Fractions and Equations
|
|
- Christiana Carroll
- 5 years ago
- Views:
Transcription
1 Frctios d Eqtios Remider of frctios We he sed frctios with mers efore: Add or strct: Chge to commo deomitor + chge oth frctios to commo deomitor of Mltipl: Ccel dow, if o c, the mltipl tops (mertors), mltipl ottoms (deomitors). ccel dow 9 8 mltipl Diide: Iert (tr pside dow) the secod frctio, chge sig to mltipl. 0 iert d frctio d chge sig to mltipl Now tret s mltiplictio. Ccel dow. mltipl: 7 0 Mied mers: If ddig or strctig, del with whole mer prts seprtel. whole mer prt first: + the s preiosl, sig commo deomitor of ow del with frctio: 6 0 +
2 Mied mers: If mltiplig or diidig, YOU MUST CHANGE THE MIXED NUMBERS TO IMPROPER (Top He) FRACTIONS. chge to improper frctios: 7 Ccel dow, where o c. 7 d mltipl 7 chge to improper frctios: iert d chge to mltipl: mltipl Chgig from frctio to mied mer: To chge improper (top he) frctio to mied mer: Diide the top (mertor) the ottom (deomitor) 9 ecse: 7 ecse: Chgig from mied mer to frctio: To chge mied mer to improper (top he) frctio: Mltipl the whole mer prt the deomitor d dd o the mertor. 8 ecse: ecse: = =
3 Algeric Frctios: Simplifig frctios: Diide the mertor d deomitor commo fctor so, 0 More emples: diide commo fctor diide commo fctor 6 6 diide commo fctor 6 diide commo fctor Tkig ot commo fctor first: m + 6 tke ot commo fctor ( m + ) ( m + ) m + + tke ot commo fctor ( + ) + + ( ) More ioled fctors ( + )( ) ( ) diide commo fctor ( ) ( + ) ( ) ( ) + ( t )( t + ) ( t + )( t ) diide commo fctor ( t + ) ( t ) ( t + ) t ( t + ) ( t ) t
4 Frther fctoristio: fctorise top ( + ) + ccel ( + ) + d + d d + fctorise top d( d + ) d + ccel d ( d + ) d + d t + t + fctorise ( t + ) ( t + ) ccel ( t + ) ( t ) + m m m fctorise m( m ) ( m ) ccel m ( m ) ( m ) m Usig differece of two sqres: fctorise ( + )( ) ccel ( + ) ( ) + m 9 m + fctorise ( m + )( m ) m + ccel ( m + ) m + ( m ) m Tr these: (coer p the swers o the right first) ( + ) + 0 ( + ) ( )( ) + d d + d + d ( + ) ( + ) d d d d d d d + ( + ) +
5 Algeric Frctios: Mltiplictio: The sme rles ppl s with mers: Ccel dow, if o c, the mltipl tops (mertors), mltipl ottoms (deomitors). m p othig will ccel so mltipl r m pr t ccel the mltipl t t k ccel the mltipl k k ccel the mltipl ccel the mltipl Diisio: The sme rles ppl s with mers: Iert (tr pside dow) the secod frctio, chge sig to mltipl. iert, ccel, mltipl iert, ccel, mltipl iert, ccel, mltipl iert, ccel, mltipl iert, ccel, mltipl iert, ccel, mltipl
6 Tr these: (coer p the swers o the right first) z z z 6 m m p r p r p t t t t t m m m p q p q p q pq q p t 7 t 7 t
7 Algeric Frctios: Additio d strctio: The sme rles ppl s with mers: Chge to commo deomitor + se commo deomitor of se commo deomitor of + se commo deomitor of se commo deomitor of m m m m m m m m m m + + se commo deomitor of 6 + ( + ) ( ) se commo deomitor of se commo deomitor of se commo deomitor of ( ) ( ) ( ) ( ) ( ) p p + se commo deomitor of p( p + ) ( + ) + ( ) ( ) ( ) p + p p p 7 p p p + p + p p p + p p + p p +
8 Eqtios with Frctios: A method: Remoe frctios Remoe rckets Use these rles: dd or strct the sme mer o ech side mltipl or diide ech side the sme mer. = mltipl oth sides 6 (l.c.m. of d ) 6 = 6 = which we c sole: = = = ( ) mltipl oth sides = ( ) ( ) = which we c sole: = = = = mltipl oth sides = = 0 which we c sole: = 0 = = mltipl oth sides = 6 ( ) = 0 9 = 0 = 9 = = mltipl oth sides = ( ) ( ) = = 8 + = 8 = = ( + ) = mltipl oth sides ( + ) = ( ) 9 + = 9 = = = 6 =
