COMPLEX FRACTIONS. section. Simplifying Complex Fractions


 Byron Dennis
 1 years ago
 Views:
Transcription
1 58 (66) Chpter 6 Rtionl Epressions undles tht they cn ttch while working together for 0 hours FIGURE FOR EXERCISE Selling. George sells one gzine suscription every 0 inutes, wheres Theres sells one every inutes. Write rtionl epression for the nuer of gzine suscriptions tht they will sell when working together for one hour Pinting. Hrry cn pint his house y hiself in 6 dys. His wife Judy cn pint the house y herself in dys. Write rtionl epression for the portion of the house tht they pint when working together for dys Driving. Jon drove for 00 iles t one speed nd then incresed her speed y 5 iles per hour nd drove 00 dditionl iles. Write rtionl epression for her totl trvel tie hours Running. Willrd jogged for iles t one speed nd then douled his speed for n dditionl ile. Write rtionl epression for his totl running tie. 7 hours GETTING MORE INVOLVED 99. Discussion. Eplin why frctions ust hve coon denointors for ddition ut not for ultipliction. 00. Discussion. Find ech infinite su nd eplin your nswer. ) ) COMPLEX FRACTIONS In this section In this section we will use the techniques of Section 6. to siplify cople frctions. As their ne suggests, cople frctions re rther essylooking epressions. Siplifying Cople Frctions Siplifying Epressions with Negtive Eponents Applictions Siplifying Cople Frctions A cople frction is frction tht hs rtionl epressions in the nuertor, the denointor, or oth. For eple,,, nd re cople frctions. In the net eple we show two ethods for siplifying cople frction.
2 6. Cople Frctions (67) 59 E X A M P L E clcultor closeup You cn use clcultor to find the vlue of cople frction. A cople frction without vriles 5 Siplify. Method A For this ethod we perfor the coputtions of the nuertor nd denointor seprtely nd then divide: Method B For this ethod we find the LCD for ll of the frctions in the cople frction. Then we ultiply the nuertor nd denointor of the cople frction y the LCD. The LCD for the denointors,,, nd 5 is 60. So we ultiply the nuertor nd denointor of the cople frction y 60: , , 60 5 In ost cses Method B of Eple is the fster ethod for siplifying cople frctions, nd we will continue to use it. E X A M P L E A cople frction with vriles Siplify. The LCD of,, nd is. Multiply the nuertor nd denointor y : helpful hint When students see ddition or sutrction in cople frction, they often convert ll of the frctions to the se denointor. This is not wrong, ut it is not necessry. Siply ultiplying ever frction y the LCD eliintes the denointors of the originl frctions. ( ) ( ) ( ) ( ) ( ) ( ) 8 Distriutive property
3 60 (68) Chpter 6 Rtionl Epressions E X A M P L E study tip Your ood for studying should tch the ood in which you re tested. Being too reled during studying will not tch the incresed level of ctivtion you ttin during test. Likewise, if you get too tensedup during test, you will not do well ecuse your testtking ood will not tch your studying ood. More coplicted denointors Siplify. Becuse 9 ( )( ) nd 6 9 ( ), the LCD is ( ) ( ). Multiply the nuertor nd denointor y the LCD: ( ) ( ) 9 ( )( ) ( ) ( ) ( ) ( ) 6 9 ( ) ( )( ) Siplify. ( ) ( )( ) ( )( ) Fctor out. ( )[ ( )] ( ) ( ) ( )( 5 ) Siplifying Epressions with Negtive Eponents Consider the epression. Using the definition of negtive eponents, we cn rewrite this epression s cople frction: The LCD for the cople frction is. Note tht could lso e otined fro the ses of the epressions with the negtive eponents. To siplify the cople frction, we could use Method B s we hve een doing. However, it is not necessry to rewrite the originl epression s cople frction. The net eple shows how to use Method B with the originl epression. E X A M P L E A cople frction with negtive eponents Siplify the cople frction. Multiply the nuertor nd denointor y, the LCD of the frctions. Reeer tht 0. ( ) ( ) () () Distriutive property () () 6
4 6. Cople Frctions (69) 6 E X A M P L E 5 helpful hint In Eples, 5, nd 6 we re siplifying the epressions without first reoving the negtive eponents to gin eperience in working with negtive eponents. Of course, ech epression with negtive eponent could e rewritten with positive eponent nd then the cople frction could e siplified s in Eples nd. A cople frction with negtive eponents Siplify the cople frction. If we rewrote,,, nd, then the denointors would e,,, nd. So the LCD is. If we ultiply the nuertor nd denointor y, the negtive eponents will e eliinted: ( ) ( ) Distriutive property 0 0 Note tht the positive eponents of re just lrge enough to eliinte ll of the negtive eponents when we ultiply. The net eple is not ectly cople frction, ut we cn use the se technique s in the previous eple. E X A M P L E 6 More negtive eponents Eliinte negtive eponents nd siplify p p q. If we ultiply the nuertor nd denointor y pq, we will eliinte the negtive eponents: p p q ( p p q ) p q pq p q p pq p q pq p q pq Applictions The net eple illustrtes how cople frctions cn occur in prole. E X A M P L E 7 An ppliction of cople frctions Estside Eleentry hs the se nuer of students s Westside Eleentry. Onehlf of the students t Estside ride uses to school, nd twothirds of the students t Westside ride uses to school. Onesith of the students t Estside re fele, nd onethird of the students t Westside re fele. If ll of the fele students ride the uses, then wht percentge of the students who ride the uses re fele? To find the required percentge, we ust divide the nuer of feles who ride the uses y the totl nuer of students who ride the uses. Let the nuer of students t Estside.
