If a is any non zero real or imaginary number and m is the positive integer, then a...

Unit 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

* 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

Summer MA Lesson 4 Section P.3. such that =, denoted by =, is the principal square root

Suer MA 00 Lesso Sectio P. I Squre Roots If b, the b is squre root of. If is oegtive rel uber, the oegtive uber b b b such tht, deoted by, is the pricipl squre root of. rdicl sig rdicl expressio rdicd

Name: 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,

Intermediate Arithmetic

Git Lerig Guides Iteredite Arithetic Nuer Syste, Surds d Idices Author: Rghu M.D. NUMBER SYSTEM Nuer syste: Nuer systes re clssified s Nturl, Whole, Itegers, Rtiol d Irrtiol uers. The syste hs ee digrticlly

M098 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

Surds, 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

SM2H. 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.

ALGEBRA. 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 +

GRAPHING 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

LAWS OF INDICES M.K. HOME TUITION. Mathematics Revision Guides Level: GCSE Higher Tier

Mthetics Revisio Guides Lws of Idices Pge of 7 Author: Mrk Kudlowski M.K. HOME TUITION Mthetics Revisio Guides Level: GCSE Higher Tier LAWS OF INDICES Versio:. Dte: 0--0 Mthetics Revisio Guides Lws of

Unit 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

Math 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

Graphing 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)

Mathematical Induction (selected questions)

Mtheticl Iductio (selected questios). () Let P() e the propositio : For P(), L.H.S. R.H.S., P() is true. Assue P() is true for soe turl uer, tht is, () For P( ),, y () By the Priciple of Mtheticl Iductio,

Algebra 2 Important Things to Know Chapters bx c can be factored into... y x 5x. 2 8x. x = a then the solutions to the equation are given by

Alger Iportt Thigs to Kow Chpters 8. Chpter - Qudrtic fuctios: The stdrd for of qudrtic fuctio is f ( ) c, where 0. c This c lso e writte s (if did equl zero, we would e left with The grph of qudrtic fuctio

, 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

[ 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

MATH 118 HW 7 KELLY DOUGAN, ANDREW KOMAR, MARIA SIMBIRSKY, BRANDEN LASKE

MATH 118 HW 7 KELLY DOUGAN, ANDREW KOMAR, MARIA SIMBIRSKY, BRANDEN LASKE Prt 1. Let be odd rime d let Z such tht gcd(, 1. Show tht if is qudrtic residue mod, the is qudrtic residue mod for y ositive iteger.

Frequency-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

Numbers (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

Linford 1. Kyle Linford. Math 211. Honors Project. Theorems to Analyze: Theorem 2.4 The Limit of a Function Involving a Radical (A4)

Liford 1 Kyle Liford Mth 211 Hoors Project Theorems to Alyze: Theorem 2.4 The Limit of Fuctio Ivolvig Rdicl (A4) Theorem 2.8 The Squeeze Theorem (A5) Theorem 2.9 The Limit of Si(x)/x = 1 (p. 85) Theorem

Repeated Root and Common Root

Repeted Root d Commo Root 1 (Method 1) Let α, β, γ e the roots of p(x) x + x + 0 (1) The α + β + γ 0, αβ + βγ + γα, αβγ - () (α - β) (α + β) - αβ (α + β) [ (βγ + γα)] + [(α + β) + γ (α + β)] +γ (α + β)

Fourier Series and Applications

9/7/9 Fourier Series d Applictios Fuctios epsio is doe to uderstd the better i powers o etc. My iportt probles ivolvig prtil dieretil equtios c be solved provided give uctio c be epressed s iiite su o

In an algebraic expression of the form (1), like terms are terms with the same power of the variables (in this case

Chpter : Algebr: A. Bckgroud lgebr: A. Like ters: I lgebric expressio of the for: () x b y c z x y o z d x... p x.. we cosider x, y, z to be vribles d, b, c, d,,, o,.. to be costts. I lgebric expressio

ALGEBRA II CHAPTER 7 NOTES. Name

ALGEBRA II CHAPTER 7 NOTES Ne Algebr II 7. th Roots d Rtiol Expoets Tody I evlutig th roots of rel ubers usig both rdicl d rtiol expoet ottio. I successful tody whe I c evlute th roots. It is iportt for