9 Some Pst Pper Qestios: Algeric Frctios. Epress s sigle frctio i its simplest form, 0. Epress s sigle frctio i its simplest form +, 0. Epress s sigle frctio i its simplest form, 0 or ( ) Soltios: ( ) ( ) ( ) ( ) ( ) More pst pper qestios o et pge
10 Some Pst Pper Qestios: Frctio Eqtios. Sole the eqtio. Sole the eqtio + = + + =, where is rel mer.. Sole lgericll the eqtio ( + ) =. Sole the eqtio + =. Sole this eqtio for : = 6. Sole lgericll, the eqtio 7. Sole lgericll, the eqtio ( ) + = ( ) m m = Soltios:... + = mltipl throghot ( + ) = = + + = mltipl throghot 6 ( + ) ( + ) = 6 = + = mltipl throghot 6 = 6 = 8. + = mltipl throghot 6 ( ) + ( ) = =. = mltipl throghot ( ) 6 = = 6. + = mltipl throghot 6 ( + ) = = 6 7. m m = mltipl throghot m = m m = 8
Appendix A Examples for Labs 1, 2, 3 1. FACTORING POLYNOMIALS
Appedi A Emples for Ls,,. FACTORING POLYNOMIALS Tere re m stdrd metods of fctorig tt ou ve lered i previous courses. You will uild o tese fctorig metods i our preclculus course to ele ou to fctor epressios
More informationAdvanced Higher Grade
Prelim Emitio / (Assessig Uits & ) MATHEMATICS Avce Higher Gre Time llowe - hors Re Crefll. Fll creit will be give ol where the soltio cotis pproprite workig.. Clcltors m be se i this pper.. Aswers obtie
More informationSM2H. Unit 2 Polynomials, Exponents, Radicals & Complex Numbers Notes. 3.1 Number Theory
SMH Uit Polyomils, Epoets, Rdicls & Comple Numbers Notes.1 Number Theory .1 Addig, Subtrctig, d Multiplyig Polyomils Notes Moomil: A epressio tht is umber, vrible, or umbers d vribles multiplied together.
More informationWestchester Community College Elementary Algebra Study Guide for the ACCUPLACER
Westchester Commuity College Elemetry Alger Study Guide for the ACCUPLACER Courtesy of Aims Commuity College The followig smple questios re similr to the formt d cotet of questios o the Accuplcer Elemetry
More informationSect Simplifying Radical Expressions. We can use our properties of exponents to establish two properties of radicals: and
128 Sect 10.3 - Simplifyig Rdicl Expressios Cocept #1 Multiplictio d Divisio Properties of Rdicls We c use our properties of expoets to estlish two properties of rdicls: () 1/ 1/ 1/ & ( Multiplictio d
More informationUnit 1. Extending the Number System. 2 Jordan School District
Uit Etedig the Number System Jord School District Uit Cluster (N.RN. & N.RN.): Etedig Properties of Epoets Cluster : Etedig properties of epoets.. Defie rtiol epoets d eted the properties of iteger epoets
More informationSurds, Indices, and Logarithms Radical
MAT 6 Surds, Idices, d Logrithms Rdicl Defiitio of the Rdicl For ll rel, y > 0, d ll itegers > 0, y if d oly if y where is the ide is the rdicl is the rdicd. Surds A umber which c be epressed s frctio
More informationCH 39 USING THE GCF TO REDUCE FRACTIONS
359 CH 39 USING THE GCF TO EDUCE FACTIONS educig Algeric Frctios M ost of us lered to reduce rithmetic frctio dividig the top d the ottom of the frctio the sme (o-zero) umer. For exmple, 30 30 5 75 75
More informationSection 3.6: Rational Exponents
CHAPTER Sectio.6: Rtiol Epoets Sectio.6: Rtiol Epoets Objectives: Covert betwee rdicl ottio d epoetil ottio. Siplif epressios with rtiol epoets usig the properties of epoets. Multipl d divide rdicl epressios
More informationAssessment Center Elementary Algebra Study Guide for the ACCUPLACER (CPT)
Assessmet Ceter Elemetr Alger Stud Guide for the ACCUPLACER (CPT) The followig smple questios re similr to the formt d cotet of questios o the Accuplcer Elemetr Alger test. Reviewig these smples will give
More informationCopyrighted by Gabriel Tang B.Ed., B.Sc. Page 1.
