7 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


 Magdalene Clark
 2 years ago
 Views:
Transcription
1 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 b) 4. Zero Expoet Rule: 0 b Examples: 0 7 b) 0 5 c) 0 5. Product Rule: m m b g b b  keep the base ad the expoets Examples: g b) x 7 g x 4. Quotiet Rule: b b m m b  keep the base ad the expoets. Examples: 4 b) x 5 x
2 m 5. Power to a Power Rule b m b  keep the base ad the expoets. Examples: 5 b) x 4 6. Product to a power: ab Note well: a b a b ad a b a b a b  raise both bases to the same power. Examples: y 4 5 b) x y 7. Quotiet to a power: a b a b  raise both bases to the same power. Examples: 5y 4 b) x y
3 ) No paretheses. ) No powers raised to a powers. ) Each based occurs oly oce. 4) No egative expoets. 5) Simplify umerical expressios. Simplifyig Expoetial Expressios Examples: Simplify the expressio. 8 4 x y 4 5 b) x y c) 7xy x y d) 6x y 5 xy e) 5xy 6 8 5xy 4 f) 00x 0x y y 6 4 g) 4x d y g 7 4 h) 5x y 8 4 5
4 Defiitio: A umber is writte i if it is of the form where a 0 ad Z. Examples : Write i stadard form (decimal form) b) c).7 0 d) Example : Write i scietific otatio. 4,970,000,000,000 b) To multiply umbers writte i scietific otatio:. Multiply the decimal parts first. (use your calculator, if ecessary). Usig the rules of expoets, multiply the powers of te. (add the expoets). Be sure your fial aswer is i scietific otatio b) c) d) e) f)
5 To divide umbers writte i scietific otatio:. Divide the decimal parts first. (use your calculator, if ecessary). Usig the rules of expoets, divide the powers of te. (subtract the expoets). Be sure your fial aswer is i scietific otatio Examples: Fid the quotiet, express aswers i scietific otatio b) c) d) e) f)
6 Guided Notes for lesso P. More with Expoets If b ad b 0, ad m, Z the the followig properties hold: 8. Fractioal Expoets: If is eve, 0 b b b If is odd, b ad If is eve, b 0 b b If is odd, b Examples: 64 b) 5 c) 6 4 d) 7 e) 64 m m 9. Fractioal Expoets: b b b m If is eve, b 0 If is odd, b ad b m m b b m If is eve, b 0 If is odd, b Examples: 7 b) 4 c) 5 8 d) 8 4 e)
7 Examples: Simplify the expressio. Express your aswer usig a radical whe your fial aswer yields a fractioal expoet. 4 5x 7x b) x y c) x 6x 7 4 d) 4 x y x y 7 4 e) x x 5 7 f) 4 x x 7
8 Examples: Factor the expressio ad simplify. This is a skill i preparatio for calculus. x x x 4 b) x 4 x 4 8x c) x8 x 8 x 8
9 Guided Notes for lesso P.7 Absolute Value Equatios Absolute Value: Defiitio: a meas the distace the umber a is from o a umber lie. Solvig Absolute Value Equatios: If X is ay algebraic expressio ad c Z, the the solutios to X care foud by solvig the equatios ad. Example Solve the equatio. x 5 b) x 7 c) x 8 4 d) x 5 e) 7 4 x f) 4 5x 7 9
10 Guided Notes for lesso P.9 Iequalities ad Absolute Value Equatios ad Iequalities To solve a iequality meas to fid all the possible values of the variable that makes the iequality true. To solve a iequality you use the same iverse operatio(s) to both sides of the iequality with oe slight glitch. Keep i mid you wat the variable all aloe o oe side of the iequality. Whe a iequality is multiplied or divided by a positive umber the iequality sig remais uchaged. Example to illustrate: 5 6 true 50 5 true 5 6 multiply by 50 5 divide by 5 0 < true < 7 true Whe a iequality is multiplied or divided by a egative umber the iequality sig is reversed. Example to illustrate: 5 6 true 50 5 true 5 6 multiply by 50 5 divide by 5 0 < false true 0 > 7 false 0 7 true Examples: Solve each of the followig ad graph your solutio o a umber lie. x 5 b) x 5 c) 4 6 d) 8 0