Assessment 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

Important Facts You Need To Know/Review:

Importt Fcts You Need To Kow/Review: Clculus: If fuctio is cotiuous o itervl I, the its grph is coected o I If f is cotiuous, d lim g Emple: lim eists, the lim lim f g f g d lim cos cos lim 3 si lim, t

Westchester 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

x 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

Section 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

Student 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

Eigenfunction Expansion. For a given function on the internal a x b the eigenfunction expansion of f(x):

Eigefuctio Epsio: For give fuctio o the iterl the eigefuctio epsio of f(): f ( ) cmm( ) m 1 Eigefuctio Epsio (Geerlized Fourier Series) To determie c s we multiply oth sides y Φ ()r() d itegrte: f ( )

a= x+1=4 Q. No. 2 Let T r be the r th term of an A.P., for r = 1,2,3,. If for some positive integers m, n. we 1 1 Option 2 1 1

Q. No. th term of the sequece, + d, + d,.. is Optio + d Optio + (- ) d Optio + ( + ) d Optio Noe of these Correct Aswer Expltio t =, c.d. = d t = + (h- )d optio (b) Q. No. Let T r be the r th term of A.P.,

LEVEL 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

Taylor series expansion of nonlinear integrodifferential equations

AMERCA JOURAL OF SCEFC AD DUSRAL RESEARCH 2, Sciece Huβ, http://www.scihu.org/ajsr SS: 253-649X doi:.525/jsir.2.2.3.376.38 yor series expsio of oier itegrodiffereti equtios Eke A.. d 2 Jckreece P. C. Deprtet

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

For 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

( 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

Math 152 Intermediate Algebra

Mth 15 Iteredite Alger Stud Guide for the Fil E You use 46 otecrd (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 ou eeded

A 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

Lincoln 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

(200 terms) equals Let f (x) = 1 + x + x 2 + +x 100 = x101 1

SECTION 5. PGE 78.. DMS: CLCULUS.... 5. 6. CHPTE 5. Sectio 5. pge 78 i + + + INTEGTION Sums d Sigm Nottio j j + + + + + i + + + + i j i i + + + j j + 5 + + j + + 9 + + 7. 5 + 6 + 7 + 8 + 9 9 i i5 8. +

INFINITE SERIES. ,... having infinite number of terms is called infinite sequence and its indicated sum, i.e., a 1

Appedix A.. Itroductio As discussed i the Chpter 9 o Sequeces d Series, sequece,,...,,... hvig ifiite umber of terms is clled ifiite sequece d its idicted sum, i.e., + + +... + +... is clled ifite series

82A Engineering Mathematics

Clss Notes 9: Power Series /) 8A Egieerig Mthetics Secod Order Differetil Equtios Series Solutio Solutio Ato Differetil Equtio =, Hoogeeous =gt), No-hoogeeous Solutio: = c + p Hoogeeous No-hoogeeous Fudetl

INTEGRATION 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)

Week 13 Notes: 1) Riemann Sum. Aim: Compute Area Under a Graph. Suppose we want to find out the area of a graph, like the one on the right:

Week 1 Notes: 1) Riem Sum Aim: Compute Are Uder Grph Suppose we wt to fid out the re of grph, like the oe o the right: We wt to kow the re of the red re. Here re some wys to pproximte the re: We cut the

A 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

Taylor Polynomials. The Tangent Line. (a, f (a)) and has the same slope as the curve y = f (x) at that point. It is the best

Tylor Polyomils Let f () = e d let p() = 1 + + 1 + 1 6 3 Without usig clcultor, evlute f (1) d p(1) Ok, I m still witig With little effort it is possible to evlute p(1) = 1 + 1 + 1 (144) + 6 1 (178) =

SUTCLIFFE 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

Approximate 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:

Calendar of first week of the school year. Monday, August 26 Full Day get books & begin Chapter 1

Gettig Strted Pcket Hoors Pre-Clculus Welcoe to Hoors Pre-Clculus. Hoors Pre-Clculus will refresh your Algebr skills, review polyoil fuctios d grphs, eplore trigooetry i depth, d give you brief itroductio

f(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

Notes 17 Sturm-Liouville Theory

ECE 638 Fll 017 Dvid R. Jckso Notes 17 Sturm-Liouville Theory Notes re from D. R. Wilto, Dept. of ECE 1 Secod-Order Lier Differetil Equtios (SOLDE) A SOLDE hs the form d y dy 0 1 p ( x) + p ( x) + p (

Content: Essential Calculus, Early Transcendentals, James Stewart, 2007 Chapter 1: Functions and Limits., in a set B.