Alger Prerequisites Chpter: Alger Review P-: Modelig the Rel World Prerequisites Chpter: Alger Review Model: - mthemticl depictio of rel world coditio. - it c e formul (equtios with meigful vriles), properly
More informationStudent Success Center Elementary Algebra Study Guide for the ACCUPLACER (CPT)
Studet Success Ceter Elemetry Algebr Study Guide for the ACCUPLACER (CPT) The followig smple questios re similr to the formt d cotet of questios o the Accuplcer Elemetry Algebr test. Reviewig these smples
More informationA GENERAL METHOD FOR SOLVING ORDINARY DIFFERENTIAL EQUATIONS: THE FROBENIUS (OR SERIES) METHOD
Diol Bgoo () A GENERAL METHOD FOR SOLVING ORDINARY DIFFERENTIAL EQUATIONS: THE FROBENIUS (OR SERIES) METHOD I. Itroductio The first seprtio of vribles (see pplictios to Newto s equtios) is ver useful method
More information8 factors of x. For our second example, let s raise a power to a power:
CH 5 THE FIVE LAWS OF EXPONENTS EXPONENTS WITH VARIABLES It s no time for chnge in tctics, in order to give us deeper understnding of eponents. For ech of the folloing five emples, e ill stretch nd squish,
More informationNumerical Integration
Numericl tegrtio Newto-Cotes Numericl tegrtio Scheme Replce complicted uctio or tulted dt with some pproimtig uctio tht is esy to itegrte d d 3-7 Roerto Muscedere The itegrtio o some uctios c e very esy
More informationNext we encountered the exponent equaled 1, so we take a leap of faith and generalize that for any x (that s not zero),
79 CH 0 MORE EXPONENTS Itroductio T his chpter is cotiutio of the epoet ides we ve used m times efore. Our gol is to comie epressios with epoets i them. First, quick review of epoets: 0 0 () () 0 ( ) 0
More informationM098 Carson Elementary and Intermediate Algebra 3e Section 10.2
M09 Crso Eleetry d Iteredite Alger e Sectio 0. Ojectives. Evlute rtiol epoets.. Write rdicls s epressios rised to rtiol epoets.. Siplify epressios with rtiol uer epoets usig the rules of epoets.. Use rtiol
More informationNumbers (Part I) -- Solutions
Ley College -- For AMATYC SML Mth Competitio Cochig Sessios v.., [/7/00] sme s /6/009 versio, with presettio improvemets Numbers Prt I) -- Solutios. The equtio b c 008 hs solutio i which, b, c re distict
More informationCHAPTER 1 INTRODUCTION NUMBER SYSTEMS AND CONVERSION
Fudmetls of Logic Desig Chp. CHAPTE /9 INTODUCTION NUMBE SYSTEMS AND CONVESION This chpter i the book icludes: Objectives Study Guide. Digitl Systems d Switchig Circuits. Number Systems d Coversio. Biry
More informationPROGRESSIONS AND SERIES
PROGRESSIONS AND SERIES A sequece is lso clled progressio. We ow study three importt types of sequeces: () The Arithmetic Progressio, () The Geometric Progressio, () The Hrmoic Progressio. Arithmetic Progressio.
More informationName: Period: Date: 2.1 Rules of Exponents
SM NOTES Ne: Period: Dte:.1 Rules of Epoets The followig properties re true for ll rel ubers d b d ll itegers d, provided tht o deoitors re 0 d tht 0 0 is ot cosidered. 1 s epoet: 1 1 1 = e.g.) 7 = 7,
More information[ 20 ] 1. Inequality exists only between two real numbers (not complex numbers). 2. If a be any real number then one and only one of there hold.
[ 0 ]. Iequlity eists oly betwee two rel umbers (ot comple umbers).. If be y rel umber the oe d oly oe of there hold.. If, b 0 the b 0, b 0.. (i) b if b 0 (ii) (iii) (iv) b if b b if either b or b b if
More informationGRAPHING LINEAR EQUATIONS. Linear Equations. x l ( 3,1 ) _x-axis. Origin ( 0, 0 ) Slope = change in y change in x. Equation for l 1.
GRAPHING LINEAR EQUATIONS Qudrt II Qudrt I ORDERED PAIR: The first umer i the ordered pir is the -coordite d the secod umer i the ordered pir is the y-coordite. (, ) Origi ( 0, 0 ) _-is Lier Equtios Qudrt
More informationIf a is any non zero real or imaginary number and m is the positive integer, then a...
Idices d Surds.. Defiitio of Idices. If is o ero re or igir uer d is the positive iteger the...... ties. Here is ced the se d the ide power or epoet... Lws of Idices. 0 0 0. where d re rtio uers where
More informationLincoln Land Community College Placement and Testing Office
Licol Ld Commuity College Plcemet d Testig Office Elemetry Algebr Study Guide for the ACCUPLACER (CPT) A totl of questios re dmiistered i this test. The first type ivolves opertios with itegers d rtiol
More information10.5 Test Info. Test may change slightly.
