Summer MA Lesson 4 Section P.3. such that =, denoted by =, is the principal square root
|
|
- August Grant
- 6 years ago
- Views:
Transcription
1 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 Ex : Evlute ech. If ot rel, write ot rel. ) b) 6 c) d) e) 9 My ties studets believe tht positive. Exie the followig. 6 ( ) 6, ot 6 6, which is ot rel. However, the pricipl squre root is lwys I geerl: Therefore, we will lwys ssue tht vribles represet positive ubers i order to void usig bsolute vlue sigs. II Other Types of Roots b es tht b If is eve, the d b ust be positive. If is odd, d b c be y rel ubers. idex If o idex is writte, the root is ssued to be squre root.
2 Ex : Evlute ech. If ot rel, write ot rel. b c ) ) 6 ) 6 d) 7 e) 0.0 f ) III The Product d Quotiet Rules of Rdicls If ll expressios represet rel ubers, b b d b b d ( b 0) b b b b Note: These properties re for ultiplictio d divisio. Siilr stteets re ot true for dditio or subtrctio. ( + b + b, for exple) Ex : Use the product or quotiet rules of rdicls (if you c) to write s oe rdicl. Siplify, if possible. ) 0 b) c ) IV Siplifyig Squre Root Rdicls A squre root is siplified whe its rdicd hs o fctors other th tht re perfect squres. Reeber:, if is ssued to be positive. We will ssue ll vribles represet positive vlues.
3 Ex : Use fctorig d the product (d/or quotiet)rule to siplify ech. ) x b) b 7 c) x x d) V Additio d Subtrctio of Squre Roots Two or ore squre roots c be cobied if they hve the se rdicd. Such rdicls re clled like rdicls. Soetie oe or ore rdicl ust be siplified i order to cobie. Ex : Siplify d cobie where possible. ) + 6
4 b) c ) 6 VI Rtiolizig Deoitors The process of rewritig squre root rdicl expressio s equivlet expressio i which the deoitor o loger cotis y rdicls is clled rtiolizig the deoitor. First, siplify y rdicls. Secodly, ultiply the uertor d deoitor by the rdicl fctor tht reis. Ex 6: Siplify by rtiolizig the deoitor. )
5 b) c) VII Cojugtes Rdicl expressios tht ivolve the su d differece of the se two ters re clled cojugtes. Exples re + d or x d + x. The product of two cojugtes will coti o rdicls! ( + b )( b ) ( ) ( b ) b I rdicl expressios with bioil (two ters) i the deoitor, to rtiolize the deoitor, ultiply uertor d deoitor by the cojugte of the deoitor. Ex 7: Rtiolize d siplify ech. ) b) +
6 VIII Rtiol Expoets Exie: ( ) ust be equivlet. d Sice both d squred equl, they Defiitio of If represets rel uber, where is iteger, the. Ex : Evlute ech, if it exists. ) 9 b) c) 6 The deoitor of the rtiol expoet becoes the idex of the rdicl. The textbook d olie hoework y use regulr frctio br for rtiol expoet or slsh frctio br. / d) e) ( ) Exie: ( ) d ( ) Therefore: ( ) or 6
7 Defiitio of If represets rel uber d is ( ) or positive rtiol uber,, the. It c be evluted or siplified by fidig the power first, the the root or by fidig the root first, the the power. Becuse you will ot hve clcultor o quizzes or your first ex, I recoed fidig the root first, the rise to the expoet power. The uertor is the expoet. or ( ) The deoitor is the idex. Ex 9: Evlute, if possible. ) b) c) 6 6 Ex 0: Evlute, if possible. ) b) ( ) 7
8 Ex : Use the properties of expoets to siplify. c) d) 6 b 9 ( x ) e) (6 x y ) ( x y ) Ex : A rectgle below hs the give width d legth. Fid the perieter (usig rdicls s eeded) d the re (usig rdicls s eeded) of this rectgle. Siplify ech. 67 Soe theticl odels y be equtios tht hve rdicl expressios. Ex : Suppose E x +. odels the uber of elderly Aerics ges 6-, i illios, for x uber of yers fter 00. Project the uber of Aerics ges 6-, i illios, i 00 d 00. Express the icrese i uber of elderly Aerics fro 00 to 00.
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
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 informationALGEBRA 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
More informationRADICALS. 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:
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 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 informationIntermediate 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
More informationIn 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
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 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 informationAlgebra 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
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 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 informationCalendar 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
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 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 informationSummer Math Requirement Algebra II Review For students entering Pre- Calculus Theory or Pre- Calculus Honors
Suer Mth Requireet Algebr II Review For studets eterig Pre- Clculus Theory or Pre- Clculus Hoors The purpose of this pcket is to esure tht studets re prepred for the quick pce of Pre- Clculus. The Topics
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 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 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 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 informationSummer MA Lesson 13 Section 1.6, Section 1.7 (part 1)
Suer MA 1500 Lesso 1 Sectio 1.6, Sectio 1.7 (part 1) I Solvig Polyoial Equatios Liear equatio ad quadratic equatios of 1 variable are specific types of polyoial equatios. Soe polyoial equatios of a higher
More informationLesson 5: Does the Order Matter?
