Mocks.ie Maths LC HL Further Calculus mocks.ie Page 1


 Roxanne Rich
 2 years ago
 Views:
Transcription
1 Maths Leavig Cert Higher Level Further Calculus Questio Paper By Cillia Fahy ad Darro Higgis Mocks.ie Maths LC HL Further Calculus mocks.ie Page
2 Further Calculus ad Series, Paper II Q8 Table of Cotets:. Maimum ad Miimum. Itegratio by Parts a. Simple Itegratio by Parts b. Itegratio by Parts with oe Part c. Double Itegratio. The Ratio Test a. Fidig from b. Dealig with factorials c. Dealig with limits d. The differece betwee 0 ad ifiity e. Ratio Test Questio 4. Maclauri Series a. Computig Maclauri Series b. Obtaiig Approimatios c. Fidig the Geeral Term ( d. Mocks.ie Maths LC HL Further Calculus mocks.ie Page
3 Maimum ad Miimum Problems: These problems have the same basis as the Ma/Mi problems that you fid i Questio 6 ad 7 of Paper. But they are more practical based ad slightly more difficult. Also you will geerally be give a diagram to help you visualise so make the most of that. Remember from Differetiatio: at a Maimum Poit Eample: Ma ad Mi Problems Part (i) at a Maimum Poit Most questios have a part (i) to them which offers you the key doig this questio. Which is epressig everythig i terms of oe variable. Leavig Cert 008, Q8 b (i) pqr is a equilateral triagle of side 6 cm. abcd is a rectagle iscribed i the triagle as show. ab = cm ad bc = y cm. Epress y i terms of. r d c p q pq =6 ab = As we ca see from the diagram ab is i pq pa = bq bq =6 bq = a {This is the lie pq without ab } b rqp cqb 60 {Equilateral triagle} Ta( cqb)=ta60= opp Ta= {Table Book, p. 6} adj bc y Ta( cqb)= bq y ( ) Mocks.ie Maths LC HL Further Calculus mocks.ie Page
4 Leavig Cert 008, Q8 b (ii) Fid the maimum possible area of abcd. (Cotiued o from the previous questio) Area of abcd = y= ( y ( ) ( ) A da da 0 {To fid ma, 0} d d d A d <0==>Ma 9 9 A= 9 cm WARNING! You ll otice that they asked for Area here, this is somethig which you might ot have dealt with sice Juior Cert. They usually ask for formulae which are somewhat basic but which you might be ot have used i some time. Check page 8 i the Table Book for these. It s importat also that you get lots of practice applyig these, eve though they may seem simple. Do t get caught out! Mocks.ie Maths LC HL Further Calculus mocks.ie Page 4
5 Itegratio by Parts: Just as Ma ad Mi problems had liks to Differetiatio questios so Itegratio by parts coects to Itegratio. The label Itegratio by Parts really says it all. You itegrate by splittig the sum up ito parts. There are three types of Itegratio by Parts Questios: udv=uv vdu Simple Itegratio by Parts Itegratio by Parts with oe part Double Itegratio by Parts Table Book, p. 6 Eplaatio ad Order: Accordig to the above formula, you let U stad for oe part of the sum ad the you let for aother part. You will uderstad this better whe you see it shortly. But there is a order i which you must do this, which is:. Logs. Iverse Trig. Algebra 4. Trigoometry 5. Epoetial Fuctio ( You assig the letter U to the above i that order. I each sum you will have two of these ad you assig U to whichever is highest rak here. Eample: Simple Itegratio by Parts Use itegratio by parts to fid stad Mocks.ie Maths LC HL Further Calculus mocks.ie Page 5
6 U 4 e d du d dv v 4 e 4 e d 4 udv uv vdu 4 4 e 4 e d 4 e d e 4 e d e C 4 6 Select U=, as is Algebra ad it is higher tha epoetial i the order Eample: Itegratio by Parts with Oe Part This is very similar to ormal itegratio by parts, but you oly have oe part which is obvious ad the other is more subtle. It may seem obvious oce you kow it but it ca easily tripe you up. Leavig Cert 00 Q8 a Use itegratio by parts to fid log e U log du d dv v d d Use the formula: udv=uv vdu {Differetiate the U to get du} {Itegrate to gai v} loge d log ( ) ( d)  d log d log ( ) C e WARNING! This is the most importat step here. Watch out for this or you ll miss it Do t forget your +C Eample: Double Itegratio by Parts Use itegratio by parts to fid Mocks.ie Maths LC HL Further Calculus mocks.ie Page 6