0.5 Test Ifo Test my chge slightly. Short swer (0 questios 6 poits ech) o Must choose your ow test o Tests my oly be used oce o Tests/types you re resposible for: Geometric (kow sum) Telescopig (kow sum)
More informationIDENTITIES FORMULA AND FACTORISATION
SPECIAL PRODUCTS AS IDENTITIES FORMULA AND FACTORISATION. Find the product of : (i) (n + ) nd ((n + 5) ( + 0.) nd ( + 0.5) (iii) (y + 0.7) nd (y + 0.) (iv) + 3 nd + 3 (v) y + 5 nd 3 y + (iv) 5 + 7 nd +
More informationdenominator, think trig! Memorize the following two formulas; you will use them often!
7. Bsic Itegrtio Rules Some itegrls re esier to evlute th others. The three problems give i Emple, for istce, hve very similr itegrds. I fct, they oly differ by the power of i the umertor. Eve smll chges
More informationLogarithmic Scales: the most common example of these are ph, sound and earthquake intensity.
Numercy Itroductio to Logrithms Logrithms re commoly credited to Scottish mthemtici med Joh Npier who costructed tle of vlues tht llowed multiplictios to e performed y dditio of the vlues from the tle.
More informationNorthwest High School s Algebra 2
Northwest High School s Algebr Summer Review Pcket 0 DUE August 8, 0 Studet Nme This pcket hs bee desiged to help ou review vrious mthemticl topics tht will be ecessr for our success i Algebr. Istructios:
More informationCalculus II Homework: The Integral Test and Estimation of Sums Page 1
Clculus II Homework: The Itegrl Test d Estimtio of Sums Pge Questios Emple (The p series) Get upper d lower bouds o the sum for the p series i= /ip with p = 2 if the th prtil sum is used to estimte the
More informationApproximate Integration
Study Sheet (7.7) Approimte Itegrtio I this sectio, we will ler: How to fid pproimte vlues of defiite itegrls. There re two situtios i which it is impossile to fid the ect vlue of defiite itegrl. Situtio:
More informationIntroduction to Algebra - Part 2
Alger Module A Introduction to Alger - Prt Copright This puliction The Northern Alert Institute of Technolog 00. All Rights Reserved. LAST REVISED Oct., 008 Introduction to Alger - Prt Sttement of Prerequisite
More informationQn Suggested Solution Marking Scheme 1 y. G1 Shape with at least 2 [2]
Mrkig Scheme for HCI 8 Prelim Pper Q Suggested Solutio Mrkig Scheme y G Shpe with t lest [] fetures correct y = f'( ) G ll fetures correct SR: The mimum poit could be i the first or secod qudrt. -itercept
More informationUnit 1 Chapter-3 Partial Fractions, Algebraic Relationships, Surds, Indices, Logarithms
Uit Chpter- Prtil Frctios, Algeric Reltioships, Surds, Idices, Logriths. Prtil Frctios: A frctio of the for 7 where the degree of the uertor is less th the degree of the deoitor is referred to s proper
More informationMathematical Notation Math Calculus for Business and Social Science
Mthemticl Nottio Mth 190 - Clculus for Busiess d Socil Sciece Use Word or WordPerfect to recrete the followig documets. Ech rticle is worth 10 poits d should e emiled to the istructor t jmes@richld.edu.
More informationf(bx) dx = f dx = dx l dx f(0) log b x a + l log b a 2ɛ log b a.
Eercise 5 For y < A < B, we hve B A f fb B d = = A B A f d f d For y ɛ >, there re N > δ >, such tht d The for y < A < δ d B > N, we hve ba f d f A bb f d l By ba A A B A bb ba fb d f d = ba < m{, b}δ
More informationPANIMALAR INSTITUTE OF TECHNOLOGY
PIT/QB/MATHEMATICS/I/MA5/M PANIMALAR INSTITUTE OF TECHNOLOGY (A Christi Miorit Istittio JAISAKTHI EDUCATIONAL TRUST (A ISO 9: 8 Certified Istittio No.: 9, Bglore Trk Rod, Vrdhrjprm, Nrthpetti, CHENNAI.
More informationLaws of Integral Indices
A Lws of Itegrl Idices A. Positive Itegrl Idices I, is clled the se, is clled the idex lso clled the expoet. mes times.... Exmple Simplify 5 6 c Solutio 8 5 6 c 6 Exmple Simplify Solutio The results i
More informationTest Info. Test may change slightly.