11 x e) 0 f) x 08 g) 6x 5 57 h) 4x 40 7x 65 i) x 7 5x j) 5x 9 i) x 8
12 Three ways to display solutios to iequalities: Set builder Notatio (used for Alg ) Iterval Notatio (used i precalc o up) Graph (used for either Alg or precalc o up) x a x b x a x b x x a x x a x x b x x b xx
13 Coectors of sets of umbers: Itersectio (ad) (what is i both) Set builder Notatio Symbol Iterval Notatio Symbol Uio (or) (what is i either or both) Set builder Notatio Iterval Notatio Graph x x 4 x 8 x x 7 x 9
14 Solvig Absolute Value Iequalities Solutio: If X is ay algebraic expressio ad c Z, the the solutios to X care foud by solvig ad. Examples: Solve the iequality ad graph the solutio set. x 5 4 b) x c) 5x 7 4
15 Solutio: If X is ay algebraic expressio ad c Z, the the solutios to X care foud by solvig ad. Examples: Solve the iequality ad graph the solutio set. x 7 0 b) 8 4 x c) 5x 7 5
16 Guided Notes for lesso P.A Radicals Date: Defiitio: a is called the th root or. a is called the ad is called the. Note Well: If is eve, a 0. If is odd, a. If a, b ad b 0, ad m, Z the the followig properties hold:. Product Rule: ab a b g. Note Well a b a b ad a b a b. Quotiet Rule: a b a b. Power Rule : a 4. Power Rule : a m a m To simplify m a 6
17 Example: Simplify the radical. Express the aswer i simplest radical form. b) 75 c) 700 d) 48 e) 75 f) 686 g) 4 50x h) x y i) x y j) 49x 64y 5 9 k) 54x 7 8y 6 Defiitio: Like radicals are radicals with the same ad Rule: You may oly add or subtract like radicals. 7
18 Example: Simplify the expressio b) c) 8 d) 8x 4 x 4 7x Example: Multiply. Express the aswer i simplest radical form, if ecessary. 8 5 g b) 8 c) 5 5 d) Defiitio: The irratioal umbers a b ad a b are called ad whe multiplied together will always yield a umber. Why? 8
19 e) 0 f) Example: Divide. Express the aswer i simplest radical form, if ecessary b) c) d)
20 Defiitio: To meas to fid a equivalet fractio with a deomiator that is a ratioal umber. Why was/is that importat? Example: Ratioalize the deomiator. 5 b) 0 c) 4 d) e) 0 0 f) x h x x h x 0
21 Guided Notes for lesso 0.5 The Biomial Theorem Defiitio: A is a two termed algebraic expressio. Defiitio: Whe ay biomial is raised to a positive itegral power, the result is called a Illustratio: Expad x y x y x y x y x y x y x x y y x x y xy x y xy y x x y x y y A few thigs you should otice i the expasio of x y ) the x s decrease i power,,, 0 ) the y s icrease i power 0,,, x x x x term by term. y y y y term by term. : ) the expoets o x ad y always add up to for each term. 4) the umber of terms (4) is oe greater tha the expoet. 5) there are coefficiets o the two middle terms Where the coefficiets of a biomial expasio come from? Defiitio: The coefficiet of ay term of biomial expasio is called a biomial coefficiet ad is foud! by r provided r, ad r. r! r! C r is used as well to deote r. A combiatio of thigs take r at a time. Examples: Evaluate by had b) 8 5 c) 7 Example: Evaluate with your calculator 9 9 b) 7 c) 5 0
22 The Biomial Theorem: For ay moomial expressio a ay moomial expressio b, ad r, : a b a b a b a b... a b a b a b Examples: Expad the expressio. (write small) x 4 b) x remember that x x( ) c) x y 5 make sure it s the ad the x that are raised to the expoets.