5 6 (60) Chpter 6 Rtionl Epressions Becuse the nuer of students t Westside is lso, we hve the totl nuer of students who ride the uses nd 6 the totl nuer of fele students. Becuse ll of the fele students ride the uses, we cn epress the percentge of riders who re fele y the following rtionl epression: 6 Multiply the nuertor nd denointor y 6, the LCD for,, nd 6: % 7 7 So % of the students who ride the uses re fele. WARMUPS True or flse? Eplin.. The LCM for,, 6, nd is 6. Flse. The LCM for,, nd 6 is 6 6. True. The LCD is the LCM of the denointors. True. 5 6 True 5. ( ) Flse 6. ( ) Flse 7. 5 Flse 8. for ny rel nuer. Flse 9. To siplify, ultiply the nuertor nd denointor y. True 0. To siplify 5, 5 ultiply the nuertor nd denointor y 5. True 6. EXERCISES Reding nd Writing After reding this section, write out the nswers to these questions. Use coplete sentences.. Wht is cople frction? A cople frction is frction tht contins frctions in the nuertor, denointor, or oth.. Wht re the two ethods for siplifying cople frctions? One ethod is to perfor the opertions in the nuertor nd then in the denointor, nd then divide the results. The other ethod is to ultiply the nuertor nd the denointor y the LCD for ll of the frctions.
6 6. Cople Frctions (6) 6 Siplify ech cople frction. Use either ethod. See Eple Siplify the cople frctions. Use Method B. See Eple. n n n n y n y w t 0... w t n y wt n y 8t w n y w t 6 z 6 z z z z y y y y y y 6 y y y 6 y y Siplify ech cople frction. See Eples y.. y y y (y )( y ) w w w w w w w w 6w w w y y.. y y ( y y )( ) y y Siplify. See Eples w y 6. z y yz wz wy wz n 0. n n n
7 6 (6) Chpter 6 Rtionl Epressions y ) y ( y ( y ) 5. ( ) y y Use clcultor to evlute ech cople frction. Round nswers to four decil plces. If your clcultor does frctions, then lso find the frctionl nswer , , , 6.79, 0 Solve ech prole. See Eple Rcil lnce. Clrksville hs three eleentry schools. Northside hs onehlf s ny students s Centrl, nd Southside hs twothirds s ny students s Centrl. Onethird of the students t Northside re Africn Aericn, threefourths of the students t Centrl re AfricnAericn, nd onesith of the students t Southside re AfricnAericn. Wht percent of the city s eleentry students re AfricnAericn? 7.% 60. Eplosive sitution. All of the eployees t Ace Eplosives re in either developent, nufcturing, or sles. Onefifth of the eployees in developent re woen, onethird of the eployees in nufcturing re woen, nd onehlf of the eployees in sles re woen. Use the ccopnying figure to deterine the percentge of workers t Ace who re woen. Wht percent of the woen t Ace re in sles? 8.%, 65.% Developent Distriution of Eployees t Ace Eplosives Mnufcturing Sles FIGURE FOR EXERCISE Averge speed. Mry drove fro Clrksville to Leesville t 5 iles per hour (ph). At Leesville she discovered tht she hd forgotten her purse. She ieditely returned to Clrksville t 55 ph. Wht ws her verge speed for the entire trip? (The nswer is not 50 ph.) 9.5 ph 6. Averge price. On her wy to New York, Jenny spent the se ount for gsoline ech of the three ties tht she filled up. She pid 99.9 cents per gllon the first tie, 09.9 cents per gllon the second tie, nd 9.9 cents per gllon the third tie. Wht ws the verge price per gllon to the nerest tenth of cent for the gsoline tht she ought? 09. cents per gllon FIGURE FOR EXERCISE 6