: Does the Order Mtter? Opeig Activity You will eed: Does the Order Mtter? sortig crds [dpted fro 5E Lesso Pl: Usig Order of Opertios to Evlute d Siplify Expressios, Pt Tyree] 1. Rerrge the crds so they
More informationCape 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
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 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 informationAlgebra 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
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 informationWeek 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
More informationLimits and an Introduction to Calculus
Nme Chpter Limits d Itroductio to Clculus Sectio. Itroductio to Limits Objective: I this lesso ou lered how to estimte limits d use properties d opertios of limits. I. The Limit Cocept d Defiitio of Limit
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 informationLAWS 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
More informationMath 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
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 information= 4 and 4 is the principal cube root of 64.
Chapter Real Numbers ad Radicals Day 1: Roots ad Radicals A2TH SWBAT evaluate radicals of ay idex Do Now: Simplify the followig: a) 2 2 b) (-2) 2 c) -2 2 d) 8 2 e) (-8) 2 f) 5 g)(-5) Based o parts a ad
More informationExponents and Radical
Expoets d Rdil Rule : If the root is eve d iside the rdil is egtive, the the swer is o rel umber, meig tht If is eve d is egtive, the Beuse rel umber multiplied eve times by itself will be lwys positive.
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 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 informationP.3 Simplifying Expressions
P3 Siplifyig Jie Esclte, the ost fous high school clculus techer of ll tie, hd ber i his clssroo tht red Clculus does t hve to be de esy, it lredy is How true tht essge is As I lredy etioed, AP Clculus
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 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 informationRULES 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.
More informationName Period ALGEBRA II Chapter 1B and 2A Notes Solving Inequalities and Absolute Value / Numbers and Functions
Nae Period ALGEBRA II Chapter B ad A Notes Solvig Iequalities ad Absolute Value / Nubers ad Fuctios SECTION.6 Itroductio to Solvig Equatios Objectives: Write ad solve a liear equatio i oe variable. Solve
More informationAppendix 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 informationMATRIX ALGEBRA, Systems Linear Equations
MATRIX ALGEBRA, Systes Lier Equtios Now we chge to the LINEAR ALGEBRA perspective o vectors d trices to reforulte systes of lier equtios. If you fid the discussio i ters of geerl d gets lost i geerlity,
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 informationImportant 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
More informationB. Examples 1. Finite Sums finite sums are an example of Riemann Sums in which each subinterval has the same length and the same x i
Mth 06 Clculus Sec. 5.: The Defiite Itegrl I. Riem Sums A. Def : Give y=f(x):. Let f e defied o closed itervl[,].. Prtitio [,] ito suitervls[x (i-),x i ] of legth Δx i = x i -x (i-). Let P deote the prtitio
More informationLecture 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
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 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 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 information0 otherwise. sin( nx)sin( kx) 0 otherwise. cos( nx) sin( kx) dx 0 for all integers n, k.
. Computtio of Fourier Series I this sectio, we compute the Fourier coefficiets, f ( x) cos( x) b si( x) d b, i the Fourier series To do this, we eed the followig result o the orthogolity of the trigoometric
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 informationMini Lecture 10.1 Radical Expressions and Functions. 81x d. x 4x 4
Mii Lecture 0. Radical Expressios ad Fuctios Learig Objectives:. Evaluate square roots.. Evaluate square root fuctios.. Fid the domai of square root fuctios.. Use models that are square root fuctios. 5.
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 informationGeometric Sequences. Geometric Sequence. Geometric sequences have a common ratio.
