numbers that have number(s) after the decimal point that repeat in a pattern.
5.5 Fractios ad Decimals Steps for Chagig a Fractio to a Decimal. Simplify the fractio, if possible. 2. Divide the umerator by the deomiator. d d Repeatig Decimals Repeatig Decimals are decimal umbers
More informationQuestion 2: How is a loan amortized?
Questio 2: How is a loa amortized? Decreasig auities may be used i auto or home loas. I these types of loas, some amout of moey is borrowed. Fixed paymets are made to pay off the loa as well as ay accrued
More informationSection 8.3 : De Moivre s Theorem and Applications
The Sectio 8 : De Moivre s Theorem ad Applicatios Let z 1 ad z be complex umbers, where z 1 = r 1, z = r, arg(z 1 ) = θ 1, arg(z ) = θ z 1 = r 1 (cos θ 1 + i si θ 1 ) z = r (cos θ + i si θ ) ad z 1 z =
More informationCHAPTER 11 Financial mathematics
CHAPTER 11 Fiacial mathematics I this chapter you will: Calculate iterest usig the simple iterest formula ( ) Use the simple iterest formula to calculate the pricipal (P) Use the simple iterest formula
More informationBasic Elements of Arithmetic Sequences and Series
MA40S PRECALCULUS UNIT G GEOMETRIC SEQUENCES CLASS NOTES (COMPLETED NO NEED TO COPY NOTES FROM OVERHEAD) Basic Elemets of Arithmetic Sequeces ad Series Objective: To establish basic elemets of arithmetic
More informationSECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES
SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES Read Sectio 1.5 (pages 5 9) Overview I Sectio 1.5 we lear to work with summatio otatio ad formulas. We will also itroduce a brief overview of sequeces,
More informationNATIONAL SENIOR CERTIFICATE GRADE 12
NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P EXEMPLAR 04 MARKS: 50 TIME: 3 hours This questio paper cosists of 8 pages ad iformatio sheet. Please tur over Mathematics/P DBE/04 NSC Grade Eemplar INSTRUCTIONS
More informationLearning outcomes. Algorithms and Data Structures. Time Complexity Analysis. Time Complexity Analysis How fast is the algorithm? Prof. Dr.
Algorithms ad Data Structures Algorithm efficiecy Learig outcomes Able to carry out simple asymptotic aalysisof algorithms Prof. Dr. Qi Xi 2 Time Complexity Aalysis How fast is the algorithm? Code the
More informationSoving Recurrence Relations
Sovig Recurrece Relatios Part 1. Homogeeous liear 2d degree relatios with costat coefficiets. Cosider the recurrece relatio ( ) T () + at ( 1) + bt ( 2) = 0 This is called a homogeeous liear 2d degree
More information5.4 Amortization. Question 1: How do you find the present value of an annuity? Question 2: How is a loan amortized?
5.4 Amortizatio Questio 1: How do you fid the preset value of a auity? Questio 2: How is a loa amortized? Questio 3: How do you make a amortizatio table? Oe of the most commo fiacial istrumets a perso
More informationFigure 40.1. Figure 40.2
40 Regular Polygos Covex ad Cocave Shapes A plae figure is said to be covex if every lie segmet draw betwee ay two poits iside the figure lies etirely iside the figure. A figure that is ot covex is called
More informationEstimating the Mean and Variance of a Normal Distribution
Estimatig the Mea ad Variace of a Normal Distributio Learig Objectives After completig this module, the studet will be able to eplai the value of repeatig eperimets eplai the role of the law of large umbers
More informationSolutions to Odd Numbered End of Chapter Exercises: Chapter 2
Itroductio to Ecoometrics (3 rd Updated Editio) by James H. Stock ad Mark W. Watso Solutios to Odd Numbered Ed of Chapter Exercises: Chapter 2 (This versio July 20, 2014) Stock/Watso  Itroductio to Ecoometrics
More informationChapter 5: Inner Product Spaces
Chapter 5: Ier Product Spaces Chapter 5: Ier Product Spaces SECION A Itroductio to Ier Product Spaces By the ed of this sectio you will be able to uderstad what is meat by a ier product space give examples