7 e si d STEP ONE: Itegrate the first time U si du cos d dv V e e d udv uv vdu si e si d e e cos d This requires itegratio by parts agai STEP TWO: Itegrate the secod time e cos d e cos d {Take out the costat for ease} U cos du si d dv e d V e Udv uv vdu cos e e cos d e ( si ) d + e (si ) d Notice that this is the eact same as your questio. This is how you kow how to stop STEP THREE: Combie agai si e si d e e cos d cos e  + e (si ) si cos e e si d = e e si cos e si = 9 e d e 9 d 0 e si d =si e cos e {Multiply by 9} e d C (si ) d si e cos e si = {Divide by 0} 0 Maipulate here: Just brig the last term across to the other side. Mocks.ie Maths LC HL Further Calculus mocks.ie Page 7
8 The Ratio Test: =R R< ==> The series is coverget R> ==> The series is diverget R= ==> The test is icoclusive You may ot uderstad how to use limits or you may but for this test a real uderstadig of limits is ot ecessary. There are three areas we eed to touch o before dealig with the Test: Fidig from Dealig with factorials Dealig with limits The differece betwee 0 ad ifiity Fidig All you do is take your (which will be the umber give to you i terms of ) ad aywhere there is a you chage to +. Eample : Mocks.ie Maths LC HL Further Calculus mocks.ie Page 8
9 Eample : Dealig with Factorials: Factorials are give the sig: So basically a eclamatio mark. Factorials work i the followig way:! =. Note: There is a butto o your calculator for factorials as well. Become familiar with it. (!) Mocks.ie Maths LC HL Further Calculus mocks.ie Page 9
10 So whatever umber the factorial is you start there ad multiply by every umber before it all the way back to. What is!? Well! = ()(... What happes frequetly i the ratio test is that you will have to cacel. Here s how: (!)= What is the differece? Well the oly differece is that has a etra (+) So = + Secod eample; Dealig with limits: Not kowig limits is ot a problem for this questio. Your oly real use of limits is at the fial part ad that is a fairly predictable system. To fid, all you do is divide every umber by the highest power of. The highest power of here is. So we divide everythig by. You the simplify. Whatever is still divide by or to ay power becomes 0. = {Highest power is } = {Highest power is } The differece betwee zero ad ifiity: The distictio betwee 0 ad pay particular attetio to it., i the ratio test questio is quite small ad therefore you eed to As Not i the Table Book Mocks.ie Maths LC HL Further Calculus mocks.ie Page 0
11 You ca see that just the positio of the 0 ca decide which is coverget ad diverget so pay particular attetio. Leavig Cert 008 Q8 a Use the ratio test to show that is coverget. I wat it to be coverget therefore I eed < where u u! ( ) 4 ( )! ( )! u!!.. u ( )! ( )! 4 4 4! 8.. ( )! lim 0 0 As 0<, the series coverges. WARNING! Pay careful attetio to the power of here whe you icrease from to + It is (+)+=++ Not ++ The Maclauri Series: This is the mai elemet of the questio. Usually a Maclauri Series of some sort will feature i this questio. We will tackle this questio i the followig way: Computig Maclauri Series Obtaiig Approimatios Fidig the Geeral Term ( Mocks.ie Maths LC HL Further Calculus mocks.ie Page
12 Computig the Maclauri Series: The Maclauri Series is foud from the followig formula: (Table Book, page 7) Maclauri Series ( ) f ' 0 f '' 0 f ''' 0 f 0 f f!!!! Usig this formula you are required to compute the followig Maclauri Series: Eample: Leavig Cert 004 Q8 b (i) Derive the first five terms of the Maclauri Series of f ' 0 f '' 0 f ''' 0 f e f!!! f e f 0 e f e f e 0 ' ' 0 f e f e 0 '' '' 0 f e f e 0 ''' ''' f e f 0 e e e Maclauri Series (First five terms) 4!! 4! Eample: log (+) Leavig Cert 005 Q8 b (i) Derive the Maclauri series for f () lup to ad icludig the term cotaiig Mocks.ie Maths LC HL Further Calculus mocks.ie Page