23 d) x 4y e) 4 x 6y Aother way to aid you with expasio of x y Example: Expad x y 5 Cosider x y x x y xy y is to use Pascal s Triagle x x y xy y usig Pascal s Triagle.
24 Cosider the expasio of x y x x y xy y. If we just wated the secod term of its expasio without expadig it, how could we fid it? It would be x y x y For ay moomial expressio a ay moomial expressio b, ad r, The rth term of a Biomial Expasio without expadig is: r r a b r Examples: Fid the give term of the epasio with out expadig it. x 5 (third term) b) 6 8 x (fourth term) c) x 5 6 (sixth term) d) x 7 9 (secod term) 5 e) 4 x y (third) 4
25 Good luck to: GMP 4: Chapter PA Test: Sectios: 0.5, P., P., ad P.9 Multiple Choice Questios: For 8, circle the best aswer to the questio. Whe you see the saw graphic, SHOW ALL WORK i the space provided. Correct aswers with o work will receive miimal credit. Icorrect aswers with work will receive partial credit. Icorrect aswers with o work will receive o credit. You may use a calculator. Each questio is worth 9 poits. ) What is the solutio of the iequality x 5? x 8 x b) x x 8 c) x x 8or x d) x x or x 8 ) What is the solutio of the iequality x 4? x 7 x b) 7 x x c) 7 x x or x d) 7 x x or x ) Which graph represets the solutio to the iequality x 7? b) c) d) 5
26 4) Which graph represets the solutio to the iequality x 7? b) c) d) 5) Evaluate the biomial coefficiet: b) 5 c) 0,40 d),68,800 6) Evaluate the biomial coefficiet: 4. 4 b) 4 c) d) 7) Expad 5x usig the biomial theorem. c) 5x 8 b) 5x 50x 60x 8 d) 6 5x 50x 6x 8 5x 0x 0x 8
27 8) Expad 4 x y usig the biomial theorem. c) x 8x y 4x y 8x y 6y b) 4 4 x 8x y 4x y xy 6y d) x x y 4x y 6x y 6y x 8x y 4x y x y 6y ) The fifth term i the expasio of 6 x y is 4 40x y b) 4 40x y c) 4 60x y d) 4 60x y 9) The last term i the expasio of x y 4 is 08xy b) 4 8y c) 4 y d) 5 4y 0) The middle term i the expasio of x y 4 is 4x y b) 4xy c) 6x y d) 6x y 7
28 ) Simplify b) 0 5 c) 5 0 d) 0 50 ) Simplify 0g 9 0 b) 0 c) 6 0 d) 40 ) Simplify b) 6 c) d) 6 4 4) Simplify a b c 4 5 5a bc 5a c b) 4 a bc 5ac c) 4 5a bc ac d) 7 5a c ac 8
29 5) Simplify b) 8 c) 4 0 d) 4 4 6) Simplify b) c) d) 5 7) Simplify b) c) 4 d) ) A Space statio rotates i order to simulate gravity, such that N, where N is the 7r umber of rotatios per miute required to simulate earth's gravity, ad r is the radius of the space statio. If the umber of rotatios per miute is 0, which of the followig is a reasoable estimate for the radius of the space statio? 5m b) 5.4m c) 0m d) 44m 9
30 Free Respose Questios: For 94. Whe you see the saw graphic, SHOW ALL WORK i the space provided. Correct aswers with o work will receive miimal credit. Icorrect aswers with work will receive partial credit. Icorrect aswers with o work will receive o credit. You may use a calculator. Each questio is worth 7.5 poits. For 9, simplify the expressio completely with oly positive expoets. 9) x 6 y 6x y 5 0) 0x y z x y z 6 9) 0) 8 9 ) 4 4x y z ) xy 6x y 5 6 ) ) ) Write i stadard form without the use of expoets. ) 4) Write 6,400,000 i scietific otatio. 4) 0