8 6.5 Solving Equtions Involving Rtionl Epressions (6) 65 GETTING MORE INVOLVED 6. Coopertive lerning. Write stepystep strtegy for siplifying cople frctions with negtive eponents. Hve clsste use your strtegy to siplify soe cople frctions fro Eercises Discussion. ) Find the ect vlue of ech epression. i) ii) 5 8, 8 ) Eplin why in ech cse the ect vlue ust e less thn. The denointor is lrger thn the nuertor in the first frction. 65. Coopertive Lerning. Work with group to siplify the cople frction. For wht vlues of is the cople frction undefined?, 0,,, In this section 6.5 SOLVING EQUATIONS INVOLVING RATIONAL EXPRESSIONS Mny proles in lger re odeled y equtions involving rtionl epressions. In this section you will lern how to solve equtions tht hve rtionl epressions, nd in Section 6.6 we will solve proles using these equtions. Multiplying y the LCD Etrneous Roots Proportions Multiplying y the LCD To solve equtions hving rtionl epressions, we ultiply ech side of the eqution y the LCD of the rtionl epressions. E X A M P L E helpful hint Note tht it is not necessry to convert ech frction into n equivlent frction with coon denointor here. Since we cn ultiply oth sides of n eqution y ny epression we choose, we choose to ultiply y the LCD. This tctic eliintes the frctions in one step nd tht is good. An eqution with rtionl epressions Solve 6. The LCD for the denointors, 6, nd is : 6 6 Multiply ech side y. Distriutive property Divide out the coon fctors. 0 Check in the originl eqution. The solution set is.
5.6 POSITIVE INTEGRAL EXPONENTS
54 (5 ) Chpter 5 Polynoils nd Eponents 5.6 POSITIVE INTEGRAL EXPONENTS In this section The product rule for positive integrl eponents ws presented in Section 5., nd the quotient rule ws presented in Section
More informationHow Simple Is Your Rational Expression? Examples
How Siple Is Your Rtionl Epression? Eples. Rtionl epressions re lebric epressions whose nuertor nd denointor 5 re polynoils. The epression,, nd re eples of rtionl y 4 epressions or lebric frctions.. A
More informationQuadratic Equations  1
Alger Module A60 Qudrtic Equtions  1 Copyright This puliction The Northern Alert Institute of Technology 00. All Rights Reserved. LAST REVISED Novemer, 008 Qudrtic Equtions  1 Sttement of Prerequisite
More informationOperations with Polynomials
38 Chpter P Prerequisites P.4 Opertions with Polynomils Wht you should lern: Write polynomils in stndrd form nd identify the leding coefficients nd degrees of polynomils Add nd subtrct polynomils Multiply
More informationChapter 9: Quadratic Equations
Chpter 9: Qudrtic Equtions QUADRATIC EQUATIONS DEFINITION + + c = 0,, c re constnts (generlly integers) ROOTS Synonyms: Solutions or Zeros Cn hve 0, 1, or rel roots Consider the grph of qudrtic equtions.
More informationBasic Math Review. Numbers. Important Properties. Absolute Value PROPERTIES OF ADDITION NATURAL NUMBERS {1, 2, 3, 4, 5, }
ƒ Bsic Mth Review Numers NATURAL NUMBERS {1,, 3, 4, 5, } WHOLE NUMBERS {0, 1,, 3, 4, } INTEGERS {, 3,, 1, 0, 1,, } The Numer Line 5 4 3 1 0 1 3 4 5 Negtive integers Positive integers RATIONAL NUMBERS All
More informationP.3 Polynomials and Factoring. P.3 an 1. Polynomial STUDY TIP. Example 1 Writing Polynomials in Standard Form. What you should learn
33337_0P03.qp 2/27/06 24 9:3 AM Chpter P Pge 24 Prerequisites P.3 Polynomils nd Fctoring Wht you should lern Polynomils An lgeric epression is collection of vriles nd rel numers. The most common type of
More informationMA 15800 Lesson 16 Notes Summer 2016 Properties of Logarithms. Remember: A logarithm is an exponent! It behaves like an exponent!