13 f '' 0 f ''' 0 f l f 0 f ' 0!! f ( ) l f 0 l 0 f '( ) f ' 0 f ''( ) f '' 0 0 f ''' f ''' 0 0 Note that to is coverted for the rest of the terms ad you apply the Chai Rule. log( ) Maclauri Series (First four terms) l 0.!! l 0 Eample: Leavig Cert 008 Q8 c (i) Derive the Maclauri series for icludig the term cotaiig up to ad f ' 0 f '' 0 f ''' 0 f f!!! cos 0... f cos f (0) cos 0 f ' si f ' 0 si 0 0 f '' cos f '' 0 cos0 f ''' si f ''' 0 si f cos f 0 cos 0 Note that it begis to repeat itself cos Maclauri Series (First five terms) (0) cos (0).!! 4! 4 cos 4! Eample: Si 4 Derive the Maclauri series for cotaiig up to ad icludig the term Mocks.ie Maths LC HL Further Calculus mocks.ie Page
14 f ' 0 f '' 0 f ''' 0 f f!!! si 0... f si f (0) si 0 0 f ' cos f ' 0 cos 0 f '' si f '' 0 si 0 0 f ''' cos f ''' 0 cos0 4 4 f si f 0 si f cos f 0 cos 0 Note the similarities betwee si Maclauri Series (First si terms) 0 ( ) 0 si (0) ().!! 4! 5! 5 si! 5! Eample: Biomial 4 5 WARNING! You ca be asked to fid or, oce you have si or cos. Simply do this by applyig the formula o p.4 Table Book. Ad lettig =A f ' 0 f '' 0 f ''' 0 f f!!! ( ) 0... f ( ) f (0) () f ' ( ) f ' 0 () f f '' ( )( ) '' 0 ( )() ( ) f f ''' ( )( )( ) ''' 0 ( )( )() ( )( ) ( ) Maclauri Series (First four terms) ( ) ( )( ) ( ) () ( ).!! Approimatios: Mocks.ie Maths LC HL Further Calculus mocks.ie Page 4
15 You ca usually be asked to fid a approimatio of the Maclauri Series. There are two types of approimatios: Geeral Approimatio Geeral Approimatio: The geeral approimatio is the simplest of approimatios ad is the same for all of the Maclauri Series above ecept for. Eample: Approimatio for l(+) Leavig Cert 005 Q8 b (ii) Use those terms to fid a approimatio for l These terms are those foud earlier i the questio: ( ) ( ) l l( ) {Use Calculator} WARNING! It is very easy to sub i = if you do t look closely. Be careful of this with every approimatio. l 0 Approimatio for These are a little differet. First of all you will probably have to sub i two values oe for the value of ad the other the value for. This series oly coverges for. You eed it to coverge ad therefore whatever you sub i for must fit this. If ot maipulate it so it will. Eample: Mocks.ie Maths LC HL Further Calculus mocks.ie Page 5
16 Leavig Cert 009 Q8 b (i)derive the first four terms of the Maclauri Series WARNING! This is somethig of a implicit approimatio. You eed to otice that here you. You will usually be asked to compute your Biomial Maclauri like this. You would solve it as we did above subbig i =. Which meas you would get: ( ) ( )( ) ( ) () ( ).!! ( ) ( ) () ( ). 4 8! ( ) () ( ). ( ) ( ) 8 6 Leavig Cert 009 Q8 b (ii)derive the first four terms of the Maclauri Series. Give that this series coverges for < <, use these four terms to fid a approimatio for, as a fractio. 7 6 But remember < so it caot be 6 6 6( ) Therefore = ( ) ( ) () ( ). ( ) ( ) {sub = } With that we have + but we wat 4 +. Multiply by A good check step is to check if o your calculator is close to your fial aswer. If they are the you should be right! Mocks.ie Maths LC HL Further Calculus mocks.ie Page 6
17 Fidig the Geeral Term: We fid the geeral term usually so we ca apply the ratio test as eplaied earlier i this chapter. There are two ways you ca fid the geeral term: How to Work it Out: It does t take much work to figure out the Geeral term o your ow so log as you kow what to look for: Begi by umberig each term accordig to order. You should see how their factorial value ad the power of relate to this umber If sigs alterate every term the you must put either a or with the geeral term. E.g. For that whe is eve the the term should be positive ad whe is odd the term should be egative. The opposite holds for List of the Geeral Terms: Workig the term out yourself ca be much easier oce you ve see the actual Geeral term for each. But if you ca t lear how to work the Geeral term out yourself the you should cosider learig these off. Not i the table book! e u! l( ) u ( ) si u cos u ( ) ( )! ( ) ( )!! r ( )( )...( r) ( ) ur Use r here as is part of the sum ( r )! Mocks.ie Maths LC HL Further Calculus mocks.ie Page 7