Review Sheet: Chpter Cotet: Essetil Clculus, Erly Trscedetls, Jmes Stewrt, 007 Chpter : Fuctios d Limits Cocepts, Defiitios, Lws, Theorems: A fuctio, f, is rule tht ssigs to ech elemet i set A ectly oe

Next 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

Order doesn t matter. There exists a number (zero) whose sum with any number is the number.

P. Real Numbers ad Their Properties Natural Numbers 1,,3. Whole Numbers 0, 1,,... Itegers..., -1, 0, 1,... Real Numbers Ratioal umbers (p/q) Where p & q are itegers, q 0 Irratioal umbers o-termiatig ad

National 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

YOUR FINAL IS THURSDAY, MAY 24 th from 10:30 to 12:15

Algebr /Trig Fil Em Study Guide (Sprig Semester) Mrs. Duphy YOUR FINAL IS THURSDAY, MAY 4 th from 10:30 to 1:15 Iformtio About the Fil Em The fil em is cumultive for secod semester, coverig Chpters, 3,

APPLICATION OF DIFFERENCE EQUATIONS TO CERTAIN TRIDIAGONAL MATRICES

Scietific Reserch of the Istitute of Mthetics d Coputer Sciece 3() 0, 5-0 APPLICATION OF DIFFERENCE EQUATIONS TO CERTAIN TRIDIAGONAL MATRICES Jolt Borows, Le Łcińs, Jowit Rychlews Istitute of Mthetics,

MA123, Chapter 9: Computing some integrals (pp )

MA13, Chpter 9: Computig some itegrls (pp. 189-05) Dte: Chpter Gols: Uderstd how to use bsic summtio formuls to evlute more complex sums. Uderstd how to compute its of rtiol fuctios t ifiity. Uderstd how

Algebra 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

Limit 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

RULES FOR MANIPULATING SURDS b. This is the addition law of surds with the same radicals. (ii)

SURDS Defiitio : Ay umer which c e expressed s quotiet m of two itegers ( 0 ), is clled rtiol umer. Ay rel umer which is ot rtiol is clled irrtiol. Irrtiol umers which re i the form of roots re clled surds.

THEORY OF EQUATIONS SYNOPSIS. Polyomil Fuctio: If,, re rel d is positive iteger, the f)x) = + x + x +.. + x is clled polyomil fuctio.. Degree of the Polyomil: The highest power of x for which the coefficiet

[Q. Booklet Number]

6 [Q. Booklet Numer] KOLKATA WB- B-J J E E - 9 MATHEMATICS QUESTIONS & ANSWERS. If C is the reflecto of A (, ) i -is d B is the reflectio of C i y-is, the AB is As : Hits : A (,); C (, ) ; B (, ) y A (,

PROGRESSIONS 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.

Section 10.2 Angles and Triangles

117 Ojective #1: Section 10.2 nges n Tringes Unerstning efinitions of ifferent types of nges. In the intersection of two ines, the nges tht re cttycorner fro ech other re vertic nges. Vertic nges wi hve

{ } { S n } is monotonically decreasing if Sn

Sequece A sequece is fuctio whose domi of defiitio is the set of turl umers. Or it c lso e defied s ordered set. Nottio: A ifiite sequece is deoted s { } S or { S : N } or { S, S, S,...} or simply s {

EXERCISE 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 + ) ()

General properties of definite integrals

Roerto s Notes o Itegrl Clculus Chpter 4: Defiite itegrls d the FTC Sectio Geerl properties of defiite itegrls Wht you eed to kow lredy: Wht defiite Riem itegrl is. Wht you c ler here: Some key properties

Chapter 7 Infinite Series

MA Ifiite Series Asst.Prof.Dr.Supree Liswdi Chpter 7 Ifiite Series Sectio 7. Sequece A sequece c be thought of s list of umbers writte i defiite order:,,...,,... 2 The umber is clled the first term, 2

Solutions 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

Indices and Logarithms

the Further Mthemtics etwork www.fmetwork.org.uk V 7 SUMMARY SHEET AS Core Idices d Logrithms The mi ides re AQA Ed MEI OCR Surds C C C C Lws of idices C C C C Zero, egtive d frctiol idices C C C C Bsic

Logarithmic 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.