More information4.1 Sigma Notation and Riemann Sums
0 the itegral. Sigma Notatio ad Riema Sums Oe strategy for calculatig the area of a regio is to cut the regio ito simple shapes, calculate the area of each simple shape, ad the add these smaller areas
More informationM06/5/MATME/SP2/ENG/TZ2/XX MATHEMATICS STANDARD LEVEL PAPER 2. Thursday 4 May 2006 (morning) 1 hour 30 minutes INSTRUCTIONS TO CANDIDATES
IB MATHEMATICS STANDARD LEVEL PAPER 2 DIPLOMA PROGRAMME PROGRAMME DU DIPLÔME DU BI PROGRAMA DEL DIPLOMA DEL BI 22067304 Thursday 4 May 2006 (morig) 1 hour 30 miutes INSTRUCTIONS TO CANDIDATES Do ot ope
More information7.6 Approximation Errors and Simpson's Rule
WileyPLUS: Home Help Contact us Logout HughesHallett, Calculus: Single and Multivariable, 4/e Calculus I, II, and Vector Calculus Reading content Integration 7.1. Integration by Substitution 7.2. Integration
More informationCS103A Handout 23 Winter 2002 February 22, 2002 Solving Recurrence Relations
CS3A Hadout 3 Witer 00 February, 00 Solvig Recurrece Relatios Itroductio A wide variety of recurrece problems occur i models. Some of these recurrece relatios ca be solved usig iteratio or some other ad
More informationS. Tanny MAT 344 Spring 1999. be the minimum number of moves required.
S. Tay MAT 344 Sprig 999 Recurrece Relatios Tower of Haoi Let T be the miimum umber of moves required. T 0 = 0, T = 7 Iitial Coditios * T = T + $ T is a sequece (f. o itegers). Solve for T? * is a recurrece,
More informationName: ID: Discussion Section:
Math 28 Midterm 3 Spring 2009 Name: ID: Discussion Section: This exam consists of 6 questions: 4 multiple choice questions worth 5 points each 2 handgraded questions worth a total of 30 points. INSTRUCTIONS:
More informationFind the inverse Laplace transform of the function F (p) = Evaluating the residues at the four simple poles, we find. residue at z = 1 is 4te t
Homework Solutios. Chater, Sectio 7, Problem 56. Fid the iverse Lalace trasform of the fuctio F () (7.6). À Chater, Sectio 7, Problem 6. Fid the iverse Lalace trasform of the fuctio F () usig (7.6). Solutio:
More informationMATH 2300 review problems for Exam 3 ANSWERS
MATH 300 review problems for Exam 3 ANSWERS. Check whether the following series converge or diverge. In each case, justify your answer by either computing the sum or by by showing which convergence test
More informationRemarques sur un beau rapport entre les series des puissances tant directes que reciproques
Aug 006 Traslatio with otes of Euler s paper Remarques sur u beau rapport etre les series des puissaces tat directes que reciproques Origially published i Memoires de l'academie des scieces de Berli 7
More information11.2 Nuclear Reactions: Fission
11.2 Nuclear Reactios: Fissio Followig Fermi s work i 1938, Otto Hah, Lise Meiter, ad Fritz Strassma discovered that whe eutros bombarded uraium atoms, the reactio produced smaller uclei that were about
More information8.1 Arithmetic Sequences
MCR3U Uit 8: Sequeces & Series Page 1 of 1 8.1 Arithmetic Sequeces Defiitio: A sequece is a comma separated list of ordered terms that follow a patter. Examples: 1, 2, 3, 4, 5 : a sequece of the first
More informationThe Euler Totient, the Möbius and the Divisor Functions
The Euler Totiet, the Möbius ad the Divisor Fuctios Rosica Dieva July 29, 2005 Mout Holyoke College South Hadley, MA 01075 1 Ackowledgemets This work was supported by the Mout Holyoke College fellowship
More informationA Recursive Formula for Moments of a Binomial Distribution
A Recursive Formula for Momets of a Biomial Distributio Árpád Béyi beyi@mathumassedu, Uiversity of Massachusetts, Amherst, MA 01003 ad Saverio M Maago smmaago@psavymil Naval Postgraduate School, Moterey,
More information1. C. The formula for the confidence interval for a population mean is: x t, which was