9. 9.6 Test Ifo Test my chge slightly. Short swer (0 questios 6 poits ech) o Must choose your ow test o Tests my oly be used oce o Tests/types you re resposible for: Geometric (kow sum) Telescopig (kow
More information( ) = A n + B ( ) + Bn
MATH 080 Test 3-SOLUTIONS Fll 04. Determie if the series is coverget or diverget. If it is coverget, fid its sum.. (7 poits) = + 3 + This is coverget geometric series where r = d
More informationFor students entering Honors Precalculus Summer Packet
Hoors PreClculus Summer Review For studets eterig Hoors Preclculus Summer Pcket The prolems i this pcket re desiged to help ou review topics from previous mthemtics courses tht re importt to our success
More informationPHY2061 Enriched Physics 2 Lecture Notes Relativity 3. Relativity 3
PHY61 Eried Psis Letre Notes Reltiit 3 Reltiit 3 Dislimer: Tese letre otes re ot met to reple te orse tetbook. Te otet m be iomplete. Some topis m be ler. Tese otes re ol met to be std id d spplemet to
More informationTHE NATIONAL UNIVERSITY OF IRELAND, CORK COLÁISTE NA hollscoile, CORCAIGH UNIVERSITY COLLEGE, CORK SUMMER EXAMINATION 2005 FIRST ENGINEERING
OLLSCOIL NA héireann, CORCAIGH THE NATIONAL UNIVERSITY OF IRELAND, CORK COLÁISTE NA hollscoile, CORCAIGH UNIVERSITY COLLEGE, CORK SUMMER EXAMINATION 2005 FIRST ENGINEERING MATHEMATICS MA008 Clculus d Lier
More informationMath 153: Lecture Notes For Chapter 1
Mth : Lecture Notes For Chpter Sectio.: Rel Nubers Additio d subtrctios : Se Sigs: Add Eples: = - - = - Diff. Sigs: Subtrct d put the sig of the uber with lrger bsolute vlue Eples: - = - = - Multiplictio
More information5.2 Exponent Properties Involving Quotients
5. Eponent Properties Involving Quotients Lerning Objectives Use the quotient of powers property. Use the power of quotient property. Simplify epressions involving quotient properties of eponents. Use
More information* power rule: * fraction raised to negative exponent: * expanded power rule:
Mth 15 Iteredite Alger Stud Guide for E 3 (Chpters 7, 8, d 9) You use 3 5 ote crd (oth sides) d scietific clcultor. You re epected to kow (or hve writte o our ote crd) foruls ou eed. Thik out rules d procedures
More informationChapter 2 Infinite Series Page 1 of 9
Chpter Ifiite eries Pge of 9 Chpter : Ifiite eries ectio A Itroductio to Ifiite eries By the ed of this sectio you will be ble to uderstd wht is met by covergece d divergece of ifiite series recogise geometric
More informationLimit of a function:
- Limit of fuctio: We sy tht f ( ) eists d is equl with (rel) umer L if f( ) gets s close s we wt to L if is close eough to (This defiitio c e geerlized for L y syig tht f( ) ecomes s lrge (or s lrge egtive
More informationBridging the gap: GCSE AS Level
Bridging the gp: GCSE AS Level CONTENTS Chpter Removing rckets pge Chpter Liner equtions Chpter Simultneous equtions 8 Chpter Fctors 0 Chpter Chnge the suject of the formul Chpter 6 Solving qudrtic equtions
More informationThe z Transform. The Discrete LTI System Response to a Complex Exponential
The Trsform The trsform geerlies the Discrete-time Forier Trsform for the etire complex ple. For the complex vrible is sed the ottio: jω x+ j y r e ; x, y Ω rg r x + y {} The Discrete LTI System Respose
More informationALGEBRA. Set of Equations. have no solution 1 b1. Dependent system has infinitely many solutions
Qudrtic Equtios ALGEBRA Remider theorem: If f() is divided b( ), the remider is f(). Fctor theorem: If ( ) is fctor of f(), the f() = 0. Ivolutio d Evlutio ( + b) = + b + b ( b) = + b b ( + b) 3 = 3 +
More informationAlgebra II, Chapter 7. Homework 12/5/2016. Harding Charter Prep Dr. Michael T. Lewchuk. Section 7.1 nth roots and Rational Exponents
Algebr II, Chpter 7 Hrdig Chrter Prep 06-07 Dr. Michel T. Lewchuk Test scores re vilble olie. I will ot discuss the test. st retke opportuit Sturd Dec. If ou hve ot tke the test, it is our resposibilit
More informationPre-Calculus - Chapter 3 Sections Notes
Pre-Clculus - Chpter 3 Sectios 3.1-3.4- Notes Properties o Epoets (Review) 1. ( )( ) = + 2. ( ) =, (c) = 3. 0 = 1 4. - = 1/( ) 5. 6. c Epoetil Fuctios (Sectio 3.1) Deiitio o Epoetil Fuctios The uctio deied