MA 5800 Lesson 6 otes Summer 06 Rememer: A logrithm is n eponent! It ehves like n eponent! In the lst lesson, we discussed four properties of logrithms. ) log 0 ) log ) log log 4) This lesson covers more
More informationChapter 6 Solving equations
Chpter 6 Solving equtions Defining n eqution 6.1 Up to now we hve looked minly t epressions. An epression is n incomplete sttement nd hs no equl sign. Now we wnt to look t equtions. An eqution hs n = sign
More informationBinary Representation of Numbers Autar Kaw
Binry Representtion of Numbers Autr Kw After reding this chpter, you should be ble to: 1. convert bse rel number to its binry representtion,. convert binry number to n equivlent bse number. In everydy
More informationTHE RATIONAL NUMBERS CHAPTER
CHAPTER THE RATIONAL NUMBERS When divided by b is not n integer, the quotient is frction.the Bbylonins, who used number system bsed on 60, epressed the quotients: 0 8 s 0 60 insted of 8 s 7 60,600 0 insted
More informationMATH 150 HOMEWORK 4 SOLUTIONS
MATH 150 HOMEWORK 4 SOLUTIONS Section 1.8 Show tht the product of two of the numbers 65 1000 8 2001 + 3 177, 79 1212 9 2399 + 2 2001, nd 24 4493 5 8192 + 7 1777 is nonnegtive. Is your proof constructive
More informationOr more simply put, when adding or subtracting quantities, their uncertainties add.
Propgtion of Uncertint through Mthemticl Opertions Since the untit of interest in n eperiment is rrel otined mesuring tht untit directl, we must understnd how error propgtes when mthemticl opertions re
More informationSection A4 Rational Expressions: Basic Operations
A Appendi A A BASIC ALGEBRA REVIEW 7. Construction. A rectngulr opentopped bo is to be constructed out of 9 by 6inch sheets of thin crdbord by cutting inch squres out of ech corner nd bending the
More informationExample 27.1 Draw a Venn diagram to show the relationship between counting numbers, whole numbers, integers, and rational numbers.
2 Rtionl Numbers Integers such s 5 were importnt when solving the eqution x+5 = 0. In similr wy, frctions re importnt for solving equtions like 2x = 1. Wht bout equtions like 2x + 1 = 0? Equtions of this
More informationReasoning to Solve Equations and Inequalities
Lesson4 Resoning to Solve Equtions nd Inequlities In erlier work in this unit, you modeled situtions with severl vriles nd equtions. For exmple, suppose you were given usiness plns for concert showing
More informationFactoring Polynomials
Fctoring Polynomils Some definitions (not necessrily ll for secondry school mthemtics): A polynomil is the sum of one or more terms, in which ech term consists of product of constnt nd one or more vribles
More informationAppendix D: Completing the Square and the Quadratic Formula. In Appendix A, two special cases of expanding brackets were considered:
Appendi D: Completing the Squre nd the Qudrtic Formul Fctoring qudrtic epressions such s: + 6 + 8 ws one of the topics introduced in Appendi C. Fctoring qudrtic epressions is useful skill tht cn help you
More informationSquare & Square Roots
Squre & Squre Roots Squre : If nuber is ultiplied by itself then the product is the squre of the nuber. Thus the squre of is x = eg. x x Squre root: The squre root of nuber is one of two equl fctors which
More informationNQF Level: 2 US No: 7480
NQF Level: 2 US No: 7480 Assessment Guide Primry Agriculture Rtionl nd irrtionl numers nd numer systems Assessor:.......................................... Workplce / Compny:.................................
More informationSquare Roots Teacher Notes
Henri Picciotto Squre Roots Techer Notes This unit is intended to help students develop n understnding of squre roots from visul / geometric point of view, nd lso to develop their numer sense round this
More informationAlgebra Review. How well do you remember your algebra?
Algebr Review How well do you remember your lgebr? 1 The Order of Opertions Wht do we men when we write + 4? If we multiply we get 6 nd dding 4 gives 10. But, if we dd + 4 = 7 first, then multiply by then
More informationSolving Linear Equations  Formulas
1. Solving Liner Equtions  Formuls Ojective: Solve liner formuls for given vrile. Solving formuls is much like solving generl liner equtions. The only difference is we will hve severl vriles in the prolem
More informationPROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY
MAT 0630 INTERNET RESOURCES, REVIEW OF CONCEPTS AND COMMON MISTAKES PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY Contents 1. ACT Compss Prctice Tests 1 2. Common Mistkes 2 3. Distributive
More informationExponential and Logarithmic Functions
Nme Chpter Eponentil nd Logrithmic Functions Section. Eponentil Functions nd Their Grphs Objective: In this lesson ou lerned how to recognize, evlute, nd grph eponentil functions. Importnt Vocbulr Define
More informationMultiplication and Division  Left to Right. Addition and Subtraction  Left to Right.