s 1. C. The formula for the cofidece iterval for a populatio mea is: x t, which was based o the sample Mea. So, x is guarateed to be i the iterval you form.. D. Use the rule : pvalue
More information58.08 g 1 mol. 1 mol
Chem 338 Homework Set #6 solutios October 17, 001 From Atkis: 6.8, 6.1, 6.14, 6.16, 6.17, 6.19, 6. 6.8) The partial molar volumes of propaoe ad trichloromethae i a mixture i which the mole fractio of CHCl
More informationTangent circles in the ratio 2 : 1. Hiroshi Okumura and Masayuki Watanabe. In this article we consider the following old Japanese geometry problem
116 Taget circles i the ratio 2 : 1 Hiroshi Okumura ad Masayuki Wataabe I this article we cosider the followig old Japaese geometry problem (see Figure 1), whose statemet i [1, p. 39] is missig the coditio
More informationφ y' CORDIC Algorithm COordinate Rotation DIgital Computer x y Example: vector rotation: Implementation cost: 2 multiplications
CORDIC Algorithm COordiate Rotatio DIgital Computer Example: vector rotatio: x' x φ y' y x' y' cos( φ) = si( φ) si( cos( φ Implemetatio cost: 2 multiplicatios φ) ) x y 2 additio or subtractio 2 si() ad
More informationWhat Is Required? You need to find the final temperature of an iron ring heated by burning alcohol. 4.95 g
Calculatig Theral Eergy i a Bob Calorieter (Studet textbook page 309) 31. Predict the fial teperature of a 5.00 10 2 g iro rig that is iitially at 25.0 C ad is heated by cobustig 4.95 g of ethaol, C 2
More informationTHE REGRESSION MODEL IN MATRIX FORM. For simple linear regression, meaning one predictor, the model is. for i = 1, 2, 3,, n
We will cosider the liear regressio model i matrix form. For simple liear regressio, meaig oe predictor, the model is i = + x i + ε i for i =,,,, This model icludes the assumptio that the ε i s are a sample
More information.04. This means $1000 is multiplied by 1.02 five times, once for each of the remaining sixmonth
Questio 1: What is a ordiary auity? Let s look at a ordiary auity that is certai ad simple. By this, we mea a auity over a fixed term whose paymet period matches the iterest coversio period. Additioally,
More informationLesson 17 Pearson s Correlation Coefficient
Outlie Measures of Relatioships Pearso s Correlatio Coefficiet (r) types of data scatter plots measure of directio measure of stregth Computatio covariatio of X ad Y uique variatio i X ad Y measurig
More informationMATHEMATICS P1 COMMON TEST JUNE 2014 NATIONAL SENIOR CERTIFICATE GRADE 12
Mathematics/P1 1 Jue 014 Commo Test MATHEMATICS P1 COMMON TEST JUNE 014 NATIONAL SENIOR CERTIFICATE GRADE 1 Marks: 15 Time: ½ hours N.B: This questio paper cosists of 7 pages ad 1 iformatio sheet. Please
More informationMATH 132: CALCULUS II SYLLABUS
MATH 32: CALCULUS II SYLLABUS Prerequisites: Successful completion of Math 3 (or its equivalent elsewhere). Math 27 is normally not a sufficient prerequisite for Math 32. Required Text: Calculus: Early
More informationThis document contains a collection of formulas and constants useful for SPC chart construction. It assumes you are already familiar with SPC.
SPC Formulas ad Tables 1 This documet cotais a collectio of formulas ad costats useful for SPC chart costructio. It assumes you are already familiar with SPC. Termiology Geerally, a bar draw over a symbol
More informationSolutions to Exercises Chapter 4: Recurrence relations and generating functions
Solutios to Exercises Chapter 4: Recurrece relatios ad geeratig fuctios 1 (a) There are seatig positios arraged i a lie. Prove that the umber of ways of choosig a subset of these positios, with o two chose
More informationSection 6.1 Radicals and Rational Exponents
Sectio 6.1 Radicals ad Ratioal Expoets Defiitio of Square Root The umber b is a square root of a if b The priciple square root of a positive umber is its positive square root ad we deote this root by usig
More informationEncrypting*a*Windows*7*Hard*Disk* with%bitlocker%disk%encryption!