18 : The Maclauri Series for is differet tha the oes we have see previously as it is formed i a differet way. As a matter of fact, you do t eve use the Maclauri Series formula at all. ta a a a Table book, page 6 This tells us that to get. All we eed do is use log divisio ad divide ito etc So d (...) d 5 7 ta {Itegrate} Mocks.ie Maths LC HL Further Calculus mocks.ie Page 8
19 Approimatig for : y ta ta yta y Importat Formula: Not i the Table Book. Although, if you look at p. 9. The formula for the agle betwee two lies is somewhat similar ad you ca lear that as a basis for this oe. Eample: Usig the first four terms of, get a approimatio for ad ta ta ( ) ta ( ) ta ( ) ta ( ) Mocks.ie Maths LC HL Further Calculus mocks.ie Page 9
20 Eample : Show. From this get a approimatio for correct to 4 decimal places. y ta ta yta y ta ta ta 5 6 = ta ta 5 6 ta ta ta 4 4 ta ta 4 4( )= 4(ta ta ) {Sub i previous appro.}.408 Mocks.ie Maths LC HL Further Calculus mocks.ie Page 0
Riemann Sums y = f (x)
Riema Sums Recall that we have previously discussed the area problem I its simplest form we ca state it this way: The Area Problem Let f be a cotiuous, oegative fuctio o the closed iterval [a, b] Fid
More informationThe geometric series and the ratio test
The geometric series ad the ratio test Today we are goig to develop aother test for covergece based o the iterplay betwee the it compariso test we developed last time ad the geometric series. A ote about
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 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 information7 b) 0. Guided Notes for lesson P.2 Properties of Exponents. If a, b, x, y and a, b, 0, and m, n Z then the following properties hold: 1 n b
Guided Notes for lesso P. Properties of Expoets If a, b, x, y ad a, b, 0, ad m, Z the the followig properties hold:. Negative Expoet Rule: b ad b b b Aswers must ever cotai egative expoets. Examples: 5
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 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 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 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 informationCALCULUS. Taimur Khalid
CALCULUS Taimur Khalid 5/20/2015 Chapter 1 What is Calculus? What is it? I m sure you ve all already heard about what calculus is from the other studets i my calculus class, but just i case you eed a refresher,
More informationORDERS OF GROWTH KEITH CONRAD
ORDERS OF GROWTH KEITH CONRAD Itroductio Gaiig a ituitive feel for the relative growth of fuctios is importat if you really wat to uderstad their behavior It also helps you better grasp topics i calculus
More information1. a n = 2. a n = 3. a n = 4. a n = 5. a n = 6. a n =
Versio PREVIEW Homework Berg (5860 This pritout should have 9 questios. Multiplechoice questios may cotiue o the ext colum or page fid all choices before aswerig. CalCb0b 00 0.0 poits Rewrite the fiite
More information4.3. 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 informationMath 475, Problem Set #6: Solutions
Math 475, Problem Set #6: Solutios A (a) For each poit (a, b) with a, b oegative itegers satisfyig ab 8, cout the paths from (0,0) to (a, b) where the legal steps from (i, j) are to (i 2, j), (i, j 2),
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 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 informationMath 115 HW #4 Solutions
Math 5 HW #4 Solutios From 2.5 8. Does the series coverge or diverge? ( ) 3 + 2 = Aswer: This is a alteratig series, so we eed to check that the terms satisfy the hypotheses of the Alteratig Series Test.