Order of Opertions r of Opertions Alger P lese Prenthesis  Do ll grouped opertions first. E cuse Eponents  Second M D er Multipliction nd Division  Left to Right. A unt S hniqu Addition nd Sutrction
More informationPolynomial Functions. Polynomial functions in one variable can be written in expanded form as ( )
Polynomil Functions Polynomil functions in one vrible cn be written in expnded form s n n 1 n 2 2 f x = x + x + x + + x + x+ n n 1 n 2 2 1 0 Exmples of polynomils in expnded form re nd 3 8 7 4 = 5 4 +
More informationSolutions to Section 1
Solutions to Section Exercise. Show tht nd. This follows from the fct tht mx{, } nd mx{, } Exercise. Show tht = { if 0 if < 0 Tht is, the bsolute vlue function is piecewise defined function. Grph this
More informationSPECIAL PRODUCTS AND FACTORIZATION
MODULE  Specil Products nd Fctoriztion 4 SPECIAL PRODUCTS AND FACTORIZATION In n erlier lesson you hve lernt multipliction of lgebric epressions, prticulrly polynomils. In the study of lgebr, we come
More informationThe Quadratic Formula and the Discriminant
99 The Qudrtic Formul nd the Discriminnt Objectives Solve qudrtic equtions by using the Qudrtic Formul. Determine the number of solutions of qudrtic eqution by using the discriminnt. Vocbulry discriminnt
More informationALGEBRAIC FRACTIONS,AND EQUATIONS AND INEQUALITIES INVOLVING FRACTIONS
CHAPTER ALGEBRAIC FRACTIONS,AND EQUATIONS AND INEQUALITIES INVOLVING FRACTIONS Although people tody re mking greter use of deciml frctions s they work with clcultors, computers, nd the metric system, common
More informationUnit 6: Exponents and Radicals
Eponents nd Rdicls : The Rel Numer Sstem Unit : Eponents nd Rdicls Pure Mth 0 Notes Nturl Numers (N):  counting numers. {,,,,, } Whole Numers (W):  counting numers with 0. {0,,,,,, } Integers (I): 
More informationMath 135 Circles and Completing the Square Examples
Mth 135 Circles nd Completing the Squre Exmples A perfect squre is number such tht = b 2 for some rel number b. Some exmples of perfect squres re 4 = 2 2, 16 = 4 2, 169 = 13 2. We wish to hve method for
More information3 The Utility Maximization Problem
3 The Utility Mxiiztion Proble We hve now discussed how to describe preferences in ters of utility functions nd how to forulte siple budget sets. The rtionl choice ssuption, tht consuers pick the best
More information10.5 Graphing Quadratic Functions
0.5 Grphing Qudrtic Functions Now tht we cn solve qudrtic equtions, we wnt to lern how to grph the function ssocited with the qudrtic eqution. We cll this the qudrtic function. Grphs of Qudrtic Functions
More informationExponentiation: Theorems, Proofs, Problems Pre/Calculus 11, Veritas Prep.
Exponentition: Theorems, Proofs, Problems Pre/Clculus, Verits Prep. Our Exponentition Theorems Theorem A: n+m = n m Theorem B: ( n ) m = nm Theorem C: (b) n = n b n ( ) n n Theorem D: = b b n Theorem E:
More informationnot to be republished NCERT POLYNOMIALS CHAPTER 2 (A) Main Concepts and Results (B) Multiple Choice Questions
POLYNOMIALS (A) Min Concepts nd Results Geometricl mening of zeroes of polynomil: The zeroes of polynomil p(x) re precisely the xcoordintes of the points where the grph of y = p(x) intersects the xxis.
More informationCS99S Laboratory 2 Preparation Copyright W. J. Dally 2001 October 1, 2001
CS99S Lortory 2 Preprtion Copyright W. J. Dlly 2 Octoer, 2 Ojectives:. Understnd the principle of sttic CMOS gte circuits 2. Build simple logic gtes from MOS trnsistors 3. Evlute these gtes to oserve logic
More informationSCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics. Basic Algebra
SCHOOL OF ENGINEERING & BUILT ENVIRONMENT Mthemtics Bsic Alger. Opertions nd Epressions. Common Mistkes. Division of Algeric Epressions. Eponentil Functions nd Logrithms. Opertions nd their Inverses. Mnipulting
More informationSirindhorn International Institute of Technology Thammasat University at Rangsit
Sirindhorn Interntionl Institute of Technology Thmmst University t Rngsit School of Informtion, Computer nd Communiction Technology COURSE : ECS 204 Bsic Electricl Engineering L INSTRUCTOR : Asst. Prof.