FOURIER SERIES PART I: DEFINITIONS AND EXAMPLES. To a 2π-periodic function f(x) we will associate a trigonometric series. a n cos(nx) + b n sin(nx),

FOURIER SERIES PART I: DEFINITIONS AND EXAMPLES To -periodic fuctio f() we will ssocite trigoometric series + cos() + b si(), or i terms of the epoetil e i, series of the form c e i. Z For most of the

UNIT 4 EXTENDING THE NUMBER SYSTEM Lesson 1: Working with the Number System Instruction

Lesso : Workig with the Nuber Syste Istructio Prerequisite Skills This lesso requires the use of the followig skills: evlutig expressios usig the order of opertios evlutig expoetil expressios ivolvig iteger

degree non-homogeneous Diophantine equation in six unknowns represented by x y 2z

Scholrs Jourl of Egieerig d Techology (SJET Sch. J. Eg. Tech., ; (A:97- Scholrs Acdeic d Scietific Pulisher (A Itertiol Pulisher for Acdeic d Scietific Resources www.sspulisher.co ISSN -X (Olie ISSN 7-9

M3P14 EXAMPLE SHEET 1 SOLUTIONS

M3P14 EXAMPLE SHEET 1 SOLUTIONS 1. Show tht for, b, d itegers, we hve (d, db) = d(, b). Sice (, b) divides both d b, d(, b) divides both d d db, d hece divides (d, db). O the other hd, there exist m d

Review of Sections

Review of Sectios.-.6 Mrch 24, 204 Abstrct This is the set of otes tht reviews the mi ides from Chpter coverig sequeces d series. The specific sectios tht we covered re s follows:.: Sequces..2: Series,

RADICALS. Upon completion, you should be able to. define the principal root of numbers. simplify radicals

RADICALS m 1 RADICALS Upo completio, you should be ble to defie the pricipl root of umbers simplify rdicls perform dditio, subtrctio, multiplictio, d divisio of rdicls Mthemtics Divisio, IMSP, UPLB Defiitio:

Northwest 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:

is an ordered list of numbers. Each number in a sequence is a term of a sequence. n-1 term

Mthemticl Ptters. Arithmetic Sequeces. Arithmetic Series. To idetify mthemticl ptters foud sequece. To use formul to fid the th term of sequece. To defie, idetify, d pply rithmetic sequeces. To defie rithmetic

Kp for a Wall with Friction with Exact Slip Surface

K for W ith Frictio ith Ect Si Surfce Frid A. Chouer, P.E., S.E. 7 Frid Chouer rights reserved Revised 7--6 Itroductio: We ko from efore ( htt://.fcsstems.com/si.df ) the est curve for K ith o frictio

y udv uv y v du 7.1 INTEGRATION BY PARTS

7. INTEGRATION BY PARTS Ever differetitio rule hs correspodig itegrtio rule. For istce, the Substitutio Rule for itegrtio correspods to the Chi Rule for differetitio. The rule tht correspods to the Product

PARTIAL DIFFERENTIAL EQUATIONS SEPARATION OF VARIABLES

Diola Bagayoko (0 PARTAL DFFERENTAL EQUATONS SEPARATON OF ARABLES. troductio As discussed i previous lectures, partial differetial equatios arise whe the depedet variale, i.e., the fuctio, varies with

The total number of permutations of S is n!. We denote the set of all permutations of S by

DETERMINNTS. DEFINITIONS Def: Let S {,,, } e the set of itegers from to, rrged i scedig order. rerrgemet jjj j of the elemets of S is clled permuttio of S. S. The totl umer of permuttios of S is!. We deote