31 ) Evaluate (express aswer i scietific otatio) 5) For 68, simplify ad express aswer i radical form, if ecessary. 6) 0x 5x 7) 7x 8x 4 5 6) 7) 8) 4 0 8x y 8)
32 For 94, solve the iequality. Graph your solutio o a umber lie. 9) 4 9 0) x 4 54 ).6 y. ) 8 k 4 ) 4 p 9 4) d d
33 For 56, solve algebraically for x. 5) x 6 8 6) 5x 0 5) 6) For 78, solve algebraically for x. Express your aswer i iterval otatio. 7) x 7 4 8) 7x 9 7) 8)
34 For 940, expad the expressio completely. 9) 4 x 5y 9) 40) x y 40) 4
35 4) Express i simplest radical form x y b) x y b) 4) Simplify the expressio b) 6 9 b) c) 5 d) 5 5 c) d) 5
36 Bous) Fid the exact value of: 7 (+5) Bous) 6
Example 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 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 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 information9.1 Simplify Radical Expressions
9.1 Simplifyig Radical Expressios (Page 1 of 20) 9.1 Simplify Radical Expressios Radical Notatio for the th Root of a If is a iteger greater tha oe, the the th root of a is the umer whose th power is
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 informationSection 6.1 Radicals and Rational Exponents
Sectio 6.1 Radicals ad Ratioal Expoets Defiitio of Square Root The umber b is a square root of a if b The priciple square root of a positive umber is its positive square root ad we deote this root by usig
More informationEssential Question How can you use properties of exponents to simplify products and quotients of radicals?
. Properties of Ratioal Expoets ad Radicals Essetial Questio How ca you use properties of expoets to simplify products ad quotiets of radicals? Reviewig Properties of Expoets Work with a parter. Let a
More informationAlgebra 1B Assignments Chapter 8: Properties of Exponents
Nae Score Algebra B Assigets Chapter 8: Properties of Expoets 8 Pages : #66 eve, 78, 9, 98 8 Pages 80: #0 eve, 9 8 Pages 6: #8 eve,  eve, 67, 68, 70, 79, 8, 90, 9 8 Pages 9: #0 eve, 7, 76,
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 informationOnestep equations. Vocabulary
Review solvig oestep equatios with itegers, fractios, ad decimals. Oestep equatios Vocabulary equatio solve solutio iverse operatio isolate the variable Additio Property of Equality Subtractio Property
More information= 2, 3, 4, etc. = { FLC Ch 7. Math 120 Intermediate Algebra Sec 7.1: Radical Expressions and Functions
Math 120 Itermediate Algebra Sec 7.1: Radical Expressios ad Fuctios idex radicad = 2,,, etc. Ex 1 For each umber, fid all of its square roots. 121 2 6 Ex 2 1 Simplify. 1 22 9 81 62 8 27 16 16 0 1 180 22
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 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 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 informationRADICALS COMMON MISTAKES
RADICALS COMMON MISTAKES 1 10/0/009 RadicalsNotatio, Defiitio, ad Simplifyig How to Uderstad the Defiitio ad Notatio Notatio: a root, radical, a radicad. Square root,, but the two is NOT writte (i.e.