More informationChapter 3 Section 3 Lesson Additional Rules for Exponents
Chpter Sectio Lesso Additiol Rules for Epoets Itroductio I this lesso we ll eie soe dditiol rules tht gover the behvior of epoets The rules should be eorized; they will be used ofte i the reiig chpters
More informationJackson 2.23 Homework Problem Solution Dr. Christopher S. Baird University of Massachusetts Lowell
Jckson.3 Homework Problem Solution Dr. Christopher S. Bird University of Msschusetts Lowell PROBLEM: A hollow cube hs conducting wlls defined by six plnes x =, y =, z =, nd x =, y =, z =. The wlls z =
More information6.3. section. Building Up the Denominator. To convert the fraction 2 3 factor 21 as 21 3 7. Because 2 3
0 (618) Chapter 6 Rational Epressions GETTING MORE INVOLVED 7. Discussion. Evaluate each epression. a) Onehalf of 1 b) Onethird of c) Onehalf of d) Onehalf of 1 a) b) c) d) 8 7. Eploration. Let R
More informationAREA OF A SURFACE OF REVOLUTION
AREA OF A SURFACE OF REVOLUTION h cut r πr h A surfce of revolution is formed when curve is rotted bout line. Such surfce is the lterl boundr of solid of revolution of the tpe discussed in Sections 7.
More informationUnderstanding 22. 23. The frictional force acting to the left is missing. It is equal in magnitude to the applied force acting to the right.
Chpter 3 Review, pges 154 159 Knowledge 1. (c) 2. () 3. (d) 4. (d) 5. (d) 6. (c) 7. (b) 8. (c) 9. Flse. One newton is equl to 1 kg /s 2. 10. Flse. A norl force is perpendiculr force cting on n object tht
More informationStrong acids and bases
Monoprotic AcidBse Equiliri (CH ) ϒ Chpter monoprotic cids A monoprotic cid cn donte one proton. This chpter includes uffers; wy to fi the ph. ϒ Chpter 11 polyprotic cids A polyprotic cid cn donte multiple
More information14.2. The Mean Value and the RootMeanSquare Value. Introduction. Prerequisites. Learning Outcomes
he Men Vlue nd the RootMenSqure Vlue 4. Introduction Currents nd voltges often vry with time nd engineers my wish to know the men vlue of such current or voltge over some prticulr time intervl. he men
More information0.2 ABSOLUTE VALUE AND DISTANCE ON THE REAL NUMBER LINE
360040_0002.q 1/3/05 11:17 AM Pge 08 08 HAPTER 0 A Preclculus Review 0.2 ABSOLUTE VALUE AND DISTANE ON THE REAL NUMBER LINE Fin the solute vlues of rel numers n unerstn the properties of solute vlue.
More informationAnswer, Key Homework 10 David McIntyre 1
Answer, Key Homework 10 Dvid McIntyre 1 This printout should hve 22 questions, check tht it is complete. Multiplechoice questions my continue on the next column or pge: find ll choices efore mking your
More informationAnswer, Key Homework 8 David McIntyre 1
Answer, Key Homework 8 Dvid McIntyre 1 This printout should hve 17 questions, check tht it is complete. Multiplechoice questions my continue on the net column or pge: find ll choices before mking your
More informationBayesian Updating with Continuous Priors Class 13, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom
Byesin Updting with Continuous Priors Clss 3, 8.05, Spring 04 Jeremy Orloff nd Jonthn Bloom Lerning Gols. Understnd prmeterized fmily of distriutions s representing continuous rnge of hypotheses for the
More information. At first sight a! b seems an unwieldy formula but use of the following mnemonic will possibly help. a 1 a 2 a 3 a 1 a 2
7 CHAPTER THREE. Cross Product Given two vectors = (,, nd = (,, in R, the cross product of nd written! is defined to e: " = (!,!,! Note! clled cross is VECTOR (unlike which is sclr. Exmple (,, " (4,5,6
More informationThe remaining two sides of the right triangle are called the legs of the right triangle.