Encrypting*a*Windows*7*Hard*Disk* with%bitlocker%disk%encryption Thisdocumentcontainsthenecessarystepstoencryptthecontentsofaharddrive usingbitlockerandwindows7. Thefollowinginstructionsarederivedfromdocumentationat:
More informationINFO  Using Functions in the Calculator
Show INFO  Using Functions in the Calculator The information in this article applies to: EV (Web) Client, version 5.0 build 1 or higher This article describes the purpose of the functions available in
More informationChapter 5 Unit 1. IET 350 Engineering Economics. Learning Objectives Chapter 5. Learning Objectives Unit 1. Annual Amount and Gradient Functions
Chapter 5 Uit Aual Amout ad Gradiet Fuctios IET 350 Egieerig Ecoomics Learig Objectives Chapter 5 Upo completio of this chapter you should uderstad: Calculatig future values from aual amouts. Calculatig
More informationSolving Logarithms and Exponential Equations
Solvig Logarithms ad Epoetial Equatios Logarithmic Equatios There are two major ideas required whe solvig Logarithmic Equatios. The first is the Defiitio of a Logarithm. You may recall from a earlier topic:
More informationDepartment of Computer Science, University of Otago
Departmet of Computer Sciece, Uiversity of Otago Techical Report OUCS200609 Permutatios Cotaiig May Patters Authors: M.H. Albert Departmet of Computer Sciece, Uiversity of Otago Micah Colema, Rya Fly
More informationNUMBERS COMMON TO TWO POLYGONAL SEQUENCES
NUMBERS COMMON TO TWO POLYGONAL SEQUENCES DIANNE SMITH LUCAS Chia Lake, Califoria a iteger, The polygoal sequece (or sequeces of polygoal umbers) of order r (where r is r > 3) may be defied recursively
More informationSEQUENCES AND SERIES CHAPTER
CHAPTER SEQUENCES AND SERIES Whe the Grat family purchased a computer for $,200 o a istallmet pla, they agreed to pay $00 each moth util the cost of the computer plus iterest had bee paid The iterest each
More informationx(x 1)(x 2)... (x k + 1) = [x] k n+m 1
1 Coutig mappigs For every real x ad positive iteger k, let [x] k deote the fallig factorial ad x(x 1)(x 2)... (x k + 1) ( ) x = [x] k k k!, ( ) k = 1. 0 I the sequel, X = {x 1,..., x m }, Y = {y 1,...,
More informationFIBONACCI NUMBERS: AN APPLICATION OF LINEAR ALGEBRA. 1. Powers of a matrix
FIBONACCI NUMBERS: AN APPLICATION OF LINEAR ALGEBRA. Powers of a matrix We begi with a propositio which illustrates the usefuless of the diagoalizatio. Recall that a square matrix A is diogaalizable if
More informationMathematical goals. Starting points. Materials required. Time needed
Level A1 of challege: C A1 Mathematical goals Startig poits Materials required Time eeded Iterpretig algebraic expressios To help learers to: traslate betwee words, symbols, tables, ad area represetatios
More informationLecture 13. Lecturer: Jonathan Kelner Scribe: Jonathan Pines (2009)
18.409 A Algorithmist s Toolkit October 27, 2009 Lecture 13 Lecturer: Joatha Keler Scribe: Joatha Pies (2009) 1 Outlie Last time, we proved the BruMikowski iequality for boxes. Today we ll go over the
More informationChapter 7: Confidence Interval and Sample Size
Chapter 7: Cofidece Iterval ad Sample Size Learig Objectives Upo successful completio of Chapter 7, you will be able to: Fid the cofidece iterval for the mea, proportio, ad variace. Determie the miimum
More informationLesson 2.4: Angle Properties in Polygons, page 99
Lesso 2.4: Agle Properties i Polygos, page 99 1. a) S(12) = 180 (12 2) S(12) = 180 (10) S(12) = 1800 A dodecago has 12 sides, so is 12. The sum of the iterior agles i a regular dodecago is 1800. S(12)
More informationClass Meeting # 16: The Fourier Transform on R n
MATH 18.152 COUSE NOTES  CLASS MEETING # 16 18.152 Itroductio to PDEs, Fall 2011 Professor: Jared Speck Class Meetig # 16: The Fourier Trasform o 1. Itroductio to the Fourier Trasform Earlier i the course,
More informationConfidence Intervals. CI for a population mean (σ is known and n > 30 or the variable is normally distributed in the.