s A geometric sequece is sequece such tht ech successive term is obtied from the previous term by multiplyig by fixed umber clled commo rtio. Exmples, 6, 8,, 6,..., 0, 0, 0, 80,... Geometric sequeces hve
More informationLinford 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
More informationis 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
More informationMA123, 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
More informationMTH 146 Class 16 Notes
MTH 46 Clss 6 Notes 0.4- Cotiued Motivtio: We ow cosider the rc legth of polr curve. Suppose we wish to fid the legth of polr curve curve i terms of prmetric equtios s: r f where b. We c view the cos si
More informationBC Calculus Review Sheet
BC Clculus Review Sheet Whe you see the words. 1. Fid the re of the ubouded regio represeted by the itegrl (sometimes 1 f ( ) clled horizotl improper itegrl). This is wht you thik of doig.... Fid the re
More informationChapter 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
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 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 informationn m CHAPTER 3 RATIONAL EXPONENTS AND RADICAL FUNCTIONS 3-1 Evaluate n th Roots and Use Rational Exponents Real nth Roots of a n th Root of a
CHAPTER RATIONAL EXPONENTS AND RADICAL FUNCTIONS Big IDEAS: 1) Usig ratioal expoets ) Performig fuctio operatios ad fidig iverse fuctios ) Graphig radical fuctios ad solvig radical equatios Sectio: Essetial
More informationFourier 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
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 informationHandout #2. Introduction to Matrix: Matrix operations & Geometric meaning
Hdout # Title: FAE Course: Eco 8/ Sprig/5 Istructor: Dr I-Mig Chiu Itroductio to Mtrix: Mtrix opertios & Geometric meig Mtrix: rectgulr rry of umers eclosed i pretheses or squre rckets It is covetiolly
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 informationStatistics for Financial Engineering Session 1: Linear Algebra Review March 18 th, 2006
Sttistics for Ficil Egieerig Sessio : Lier Algebr Review rch 8 th, 6 Topics Itroductio to trices trix opertios Determits d Crmer s rule Eigevlues d Eigevectors Quiz The cotet of Sessio my be fmilir to
More informationDefinite Integral. The Left and Right Sums
Clculus Li Vs Defiite Itegrl. The Left d Right Sums The defiite itegrl rises from the questio of fidig the re betwee give curve d x-xis o itervl. The re uder curve c be esily clculted if the curve is give
More information1. (25 points) Use the limit definition of the definite integral and the sum formulas to compute. [1 x + x2
Mth 3, Clculus II Fil Exm Solutios. (5 poits) Use the limit defiitio of the defiite itegrl d the sum formuls to compute 3 x + x. Check your swer by usig the Fudmetl Theorem of Clculus. Solutio: The limit
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 informationHow Euler Did It. by Ed Sandifer. Theorema Arithmeticum. March, 2005
How Euler Did It Theore Aritheticu Mrch, 005 by Ed Sdifer Euler s 748 textbook, the Itroductio i lysi ifiitoru, ws oe of the ost ifluetil thetics books of ll tie Joh Blto s excellet trsltio, is vilble
More informationApproximations of Definite Integrals
Approximtios of Defiite Itegrls So fr we hve relied o tiderivtives to evlute res uder curves, work doe by vrible force, volumes of revolutio, etc. More precisely, wheever we hve hd to evlute defiite itegrl
More informationTaylor 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) =
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 informationInfinite Series Sequences: terms nth term Listing Terms of a Sequence 2 n recursively defined n+1 Pattern Recognition for Sequences Ex:
Ifiite Series Sequeces: A sequece i defied s fuctio whose domi is the set of positive itegers. Usully it s esier to deote sequece i subscript form rther th fuctio ottio.,, 3, re the terms of the sequece
More informationSchrödinger Equation Via Laplace-Beltrami Operator
IOSR Jourl of Mthemtics (IOSR-JM) e-issn: 78-578, p-issn: 39-765X. Volume 3, Issue 6 Ver. III (Nov. - Dec. 7), PP 9-95 www.iosrjourls.org Schrödiger Equtio Vi Lplce-Beltrmi Opertor Esi İ Eskitşçioğlu,
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 informationThe 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
More informationMATH 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.
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 informationFOURIER 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
More information10.5 Power Series. In this section, we are going to start talking about power series. A power series is a series of the form
0.5 Power Series I the lst three sectios, we ve spet most of tht time tlkig bout how to determie if series is coverget or ot. Now it is time to strt lookig t some specific kids of series d we will evetully
More information7-1: Zero and Negative Exponents
7-: Zero nd Negtive Exponents Objective: To siplify expressions involving zero nd negtive exponents Wr Up:.. ( ).. 7.. Investigting Zero nd Negtive Exponents: Coplete the tble. Write non-integers s frctions
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 informationz line a) Draw the single phase equivalent circuit. b) Calculate I BC.
ECE 2260 F 08 HW 7 prob 4 solutio EX: V gyb' b' b B V gyc' c' c C = 101 0 V = 1 + j0.2 Ω V gyb' = 101 120 V = 6 + j0. Ω V gyc' = 101 +120 V z LΔ = 9 j1.5 Ω ) Drw the sigle phse equivlet circuit. b) Clculte
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 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 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 information,... are the terms of the sequence. If the domain consists of the first n positive integers only, the sequence is a finite sequence.
Chpter 9 & 0 FITZGERALD MAT 50/5 SECTION 9. Sequece Defiitio A ifiite sequece is fuctio whose domi is the set of positive itegers. The fuctio vlues,,, 4,...,,... re the terms of the sequece. If the domi
More informationThe picture in figure 1.1 helps us to see that the area represents the distance traveled. Figure 1: Area represents distance travelled
1 Lecture : Area Area ad distace traveled Approximatig area by rectagles Summatio The area uder a parabola 1.1 Area ad distace Suppose we have the followig iformatio about the velocity of a particle, how
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 informationYOUR 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,
More informationTHEORY 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
More informationContent: 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
More informationVectors. Vectors in Plane ( 2
Vectors Vectors i Ple ( ) The ide bout vector is to represet directiol force Tht mes tht every vector should hve two compoets directio (directiol slope) d mgitude (the legth) I the ple we preset vector
More information8.3 Sequences & Series: Convergence & Divergence
8.3 Sequeces & Series: Covergece & Divergece A sequece is simply list of thigs geerted by rule More formlly, sequece is fuctio whose domi is the set of positive itegers, or turl umbers,,,3,. The rge of
More information