10 MODULE 6. RADICAL EXPRESSIONS 6 Pythgoren Theorem The Pythgoren Theorem An ngle tht mesures 90 degrees is lled right ngle. If one of the ngles of tringle is right ngle, then the tringle is lled right
More informationSequences and Series
Centre for Eduction in Mthemtics nd Computing Euclid eworkshop # 5 Sequences nd Series c 014 UNIVERSITY OF WATERLOO While the vst mjority of Euclid questions in this topic re use formule for rithmetic
More information1. Find the zeros Find roots. Set function = 0, factor or use quadratic equation if quadratic, graph to find zeros on calculator
AP Clculus Finl Review Sheet When you see the words. This is wht you think of doing. Find the zeros Find roots. Set function =, fctor or use qudrtic eqution if qudrtic, grph to find zeros on clcultor.
More informationMathematics Higher Level
Mthemtics Higher Level Higher Mthemtics Exmintion Section : The Exmintion Mthemtics Higher Level. Structure of the exmintion pper The Higher Mthemtics Exmintion is divided into two ppers s detiled below:
More informationIn this section make precise the idea of a matrix inverse and develop a method to find the inverse of a given square matrix when it exists.
Mth 52 Sec S060/S0602 Notes Mtrices IV 5 Inverse Mtrices 5 Introduction In our erlier work on mtrix multipliction, we sw the ide of the inverse of mtrix Tht is, for squre mtrix A, there my exist mtrix
More information4.0 5Minute Review: Rational Functions
mth 130 dy 4: working with limits 1 40 5Minute Review: Rtionl Functions DEFINITION A rtionl function 1 is function of the form y = r(x) = p(x) q(x), 1 Here the term rtionl mens rtio s in the rtio of two
More informationVariable Dry Run (for Python)
Vrile Dr Run (for Pthon) Age group: Ailities ssumed: Time: Size of group: Focus Vriles Assignment Sequencing Progrmming 7 dult Ver simple progrmming, sic understnding of ssignment nd vriles 2050 minutes
More informationGraphs on Logarithmic and Semilogarithmic Paper
0CH_PHClter_TMSETE_ 3//00 :3 PM Pge Grphs on Logrithmic nd Semilogrithmic Pper OBJECTIVES When ou hve completed this chpter, ou should be ble to: Mke grphs on logrithmic nd semilogrithmic pper. Grph empiricl
More informationThe AVL Tree Rotations Tutorial
The AVL Tree Rottions Tutoril By John Hrgrove Version 1.0.1, Updted Mr222007 Astrt I wrote this doument in n effort to over wht I onsider to e drk re of the AVL Tree onept. When presented with the tsk
More informationEcon 4721 Money and Banking Problem Set 2 Answer Key
Econ 472 Money nd Bnking Problem Set 2 Answer Key Problem (35 points) Consider n overlpping genertions model in which consumers live for two periods. The number of people born in ech genertion grows in
More informationMaths Assessment Year 4: Number and Place Value
Nme: Mths Assessment Yer 4: Numer nd Plce Vlue 1. Count in multiples of 6, 7, 9, 25 nd 1 000; find 1 000 more or less thn given numer. 2. Find 1,000 more or less thn given numer. 3. Count ckwrds through
More informationChapter 5: Elasticity. measures how strongly people respond to changes in prices and changes in income.
Chpter 5: Elsticity Elsticity responsiveness mesures how strongly people respond to chnges in prices nd chnges in income. Exmples of questions tht elsticity helps nswer Wht hppens to ttendnce t your museum
More informationQuadratic Equations. Math 99 N1 Chapter 8
Qudrtic Equtions Mth 99 N1 Chpter 8 1 Introduction A qudrtic eqution is n eqution where the unknown ppers rised to the second power t most. In other words, it looks for the vlues of x such tht second degree
More informationUniform convergence and its consequences
Uniform convergence nd its consequences The following issue is centrl in mthemtics: On some domin D, we hve sequence of functions {f n }. This mens tht we relly hve n uncountble set of ordinry sequences,
More informationNet Change and Displacement
mth 11, pplictions motion: velocity nd net chnge 1 Net Chnge nd Displcement We hve seen tht the definite integrl f (x) dx mesures the net re under the curve y f (x) on the intervl [, b] Any prt of the
More informationThe area of the larger square is: IF it s a right triangle, THEN + =
8.1 Pythgoren Theorem nd 2D Applitions The Pythgoren Theorem sttes tht IF tringle is right tringle, THEN the sum of the squres of the lengths of the legs equls the squre of the hypotenuse lengths. Tht
More informationAssuming all values are initially zero, what are the values of A and B after executing this Verilog code inside an always block? C=1; A <= C; B = C;
B26 Appendix B The Bsics of Logic Design Check Yourself ALU n [Arthritic Logic Unit or (rre) Arithmetic Logic Unit] A rndomnumer genertor supplied s stndrd with ll computer systems Stn KellyBootle,
More informationLECTURE #05. Learning Objective. To describe the geometry in and around a unit cell in terms of directions and planes.