More informationSUTCLIFFE S NOTES: CALCULUS 2 SWOKOWSKI S CHAPTER 11
UTCLIFFE NOTE: CALCULU WOKOWKI CHAPTER Ifiite eries Coverget or Diverget eries Cosider the sequece If we form the ifiite sum 0, 00, 000, 0 00 000, we hve wht is clled ifiite series We wt to fid the sum
More informationSolutions to Problem Set 7
8.0 Clculus Jso Strr Due by :00pm shrp Fll 005 Fridy, Dec., 005 Solutios to Problem Set 7 Lte homework policy. Lte work will be ccepted oly with medicl ote or for other Istitute pproved reso. Coopertio
More informationSCHOOL OF MATHEMATICS AND STATISTICS. Mathematics II (Materials)
Dt proie: Form Sheet MAS5 SCHOOL OF MATHEMATICS AND STATISTICS Mthemtics II (Mteris) Atm Semester -3 hors Mrks wi e wre or swers to qestios i Sectio A or or est THREE swers to qestios i Sectio. Sectio
More informationGraphing Review Part 3: Polynomials
Grphig Review Prt : Polomils Prbols Recll, tht the grph of f ( ) is prbol. It is eve fuctio, hece it is smmetric bout the bout the -is. This mes tht f ( ) f ( ). Its grph is show below. The poit ( 0,0)
More informationEdexcel Core 1 Help Guide
Edexcel Core Help Gide Steve Bldes 0 www.mths.com www.strmths.com Aim The im of this booklet is to llow ppils opportity to brek dow topics i Core ito lgorithmic process. It cold be viewed s checklist of
More informationA Level Mathematics Transition Work. Summer 2018
A Level Mthetics Trsitio Work Suer 08 A Level Mthetics Trsitio A level thetics uses y of the skills you developed t GCSE. The big differece is tht you will be expected to recogise where you use these skills
More informationINTEGRATION TECHNIQUES (TRIG, LOG, EXP FUNCTIONS)
Mthemtics Revisio Guides Itegrtig Trig, Log d Ep Fuctios Pge of MK HOME TUITION Mthemtics Revisio Guides Level: AS / A Level AQA : C Edecel: C OCR: C OCR MEI: C INTEGRATION TECHNIQUES (TRIG, LOG, EXP FUNCTIONS)
More informationSection IV.6: The Master Method and Applications
Sectio IV.6: The Mster Method d Applictios Defiitio IV.6.1: A fuctio f is symptoticlly positive if d oly if there exists rel umer such tht f(x) > for ll x >. A cosequece of this defiitio is tht fuctio
More informationLEVEL I. ,... if it is known that a 1
LEVEL I Fid the sum of first terms of the AP, if it is kow tht + 5 + 0 + 5 + 0 + = 5 The iterior gles of polygo re i rithmetic progressio The smllest gle is 0 d the commo differece is 5 Fid the umber of
More informationFrequency-domain Characteristics of Discrete-time LTI Systems
requecy-domi Chrcteristics of Discrete-time LTI Systems Prof. Siripog Potisuk LTI System descriptio Previous bsis fuctio: uit smple or DT impulse The iput sequece is represeted s lier combitio of shifted
More informationMATHEMATICS AND STATISTICS 1.2
MATHEMATICS AND STATISTICS. Apply lgebric procedures in solving problems Eternlly ssessed 4 credits Electronic technology, such s clcultors or computers, re not permitted in the ssessment of this stndr
More information( a n ) converges or diverges.
Chpter Ifiite Series Pge of Sectio E Rtio Test Chpter : Ifiite Series By the ed of this sectio you will be ble to uderstd the proof of the rtio test test series for covergece by pplyig the rtio test pprecite
More informationMath 3B Midterm Review
Mth 3B Midterm Review Writte by Victori Kl vtkl@mth.ucsb.edu SH 643u Office Hours: R 11:00 m - 1:00 pm Lst updted /15/015 Here re some short otes o Sectios 7.1-7.8 i your ebook. The best idictio of wht
More information1. Twelve less than five times a number is thirty three. What is the number
Alger 00 Midterm Review Nme: Dte: Directions: For the following prolems, on SEPARATE PIECE OF PAPER; Define the unknown vrile Set up n eqution (Include sketch/chrt if necessr) Solve nd show work Answer
More informationChapter Real Numbers
Chpter. - Rel Numbers Itegers: coutig umbers, zero, d the egtive of the coutig umbers. ex: {,-3, -, -,,,, 3, } Rtiol Numbers: quotiets of two itegers with ozero deomitor; termitig or repetig decimls. ex:
More information2.4 Linear Inequalities and Interval Notation