Cofidece Itervals A cofidece iterval is a iterval whose purpose is to estimate a parameter (a umber that could, i theory, be calculated from the populatio, if measuremets were available for the whole populatio).
More information= 1 lim sup{ sn : n > N} )
ATH 104, SUER 2006, HOEWORK 4 SOLUTION BENJAIN JOHNSON Due July 12 Assgmet: Secto 11: 11.4(b)(c), 11.8 Secto 12: 12.6(c), 12.12, 12.13 Secto 13: 13.1 Secto 11 11.4 Cosder the sequeces s = cos ( ) π 3,
More informationCase Study. Normal and t Distributions. Density Plot. Normal Distributions
Case Study Normal ad t Distributios Bret Halo ad Bret Larget Departmet of Statistics Uiversity of Wiscosi Madiso October 11 13, 2011 Case Study Body temperature varies withi idividuals over time (it ca
More information1 Computing the Standard Deviation of Sample Means
Computig the Stadard Deviatio of Sample Meas Quality cotrol charts are based o sample meas ot o idividual values withi a sample. A sample is a group of items, which are cosidered all together for our aalysis.
More informationSnap. Jenine's formula. The SNAP probability is
Sap The game of SNAP is played with stadard decks of cards. The decks are shuffled ad cards are dealt simultaeously from the top of each deck. SNAP is called if the two dealt cards are idetical (value
More information3. Covariance and Correlation
Virtual Laboratories > 3. Expected Value > 1 2 3 4 5 6 3. Covariace ad Correlatio Recall that by takig the expected value of various trasformatios of a radom variable, we ca measure may iterestig characteristics
More informationMATHEMATICS SYLLABUS SECONDARY 7th YEAR
Europe Schools Office of the SecretryGeerl Pedgogicl developmet Uit Ref.: 201101D41e2 Orig.: DE MATHEMATICS SYLLABUS SECONDARY 7th YEAR Stdrd level 5 period/week course Approved y the Joit Techig
More informationReview for 1 sample CI Name. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Review for 1 sample CI Name MULTIPLE CHOICE. Choose the oe alterative that best completes the statemet or aswers the questio. Fid the margi of error for the give cofidece iterval. 1) A survey foud that
More informationLies My Calculator and Computer Told Me
Lies My Calculator and Computer Told Me 2 LIES MY CALCULATOR AND COMPUTER TOLD ME Lies My Calculator and Computer Told Me See Section.4 for a discussion of graphing calculators and computers with graphing
More informationNumber Systems I. CIS0082 Logic and Foundations of Mathematics. David Goodwin. 11:00, Tuesday 18 th October
Number Systems I CIS0082 Logic and Foundations of Mathematics David Goodwin david.goodwin@perisic.com 11:00, Tuesday 18 th October 2011 Outline 1 Number systems Numbers Natural numbers Integers Rational
More informationNATIONAL SENIOR CERTIFICATE GRADE 11
NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P EXEMPLAR 007 MARKS: 50 TIME: 3 hours This questio paper cosists of pages, 4 diagram sheets ad a page formula sheet. Please tur over Mathematics/P DoE/Exemplar
More informationThe second difference is the sequence of differences of the first difference sequence, 2
Differece Equatios I differetial equatios, you look for a fuctio that satisfies ad equatio ivolvig derivatives. I differece equatios, istead of a fuctio of a cotiuous variable (such as time), we look for
More informationAFM Ch.12  Practice Test
AFM Ch.2  Practice Test Multiple Choice Identify the choice that best completes the statement or answers the question.. Form a sequence that has two arithmetic means between 3 and 89. a. 3, 33, 43, 89
More information1 Correlation and Regression Analysis
1 Correlatio ad Regressio Aalysis I this sectio we will be ivestigatig the relatioship betwee two cotiuous variable, such as height ad weight, the cocetratio of a ijected drug ad heart rate, or the cosumptio
More informationCHAPTER 3 THE TIME VALUE OF MONEY
CHAPTER 3 THE TIME VALUE OF MONEY OVERVIEW A dollar i the had today is worth more tha a dollar to be received i the future because, if you had it ow, you could ivest that dollar ad ear iterest. Of all
More informationDetermining the sample size
Determiig the sample size Oe of the most commo questios ay statisticia gets asked is How large a sample size do I eed? Researchers are ofte surprised to fid out that the aswer depeds o a umber of factors
More information