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 informationhp calculators HP 30S Base Conversions Numbers in Different Bases Practice Working with Numbers in Different Bases
Numbers i Differet Bases Practice Workig with Numbers i Differet Bases Numbers i differet bases Our umber system (called HiduArabic) is a decimal system (it s also sometimes referred to as deary system)
More informationChapter Eleven. Taylor Series. (x a) k. c k. k= 0
Chapter Eleve Taylor Series 111 Power Series Now that we are kowledgeable about series, we ca retur to the problem of ivestigatig the approximatio of fuctios by Taylor polyomials of higher ad higher degree
More information8 The Poisson Distribution
8 The Poisso Distributio Let X biomial, p ). Recall that this meas that X has pmf ) p,p k) p k k p ) k for k 0,,...,. ) Agai, thik of X as the umber of successes i a series of idepedet experimets, each
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 informationhttp://www.webassign.net/v4cgijeff.downs@wnc/control.pl
Assigmet Previewer http://www.webassig.et/vcgijeff.dows@wc/cotrol.pl of // : PM Practice Eam () Questio Descriptio Eam over chapter.. Questio DetailsLarCalc... [] Fid the geeral solutio of the differetial
More informationUnit 8 Rational Functions
Uit 8 Ratioal Fuctios Algebraic Fractios: Simplifyig Algebraic Fractios: To simplify a algebraic fractio meas to reduce it to lowest terms. This is doe by dividig out the commo factors i the umerator ad
More informationwhen n = 1, 2, 3, 4, 5, 6, This list represents the amount of dollars you have after n days. Note: The use of is read as and so on.
Geometric eries Before we defie what is meat by a series, we eed to itroduce a related topic, that of sequeces. Formally, a sequece is a fuctio that computes a ordered list. uppose that o day 1, you have
More informationStrategy for Testing Series
Strategy for Testig Series We ow have several ways of testig a series for covergece or divergece; the problem is to decide which test to use o which series. I this respect testig series is similar to itegratig
More informationMath 105: Review for Final Exam, Part II  SOLUTIONS
Math 5: Review for Fial Exam, Part II  SOLUTIONS. Cosider the fuctio fx) =x 3 l x o the iterval [/e, e ]. a) Fid the x ad ycoordiates of ay ad all local extrema ad classify each as a local maximum or
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 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 information3.2 Introduction to Infinite Series
3.2 Itroductio to Ifiite Series May of our ifiite sequeces, for the remaider of the course, will be defied by sums. For example, the sequece S m := 2. () is defied by a sum. Its terms (partial sums) are
More information8.5 Alternating infinite series
65 8.5 Alteratig ifiite series I the previous two sectios we cosidered oly series with positive terms. I this sectio we cosider series with both positive ad egative terms which alterate: positive, egative,
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 informationTHE ARITHMETIC OF INTEGERS.  multiplication, exponentiation, division, addition, and subtraction
THE ARITHMETIC OF INTEGERS  multiplicatio, expoetiatio, divisio, additio, ad subtractio What to do ad what ot to do. THE INTEGERS Recall that a iteger is oe of the whole umbers, which may be either positive,
More informationLesson 12. Sequences and Series
Retur to List of Lessos Lesso. Sequeces ad Series A ifiite sequece { a, a, a,... a,...} ca be thought of as a list of umbers writte i defiite order ad certai patter. It is usually deoted by { a } =, or
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 informationSection 9.2 Series and Convergence
Sectio 9. Series ad Covergece Goals of Chapter 9 Approximate Pi Prove ifiite series are aother importat applicatio of limits, derivatives, approximatio, slope, ad cocavity of fuctios. Fid challegig atiderivatives
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 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 informationARITHMETIC AND GEOMETRIC PROGRESSIONS
Arithmetic Ad Geometric Progressios Sequeces Ad ARITHMETIC AND GEOMETRIC PROGRESSIONS Successio of umbers of which oe umber is desigated as the first, other as the secod, aother as the third ad so o gives
More informationFourier Series and the Wave Equation Part 2
Fourier Series ad the Wave Equatio Part There are two big ideas i our work this week. The first is the use of liearity to break complicated problems ito simple pieces. The secod is the use of the symmetries
More informationTo get the next Fibonacci number, you add the previous two. numbers are defined by the recursive formula. F 1 F n+1
Liear Algebra Notes Chapter 6 FIBONACCI NUMBERS The Fiboacci umbers are F, F, F 2, F 3 2, F 4 3, F, F 6 8, To get the ext Fiboacci umber, you add the previous two umbers are defied by the recursive formula
More informationREVISION SHEET FP2 (AQA) CALCULUS. x x π π. Standard Calculus of Inverse Trig and Hyperbolic Trig Functions = + arcsin x = + ar sinh x
the Further Mathematics etwork www.fmetwork.org.uk V 07 REVISION SHEET FP (AQA) CALCULUS The mai ideas are: Calculus usig iverse trig fuctios & hperbolic trig fuctios ad their iverses. Calculatig arc legths.