Cape Cod Community College

Cpe Cod Couity College Deprtetl Syllus Prepred y the Deprtet of Mthetics Dte of Deprtetl Approvl: Noveer, 006 Dte pproved y Curriculu d Progrs: Jury 9, 007 Effective: Fll 007 1. Course Nuer: MAT110 Course

Algebra 2 Readiness Summer Packet El Segundo High School

Algebr Rediess Suer Pcket El Segudo High School This pcket is desiged for those who hve copleted Geoetry d will be erolled i Algebr (CP or H) i the upcoig fll seester. Suer Pcket Algebr II Welcoe to Algebr

Copyrighted 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

EXPONENTS AND LOGARITHMS

978--07-6- Mthemtis Stdrd Level for IB Diplom Eerpt EXPONENTS AND LOGARITHMS WHAT YOU NEED TO KNOW The rules of epoets: m = m+ m = m ( m ) = m m m = = () = The reltioship etwee epoets d rithms: = g where

Lecture 2. Rational Exponents and Radicals. 36 y. b can be expressed using the. Rational Exponent, thus b. b can be expressed using the

Lecture. Rtiol Epoets d Rdicls Rtiol Epoets d Rdicls Lier Equtios d Iequlities i Oe Vrile Qudrtic Equtios Appedi A6 Nth Root - Defiitio Rtiol Epoets d Rdicls For turl umer, c e epressed usig the r is th

Objective Mathematics

. o o o o {cos 4 cos 9 si cos 65 } si 7º () cos 6º si 8º. If x R oe of these, the mximum vlue of the expressio si x si x.cos x c cos x ( c) is : () c c c c c c. If ( cos )cos cos ; 0, the vlue of 4. The

COMP 2804 Solutions Assignment 1

COMP 2804 Solutios Assiget 1 Questio 1: O the first page of your assiget, write your ae ad studet uber Solutio: Nae: Jaes Bod Studet uber: 007 Questio 2: I Tic-Tac-Toe, we are give a 3 3 grid, cosistig

BINOMIAL THEOREM SOLUTION. 1. (D) n. = (C 0 + C 1 x +C 2 x C n x n ) (1+ x+ x 2 +.)

BINOMIAL THEOREM SOLUTION. (D) ( + + +... + ) (+ + +.) The coefficiet of + + + +... + fo. Moeove coefficiet of is + + + +... + if >. So. (B)... e!!!! The equied coefficiet coefficiet of i e -.!...!. (A),

MATH 174: Numerical Analysis. Lecturer: Jomar F. Rabajante 1 st Sem AY

MATH 74: Numeric Aysis Lecturer: Jomr F. Rbjte st Sem AY - INTERPOLATION THEORY We wt to seect fuctio p from give css of fuctios i such wy tht the grph of y=p psses through fiite set of give dt poits odes.

Solutions to RSPL/1. log 3. When x = 1, t = 0 and when x = 3, t = log 3 = sin(log 3) 4. Given planes are 2x + y + 2z 8 = 0, i.e.

olutios to RPL/. < F < F< Applig C C + C, we get F < 5 F < F< F, $. f() *, < f( h) f( ) h Lf () lim lim lim h h " h h " h h " f( + h) f( ) h Rf () lim lim lim h h " h h " h h " Lf () Rf (). Hee, differetile

Fast Fourier Transform 1) Legendre s Interpolation 2) Vandermonde Matrix 3) Roots of Unity 4) Polynomial Evaluation

Algorithm Desig d Alsis Victor Admchi CS 5-45 Sprig 4 Lecture 3 J 7, 4 Cregie Mello Uiversit Outlie Fst Fourier Trsform ) Legedre s Iterpoltio ) Vdermode Mtri 3) Roots of Uit 4) Polomil Evlutio Guss (777

CH 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

Advanced Algorithmic Problem Solving Le 6 Math and Search

Advced Algorithmic Prolem Solvig Le Mth d Serch Fredrik Heitz Dept of Computer d Iformtio Sciece Liköpig Uiversity Outlie Arithmetic (l. d.) Solvig lier equtio systems (l. d.) Chiese remider theorem (l.5