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 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 informationTHE LEAST SQUARES REGRESSION LINE and R 2
THE LEAST SQUARES REGRESSION LINE ad R M358K I. Recall from p. 36 that the least squares regressio lie of y o x is the lie that makes the sum of the squares of the vertical distaces of the data poits from
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 informationUnderstanding Rational Exponents and Radicals
x Locker LESSON. Uderstadig Ratioal Expoets ad Radicals Name Class Date. Uderstadig Ratioal Expoets ad Radicals Essetial Questio: How are radicals ad ratioal expoets related? A..A simplify umerical radical
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 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 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 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 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 informationContinued Fractions continued. 3. Best rational approximations
Cotiued Fractios cotiued 3. Best ratioal approximatios We hear so much about π beig approximated by 22/7 because o other ratioal umber with deomiator < 7 is closer to π. Evetually 22/7 is defeated by 333/06
More informationLiteral Equations and Formulas
. Literal Equatios ad Formulas. OBJECTIVE 1. Solve a literal equatio for a specified variable May problems i algebra require the use of formulas for their solutio. Formulas are simply equatios that express
More informationRADICALS AND SOLVING QUADRATIC EQUATIONS
RADICALS AND SOLVING QUADRATIC EQUATIONS Evaluate Roots Overview of Objectives, studets should be able to:. Evaluate roots a. Siplify expressios of the for a b. Siplify expressios of the for a. Evaluate
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 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 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 information1 Correlation and Regression Analysis
1 Correlatio ad Regressio Aalysis I this sectio we will be ivestigatig the relatioship betwee two cotiuous variable, such as height ad weight, the cocetratio of a ijected drug ad heart rate, or the cosumptio
More informationMath Discrete Math Combinatorics MULTIPLICATION PRINCIPLE:
Math 355  Discrete Math 4.14.4 Combiatorics Notes MULTIPLICATION PRINCIPLE: If there m ways to do somethig ad ways to do aother thig the there are m ways to do both. I the laguage of set theory: Let
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 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 informationMocks.ie Maths LC HL Further Calculus mocks.ie Page 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 Further Calculus ad Series, Paper II Q8 Table of Cotets:.
More information4.1 Sigma Notation and Riemann Sums
0 the itegral. Sigma Notatio ad Riema Sums Oe strategy for calculatig the area of a regio is to cut the regio ito simple shapes, calculate the area of each simple shape, ad the add these smaller areas
More 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 information5.3. Generalized Permutations and Combinations
53 GENERALIZED PERMUTATIONS AND COMBINATIONS 73 53 Geeralized Permutatios ad Combiatios 53 Permutatios with Repeated Elemets Assume that we have a alphabet with letters ad we wat to write all possible
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 informationArithmetic Sequences
. Arithmetic Sequeces Essetial Questio How ca you use a arithmetic sequece to describe a patter? A arithmetic sequece is a ordered list of umbers i which the differece betwee each pair of cosecutive terms,
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 informationMATH 083 Final Exam Review
MATH 08 Fial Eam Review Completig the problems i this review will greatly prepare you for the fial eam Calculator use is ot required, but you are permitted to use a calculator durig the fial eam period
More information23 The Remainder and Factor Theorems
 The Remaider ad Factor Theorems Factor each polyomial completely usig the give factor ad log divisio 1 x + x x 60; x + So, x + x x 60 = (x + )(x x 15) Factorig the quadratic expressio yields x + x x
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 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 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 information1.3 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 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 informationEngineering 323 Beautiful Homework Set 3 1 of 7 Kuszmar Problem 2.51
Egieerig 33 eautiful Homewor et 3 of 7 Kuszmar roblem.5.5 large departmet store sells sport shirts i three sizes small, medium, ad large, three patters plaid, prit, ad stripe, ad two sleeve legths log
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 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 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 informationThe Field Q of Rational Numbers
Chapter 3 The Field Q of Ratioal Numbers I this chapter we are goig to costruct the ratioal umber from the itegers. Historically, the positive ratioal umbers came first: the Babyloias, Egyptias ad Grees
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 information1. MATHEMATICAL INDUCTION
1. MATHEMATICAL INDUCTION EXAMPLE 1: Prove that for ay iteger 1. Proof: 1 + 2 + 3 +... + ( + 1 2 (1.1 STEP 1: For 1 (1.1 is true, sice 1 1(1 + 1. 2 STEP 2: Suppose (1.1 is true for some k 1, that is 1
More informationMeasures of Spread and Boxplots Discrete Math, Section 9.4
Measures of Spread ad Boxplots Discrete Math, Sectio 9.4 We start with a example: Example 1: Comparig Mea ad Media Compute the mea ad media of each data set: S 1 = {4, 6, 8, 10, 1, 14, 16} S = {4, 7, 9,
More informationChapter One BASIC MATHEMATICAL TOOLS
Chapter Oe BAIC MATHEMATICAL TOOL As the reader will see, the study of the time value of moey ivolves substatial use of variables ad umbers that are raised to a power. The power to which a variable is
More informationHomework 1 Solutions
Homewor 1 Solutios Math 171, Sprig 2010 Please sed correctios to herya@math.staford.edu 2.2. Let h : X Y, g : Y Z, ad f : Z W. Prove that (f g h = f (g h. Solutio. Let x X. Note that ((f g h(x = (f g(h(x
More informationTILE PATTERNS & GRAPHING
TILE PATTERNS & GRAPHING LESSON 1 THE BIG IDEA Tile patters provide a meaigful cotext i which to geerate equivalet algebraic expressios ad develop uderstadig of the cocept of a variable. Such patters are
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 information7.1 Finding Rational Solutions of Polynomial Equations
4 Locker LESSON 7. Fidig Ratioal Solutios of Polyomial Equatios Name Class Date 7. Fidig Ratioal Solutios of Polyomial Equatios Essetial Questio: How do you fid the ratioal roots of a polyomial equatio?