More informationUSING STATISTICAL FUNCTIONS ON A SCIENTIFIC CALCULATOR
USING STATISTICAL FUNCTIONS ON A SCIENTIFIC CALCULATOR Objective:. Improve calculator skills eeded i a multiple choice statistical eamiatio where the eam allows the studet to use a scietific calculator..
More informationNumerical Solution of Equations
School of Mechaical Aerospace ad Civil Egieerig Numerical Solutio of Equatios T J Craft George Begg Buildig, C4 TPFE MSc CFD Readig: J Ferziger, M Peric, Computatioal Methods for Fluid Dyamics HK Versteeg,
More information1 The Binomial Theorem: Another Approach
The Biomial Theorem: Aother Approach Pascal s Triagle I class (ad i our text we saw that, for iteger, the biomial theorem ca be stated (a + b = c a + c a b + c a b + + c ab + c b, where the coefficiets
More informationMath 152 Final Exam Review
Math 5 Fial Eam Review Problems Math 5 Fial Eam Review Problems appearig o your iclass fial will be similar to those here but will have umbers ad fuctios chaged. Here is a eample of the way problems selected
More informationHere are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed.
This documet was writte ad copyrighted by Paul Dawkis. Use of this documet ad its olie versio is govered by the Terms ad Coditios of Use located at http://tutorial.math.lamar.edu/terms.asp. The olie versio
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 informationModule 4: Mathematical Induction
Module 4: Mathematical Iductio Theme 1: Priciple of Mathematical Iductio Mathematical iductio is used to prove statemets about atural umbers. As studets may remember, we ca write such a statemet as a predicate
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 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 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 informationAlgebra Work Sheets. Contents
The work sheets are grouped accordig to math skill. Each skill is the arraged i a sequece of work sheets that build from simple to complex. Choose the work sheets that best fit the studet s eed ad will
More informationHW 1 Solutions Math 115, Winter 2009, Prof. Yitzhak Katznelson
HW Solutios Math 5, Witer 2009, Prof. Yitzhak Katzelso.: Prove 2 + 2 2 +... + 2 = ( + )(2 + ) for all atural umbers. The proof is by iductio. Call the th propositio P. The basis for iductio P is the statemet
More informationEquation of a line. Line in coordinate geometry. Slopeintercept form ( 斜 截 式 ) Intercept form ( 截 距 式 ) Pointslope form ( 點 斜 式 )
Chapter : Liear Equatios Chapter Liear Equatios Lie i coordiate geometr I Cartesia coordiate sstems ( 卡 笛 兒 坐 標 系 統 ), a lie ca be represeted b a liear equatio, i.e., a polomial with degree. But before
More informationNATIONAL SENIOR CERTIFICATE GRADE 12
NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P NOVEMBER 009() MARKS: 50 TIME: 3 hours This questio paper cosists of 0 pages, a iformatio sheet ad diagram sheet. Please tur over Mathematics/P DoE/November
More informationReview for College Algebra Final Exam
Review for College Algebra Fial Exam (Please remember that half of the fial exam will cover chapters 14. This review sheet covers oly the ew material, from chapters 5 ad 7.) 5.1 Systems of equatios i
More informationIntro to Sequences / Arithmetic Sequences and Series Levels
Itro to Sequeces / Arithmetic Sequeces ad Series Levels Level : pg. 569: #7, 0, 33 Pg. 575: #, 7, 8 Pg. 584: #8, 9, 34, 36 Levels, 3, ad 4(Fiboacci Sequece Extesio) See Hadout Check for Uderstadig Level
More informationSum of Exterior Angles of Polygons TEACHER NOTES
Sum of Exterior Agles of Polygos TEACHER NOTES Math Objectives Studets will determie that the iterior agle of a polygo ad a exterior agle of a polygo form a liear pair (i.e., the two agles are supplemetary).