.4 Liner Inequlities nd Intervl Nottion We wnt to solve equtions tht hve n inequlity symol insted of n equl sign. There re four inequlity symols tht we will look t: Less thn , Less thn or
More informationAccuplacer Elementary Algebra Study Guide
Testig Ceter Studet Suess Ceter Aupler Elemetry Alger Study Guide The followig smple questios re similr to the formt d otet of questios o the Aupler Elemetry Alger test. Reviewig these smples will give
More informationMath 154B Elementary Algebra-2 nd Half Spring 2015
Mth 154B Elementry Alger- nd Hlf Spring 015 Study Guide for Exm 4, Chpter 9 Exm 4 is scheduled for Thursdy, April rd. You my use " x 5" note crd (oth sides) nd scientific clcultor. You re expected to know
More informationConvergence of the FEM. by Hui Zhang Jorida Kushova Ruwen Jung
Covergece o te FEM by Hi Zg Jorid Ksov Rwe Jg I order to proo FEM soltios to be coverget, mesremet or teir qlity is reqired. A simple pproc i ect soltio is ccessible is to qtiy te error betwee FEMd te
More informationIntroduction to Digital Signal Processing(DSP)
Forth Clss Commictio II Electricl Dept Nd Nsih Itrodctio to Digitl Sigl ProcessigDSP Recet developmets i digitl compters ope the wy to this sject The geerl lock digrm of DSP system is show elow: Bd limited
More informationx x x a b) Math 233B Intermediate Algebra Fall 2012 Final Exam Study Guide
Mth B Iteredite Alger Fll 0 Fil E Stud Guide The fil e is o Thursd, Deceer th fro :00p :00p. You re llowed scietific clcultor d 4" 6" ide crd for otes. O our ide crd e sure to write foruls ou eeded for
More informationfractions Let s Learn to
5 simple lgebric frctions corne lens pupil retin Norml vision light focused on the retin concve lens Shortsightedness (myopi) light focused in front of the retin Corrected myopi light focused on the retin
More informationn 2 + 3n + 1 4n = n2 + 3n + 1 n n 2 = n + 1
Ifiite Series Some Tests for Divergece d Covergece Divergece Test: If lim u or if the limit does ot exist, the series diverget. + 3 + 4 + 3 EXAMPLE: Show tht the series diverges. = u = + 3 + 4 + 3 + 3
More informationCrushed Notes on MATH132: Calculus
Mth 13, Fll 011 Siyg Yg s Outlie Crushed Notes o MATH13: Clculus The otes elow re crushed d my ot e ect This is oly my ow cocise overview of the clss mterils The otes I put elow should ot e used to justify
More informationChapter 5. The Riemann Integral. 5.1 The Riemann integral Partitions and lower and upper integrals. Note: 1.5 lectures
Chpter 5 The Riem Itegrl 5.1 The Riem itegrl Note: 1.5 lectures We ow get to the fudmetl cocept of itegrtio. There is ofte cofusio mog studets of clculus betwee itegrl d tiderivtive. The itegrl is (iformlly)
More informationA General Construction Method of Simultaneous Confounding in
hk Uiv J Sci : - Jly A Geerl Costrctio ethod of Simlteos Cofodig i - Fctoril Eerimet A Jlil ertmet of Sttistics iosttistics d formtics Uiversity of hk hk gldesh E-mil: mjlil@ivdhked Received o Acceted
More information1. Extend QR downwards to meet the x-axis at U(6, 0). y
In the digrm, two stright lines re to be drwn through so tht the lines divide the figure OPQRST into pieces of equl re Find the sum of the slopes of the lines R(6, ) S(, ) T(, 0) Determine ll liner functions
More information( x y ) x y. a b. a b. Chapter 2Properties of Exponents and Scientific Notation. x x. x y, Example: (x 2 )(x 4 ) = x 6.
Chpter Properties of Epoets d Scietific Nottio Epoet - A umer or symol, s i ( + y), plced to the right of d ove other umer, vrile, or epressio (clled the se), deotig the power to which the se is to e rised.
More informationMathematical Notation Math Calculus & Analytic Geometry I
Mthemticl Nottio Mth - Clculus & Alytic Geometry I Use Wor or WorPerect to recrete the ollowig ocumets. Ech rticle is worth poits shoul e emile to the istructor t jmes@richl.eu. Type your me t the top
More informationCALCULUS I. Extras. Paul Dawkins
CALCULUS I Extrs Pul Dwkis Clculus I Tle of Cotets Prefce... ii Extrs... Itroductio... 3 Proof of Vrious Limit Properties... 4 Proof of Vrious Derivtive Fcts/Formuls/Properties... 0 Proof of Trig Limits...