More informationGrade 7. Strand: Number Specific Learning Outcomes It is expected that students will:
Strad: Number Specific Learig Outcomes It is expected that studets will: 7.N.1. Determie ad explai why a umber is divisible by 2, 3, 4, 5, 6, 8, 9, or 10, ad why a umber caot be divided by 0. [C, R] [C]
More informationAP Calculus BC 2003 Scoring Guidelines Form B
AP Calculus BC Scorig Guidelies Form B The materials icluded i these files are iteded for use by AP teachers for course ad exam preparatio; permissio for ay other use must be sought from the Advaced Placemet
More informationInfinite Sequences and Series
CHAPTER 4 Ifiite Sequeces ad Series 4.1. Sequeces A sequece is a ifiite ordered list of umbers, for example the sequece of odd positive itegers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29...
More 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 informationChapter 9: Correlation and Regression: Solutions
Chapter 9: Correlatio ad Regressio: Solutios 9.1 Correlatio I this sectio, we aim to aswer the questio: Is there a relatioship betwee A ad B? Is there a relatioship betwee the umber of emploee traiig hours
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 informationYour grandmother and her financial counselor
Sectio 10. Arithmetic Sequeces 963 Objectives Sectio 10. Fid the commo differece for a arithmetic sequece. Write s of a arithmetic sequece. Use the formula for the geeral of a arithmetic sequece. Use the
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 informationAlgebra Vocabulary List (Definitions for Middle School Teachers)
Algebra Vocabulary List (Defiitios for Middle School Teachers) A Absolute Value Fuctio The absolute value of a real umber x, x is xifx 0 x = xifx < 0 http://www.math.tamu.edu/~stecher/171/f02/absolutevaluefuctio.pdf
More informationMath 114 Intermediate Algebra Integral Exponents & Fractional Exponents (10 )
Math 4 Math 4 Itermediate Algebra Itegral Epoets & Fractioal Epoets (0 ) Epoetial Fuctios Epoetial Fuctios ad Graphs I. Epoetial Fuctios The fuctio f ( ) a, where is a real umber, a 0, ad a, is called
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 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 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 informationUC Berkeley Department of Electrical Engineering and Computer Science. EE 126: Probablity and Random Processes. Solutions 9 Spring 2006
Exam format UC Bereley Departmet of Electrical Egieerig ad Computer Sciece EE 6: Probablity ad Radom Processes Solutios 9 Sprig 006 The secod midterm will be held o Wedesday May 7; CHECK the fial exam
More informationMath C067 Sampling Distributions
Math C067 Samplig Distributios Sample Mea ad Sample Proportio Richard Beigel Some time betwee April 16, 2007 ad April 16, 2007 Examples of Samplig A pollster may try to estimate the proportio of voters
More information1 StateSpace Canonical Forms
StateSpace Caoical Forms For ay give system, there are essetially a ifiite umber of possible state space models that will give the idetical iput/output dyamics Thus, it is desirable to have certai stadardized
More informationFOUNDATIONS OF MATHEMATICS AND PRECALCULUS GRADE 10
FOUNDATIONS OF MATHEMATICS AND PRECALCULUS GRADE 10 [C] Commuicatio Measuremet A1. Solve problems that ivolve liear measuremet, usig: SI ad imperial uits of measure estimatio strategies measuremet strategies.