More information1 n. n > dt. t < n 1 + n=1
Math 05 otes C. Pomerace The harmoic sum The harmoic sum is the sum of recirocals of the ositive itegers. We kow from calculus that it diverges, this is usually doe by the itegral test. There s a more
More informationChapter Gaussian Elimination
Chapter 04.06 Gaussia Elimiatio After readig this chapter, you should be able to:. solve a set of simultaeous liear equatios usig Naïve Gauss elimiatio,. lear the pitfalls of the Naïve Gauss elimiatio
More informationSearching Algorithm Efficiencies
Efficiecy of Liear Search Searchig Algorithm Efficiecies Havig implemeted the liear search algorithm, how would you measure its efficiecy? A useful measure (or metric) should be geeral, applicable to ay
More informationBINOMIAL EXPANSIONS 12.5. In this section. Some Examples. Obtaining the Coefficients
652 (1226) Chapter 12 Sequeces ad Series 12.5 BINOMIAL EXPANSIONS I this sectio Some Examples Otaiig the Coefficiets The Biomial Theorem I Chapter 5 you leared how to square a iomial. I this sectio you
More informationIncremental calculation of weighted mean and variance
Icremetal calculatio of weighted mea ad variace Toy Fich faf@cam.ac.uk dot@dotat.at Uiversity of Cambridge Computig Service February 009 Abstract I these otes I eplai how to derive formulae for umerically
More informationThe Limit of a Sequence
3 The Limit of a Sequece 3. Defiitio of limit. I Chapter we discussed the limit of sequeces that were mootoe; this restrictio allowed some shortcuts ad gave a quick itroductio to the cocept. But may importat
More informationI. Chisquared Distributions
1 M 358K Supplemet to Chapter 23: CHISQUARED DISTRIBUTIONS, TDISTRIBUTIONS, AND DEGREES OF FREEDOM To uderstad tdistributios, we first eed to look at aother family of distributios, the chisquared distributios.
More information1.3 Binomial Coefficients
18 CHAPTER 1. COUNTING 1. Biomial Coefficiets I this sectio, we will explore various properties of biomial coefficiets. Pascal s Triagle Table 1 cotais the values of the biomial coefficiets ( ) for 0to
More informationRadicals and Fractional Exponents
Radicals ad Roots Radicals ad Fractioal Expoets I math, may problems will ivolve what is called the radical symbol, X is proouced the th root of X, where is or greater, ad X is a positive umber. What it
More information2.3. GEOMETRIC SERIES
6 CHAPTER INFINITE SERIES GEOMETRIC SERIES Oe of the most importat types of ifiite series are geometric series A geometric series is simply the sum of a geometric sequece, Fortuately, geometric series
More informationSolving Inequalities
Solvig Iequalities Say Thaks to the Authors Click http://www.ck12.org/saythaks (No sig i required) To access a customizable versio of this book, as well as other iteractive cotet, visit www.ck12.org CK12
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 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 informationSpss Lab 7: Ttests Section 1
Spss Lab 7: Ttests Sectio I this lab, we will be usig everythig we have leared i our text ad applyig that iformatio to uderstad ttests for parametric ad oparametric data. THERE WILL BE TWO SECTIONS FOR
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 informationAQA STATISTICS 1 REVISION NOTES
AQA STATISTICS 1 REVISION NOTES AVERAGES AND MEASURES OF SPREAD www.mathsbox.org.uk Mode : the most commo or most popular data value the oly average that ca be used for qualitative data ot suitable if
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 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 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 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 informationConcept #1. Goals for Presentation. I m going to be a mathematics teacher: Where did this stuff come from? Why didn t I know this before?