More informationEXERCISE a a a 5. + a 15 NEETIIT.COM
- Dowlod our droid App. Sigle choice Type Questios EXERCISE -. The first term of A.P. of cosecutive iteger is p +. The sum of (p + ) terms of this series c be expressed s () (p + ) () (p + ) (p + ) ()
More informationNational Quali cations AHEXEMPLAR PAPER ONLY
Ntiol Quli ctios AHEXEMPLAR PAPER ONLY EP/AH/0 Mthemtics Dte Not pplicble Durtio hours Totl mrks 00 Attempt ALL questios. You my use clcultor. Full credit will be give oly to solutios which coti pproprite
More informationHIGHER SCHOOL CERTIFICATE EXAMINATION MATHEMATICS 3 UNIT (ADDITIONAL) AND 3/4 UNIT (COMMON) Time allowed Two hours (Plus 5 minutes reading time)
HIGHER SCHOOL CERTIFICATE EXAMINATION 999 MATHEMATICS 3 UNIT (ADDITIONAL) AND 3/ UNIT (COMMON) Time llowed Two hours (Plus 5 miutes redig time) DIRECTIONS TO CANDIDATES Attempt ALL questios. ALL questios
More information, we would have a series, designated as + j 1
Clculus sectio 9. Ifiite Series otes by Ti Pilchowski A sequece { } cosists of ordered set of ubers. If we were to begi ddig the ubers of sequece together s we would hve series desigted s. Ech iteredite
More information1.3 Continuous Functions and Riemann Sums
mth riem sums, prt 0 Cotiuous Fuctios d Riem Sums I Exmple we sw tht lim Lower() = lim Upper() for the fuctio!! f (x) = + x o [0, ] This is o ccidet It is exmple of the followig theorem THEOREM Let f be
More informationA-Level Mathematics Transition Task (compulsory for all maths students and all further maths student)
A-Level Mthemtics Trnsition Tsk (compulsory for ll mths students nd ll further mths student) Due: st Lesson of the yer. Length: - hours work (depending on prior knowledge) This trnsition tsk provides revision
More informationSection 6.3: Geometric Sequences
40 Chpter 6 Sectio 6.: Geometric Sequeces My jobs offer ul cost-of-livig icrese to keep slries cosistet with ifltio. Suppose, for exmple, recet college grdute fids positio s sles mger erig ul slry of $6,000.
More information8Bindi is 18 years old and
Expoetil fuctios 8Bidi is 8 ers old d ivestig $0 000 i fixed term deposit pig 6% p.. compoud iterest. Whe Bidi hs $0 000 she iteds to put deposit o home. How log will it tke for Bidi s $0 000 to grow to
More informationThomas Whitham Sixth Form
Thoms Whithm Sith Form Pure Mthemtics Unit C Alger Trigonometry Geometry Clculus Vectors Trigonometry Compound ngle formule sin sin cos cos Pge A B sin Acos B cos Asin B A B sin Acos B cos Asin B A B cos
More informationPROPERTIES OF AN EULER SQUARE
PROPERTIES OF N EULER SQURE bout 0 the mathematicia Leoard Euler discussed the properties a x array of letters or itegers ow kow as a Euler or Graeco-Lati Square Such squares have the property that every
More informationBy the end of this set of exercises, you should be able to. reduce an algebraic fraction to its simplest form
ALGEBRAIC OPERATIONS By the end of this set of eercises, you should be ble to () (b) (c) reduce n lgebric frction to its simplest form pply the four rules to lgebric frctions chnge the subject of formul
More informationAP Calculus BC Formulas, Definitions, Concepts & Theorems to Know
P Clls BC Formls, Deiitios, Coepts & Theorems to Kow Deiitio o e : solte Vle: i 0 i 0 e lim Deiitio o Derivtive: h lim h0 h ltertive orm o De o Derivtive: lim Deiitio o Cotiity: is otios t i oly i lim
More informationChapter Real Numbers
Chpter. - Rel Numbers Itegers: coutig umbers, zero, d the egtive of the coutig umbers. ex: {,-3, -, -, 0,,, 3, } Rtiol Numbers: quotiets of two itegers with ozero deomitor; termitig or repetig decimls.
More informationEVALUATING DEFINITE INTEGRALS
Chpter 4 EVALUATING DEFINITE INTEGRALS If the defiite itegrl represets re betwee curve d the x-xis, d if you c fid the re by recogizig the shpe of the regio, the you c evlute the defiite itegrl. Those
More information1 Tangent Line Problem
October 9, 018 MAT18 Week Justi Ko 1 Tget Lie Problem Questio: Give the grph of fuctio f, wht is the slope of the curve t the poit, f? Our strteg is to pproimte the slope b limit of sect lies betwee poits,
More informationM344 - ADVANCED ENGINEERING MATHEMATICS
M3 - ADVANCED ENGINEERING MATHEMATICS Lecture 18: Lplce s Eqution, Anltic nd Numericl Solution Our emple of n elliptic prtil differentil eqution is Lplce s eqution, lso clled the Diffusion Eqution. If
More information