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 informationLecture Notes CMSC 251
We have this messy summatio to solve though First observe that the value remais costat throughout the sum, ad so we ca pull it out frot Also ote that we ca write 3 i / i ad (3/) i T () = log 3 (log ) 1
More informationSEQUENCES AND SERIES CHAPTER
CHAPTER SEQUENCES AND SERIES Whe the Grat family purchased a computer for $,200 o a istallmet pla, they agreed to pay $00 each moth util the cost of the computer plus iterest had bee paid The iterest each
More 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 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 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 informationThe Field of Complex Numbers
The Field of Complex Numbers S. F. Ellermeyer The costructio of the system of complex umbers begis by appedig to the system of real umbers a umber which we call i with the property that i = 1. (Note that
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 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 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 informationWinter Camp 2012 Sequences Alexander Remorov. Sequences. Alexander Remorov
Witer Camp 202 Sequeces Alexader Remorov Sequeces Alexader Remorov alexaderrem@gmail.com Warmup Problem : Give a positive iteger, cosider a sequece of real umbers a 0, a,..., a defied as a 0 = 2 ad =
More informationConfidence Intervals for the Mean of Nonnormal Data Class 23, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom
Cofidece Itervals for the Mea of Noormal Data Class 23, 8.05, Sprig 204 Jeremy Orloff ad Joatha Bloom Learig Goals. Be able to derive the formula for coservative ormal cofidece itervals for the proportio
More information3. Covariance and Correlation
Virtual Laboratories > 3. Expected Value > 1 2 3 4 5 6 3. Covariace ad Correlatio Recall that by takig the expected value of various trasformatios of a radom variable, we ca measure may iterestig characteristics
More informationLaws of Exponents. net effect is to multiply with 2 a total of 3 + 5 = 8 times
The Mathematis 11 Competey Test Laws of Expoets (i) multipliatio of two powers: multiply by five times 3 x = ( x x ) x ( x x x x ) = 8 multiply by three times et effet is to multiply with a total of 3
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 informationNPTEL STRUCTURAL RELIABILITY
NPTEL Course O STRUCTURAL RELIABILITY Module # 0 Lecture 1 Course Format: Web Istructor: Dr. Aruasis Chakraborty Departmet of Civil Egieerig Idia Istitute of Techology Guwahati 1. Lecture 01: Basic Statistics
More information3.1 Measures of Central Tendency. Introduction 5/28/2013. Data Description. Outline. Objectives. Objectives. Traditional Statistics Average
5/8/013 C H 3A P T E R Outlie 3 1 Measures of Cetral Tedecy 3 Measures of Variatio 3 3 3 Measuresof Positio 3 4 Exploratory Data Aalysis Copyright 013 The McGraw Hill Compaies, Ic. C H 3A P T E R Objectives
More informationIntroductory Explorations of the Fourier Series by
page Itroductory Exploratios of the Fourier Series by Theresa Julia Zieliski Departmet of Chemistry, Medical Techology, ad Physics Momouth Uiversity West Log Brach, NJ 7764898 tzielis@momouth.edu Copyright
More information1. Strong vs regular indiction 2. Strong induction examples: ! Divisibility by a prime! Recursion sequence: product of fractions
Today s Topics: CSE 0: Discrete Mathematics for Computer Sciece Prof. Miles Joes 1. Strog vs regular idictio. Strog iductio examples:! Divisibility by a prime! Recursio sequece: product of fractios 3 4
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 information