I m goig to be a mathematics teacher: Why did t I kow this before? Steve Williams Associate Professor of Mathematics/ Coordiator of Secodary Mathematics Educatio Lock Have Uiversity of PA swillia@lhup.edu
More informationPower Factor in Electrical Power Systems with NonLinear Loads
Power Factor i Electrical Power Systems with NoLiear Loads By: Gozalo Sadoval, ARTECHE / INELAP S.A. de C.V. Abstract. Traditioal methods of Power Factor Correctio typically focus o displacemet power
More informationsum of all values n x = the number of values = i=1 x = n n. When finding the mean of a frequency distribution the mean is given by
Statistics Module Revisio Sheet The S exam is hour 30 miutes log ad is i two sectios Sectio A 3 marks 5 questios worth o more tha 8 marks each Sectio B 3 marks questios worth about 8 marks each You are
More informationCHAPTER 7: Central Limit Theorem: CLT for Averages (Means)
CHAPTER 7: Cetral Limit Theorem: CLT for Averages (Meas) X = the umber obtaied whe rollig oe six sided die oce. If we roll a six sided die oce, the mea of the probability distributio is X P(X = x) Simulatio:
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 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 informationDivide and Conquer, Solving Recurrences, Integer Multiplication Scribe: Juliana Cook (2015), V. Williams Date: April 6, 2016
CS 6, Lecture 3 Divide ad Coquer, Solvig Recurreces, Iteger Multiplicatio Scribe: Juliaa Cook (05, V Williams Date: April 6, 06 Itroductio Today we will cotiue to talk about divide ad coquer, ad go ito
More informationSum and Product Rules. Combinatorics. Some Subtler Examples
Combiatorics Sum ad Product Rules Problem: How to cout without coutig. How do you figure out how may thigs there are with a certai property without actually eumeratig all of them. Sometimes this requires
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 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 information3. Greatest Common Divisor  Least Common Multiple
3 Greatest Commo Divisor  Least Commo Multiple Defiitio 31: The greatest commo divisor of two atural umbers a ad b is the largest atural umber c which divides both a ad b We deote the greatest commo gcd
More information7) an = 7 n 7n. Solve the problem. Answer the question. n=1. Solve the problem. Answer the question. 16) an =
Eam Name MULTIPLE CHOICE. Choose the oe alterative that best comletes the statemet or aswers the questio. ) Use series to estimate the itegral's value to withi a error of magitude less tha .. l( + )d..79.9.77
More informationMESSAGE TO TEACHERS: NOTE TO EDUCATORS:
MESSAGE TO TEACHERS: NOTE TO EDUCATORS: Attached herewith, please fid suggested lesso plas for term 1 of MATHEMATICS Grade 12. Please ote that these lesso plas are to be used oly as a guide ad teachers
More informationNATIONAL SENIOR CERTIFICATE GRADE 12
NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P FEBRUARY/MARCH 009 MARKS: 50 TIME: 3 hours This questio paper cosists of 0 pages, a iformatio sheet ad 3 diagram sheets. Please tur over Mathematics/P DoE/Feb.
More informationProperties of MLE: consistency, asymptotic normality. Fisher information.
Lecture 3 Properties of MLE: cosistecy, asymptotic ormality. Fisher iformatio. I this sectio we will try to uderstad why MLEs are good. Let us recall two facts from probability that we be used ofte throughout
More informationCS103X: Discrete Structures Homework 4 Solutions
CS103X: Discrete Structures Homewor 4 Solutios Due February 22, 2008 Exercise 1 10 poits. Silico Valley questios: a How may possible sixfigure salaries i whole dollar amouts are there that cotai at least
More information8.3 POLAR FORM AND DEMOIVRE S THEOREM
SECTION 8. POLAR FORM AND DEMOIVRE S THEOREM 48 8. POLAR FORM AND DEMOIVRE S THEOREM Figure 8.6 (a, b) b r a 0 θ Complex Number: a + bi Rectagular Form: (a, b) Polar Form: (r, θ) At this poit you ca add,
More informationHow to set up your GMC Online account
How to set up your GMC Olie accout Mai title Itroductio GMC Olie is a secure part of our website that allows you to maage your registratio with us. Over 100,000 doctors already use GMC Olie. We wat every
More informationConfidence Intervals and Sample Size
8/7/015 C H A P T E R S E V E N Cofidece Itervals ad Copyright 015 The McGrawHill Compaies, Ic. Permissio required for reproductio or display. 1 Cofidece Itervals ad Outlie 71 Cofidece Itervals for the
More informationA Resource for Freestanding Mathematics Qualifications Working with %
Ca you aswer these questios? A savigs accout gives % iterest per aum.. If 000 is ivested i this accout, how much will be i the accout at the ed of years? A ew car costs 16 000 ad its value falls by 1%
More information