LECTURE #05 Chpter 3: Lttice Positions, Directions nd Plnes Lerning Objective To describe the geometr in nd round unit cell in terms of directions nd plnes. 1 Relevnt Reding for this Lecture... Pges 6483.
More information1.2 The Integers and Rational Numbers
.2. THE INTEGERS AND RATIONAL NUMBERS.2 The Integers n Rtionl Numers The elements of the set of integers: consist of three types of numers: Z {..., 5, 4, 3, 2,, 0,, 2, 3, 4, 5,...} I. The (positive) nturl
More informationSmall Business Networking
Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd processes. Introducing technology
More informationIntegration. 148 Chapter 7 Integration
48 Chpter 7 Integrtion 7 Integrtion t ech, by supposing tht during ech tenth of second the object is going t constnt speed Since the object initilly hs speed, we gin suppose it mintins this speed, but
More informationD e c i m a l s DECIMALS.
D e i m l s DECIMALS www.mthletis.om.u Deimls DECIMALS A deiml numer is sed on ple vlue. 214.84 hs 2 hundreds, 1 ten, 4 units, 8 tenths nd 4 hundredths. Sometimes different 'levels' of ple vlue re needed
More information9 CONTINUOUS DISTRIBUTIONS
9 CONTINUOUS DISTIBUTIONS A rndom vrible whose vlue my fll nywhere in rnge of vlues is continuous rndom vrible nd will be ssocited with some continuous distribution. Continuous distributions re to discrete
More informationSect 8.3 Triangles and Hexagons
13 Objective 1: Sect 8.3 Tringles nd Hexgons Understnding nd Clssifying Different Types of Polygons. A Polygon is closed twodimensionl geometric figure consisting of t lest three line segments for its
More informationFormal Languages and Automata Exam
Forml Lnguges nd Automt Exm Fculty of Computers & Informtion Deprtment: Computer Science Grde: Third Course code: CSC 34 Totl Mrk: 8 Dte: 23//2 Time: 3 hours Answer the following questions: ) Consider
More informationThe Chain Rule. rf dx. t t lim " (x) dt " (0) dx. df dt = df. dt dt. f (r) = rf v (1) df dx
The Chin Rule The Chin Rule In this section, we generlize the chin rule to functions of more thn one vrible. In prticulr, we will show tht the product in the singlevrible chin rule extends to n inner
More informationCONIC SECTIONS. Chapter 11
CONIC SECTIONS Chpter 11 11.1 Overview 11.1.1 Sections of cone Let l e fied verticl line nd m e nother line intersecting it t fied point V nd inclined to it t n ngle α (Fig. 11.1). Fig. 11.1 Suppose we
More informationRate and Activation Energy of the Iodination of Acetone
nd Activtion Energ of the Iodintion of Acetone rl N. eer Dte of Eperiment: //00 Florence F. Ls (prtner) Abstrct: The rte, rte lw nd ctivtion energ of the iodintion of cetone re detered b observing the
More informationSmall Business Networking
Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd business. Introducing technology
More information9.3. The Scalar Product. Introduction. Prerequisites. Learning Outcomes
The Sclr Product 9.3 Introduction There re two kinds of multipliction involving vectors. The first is known s the sclr product or dot product. This is soclled becuse when the sclr product of two vectors
More informationHomework 3 Solutions
CS 341: Foundtions of Computer Science II Prof. Mrvin Nkym Homework 3 Solutions 1. Give NFAs with the specified numer of sttes recognizing ech of the following lnguges. In ll cses, the lphet is Σ = {,1}.
More informationHomework 10. Problems: 19.29, 19.63, 20.9, 20.68
Homework 0 Prolems: 9.29, 9.63, 20.9, 20.68 Prolem 9.29 An utomoile tire is inlted with ir originlly t 0 º nd norml tmospheric pressure. During the process, the ir is compressed to 28% o its originl volume
More informationand thus, they are similar. If k = 3 then the Jordan form of both matrices is
Homework ssignment 11 Section 7. pp. 24925 Exercise 1. Let N 1 nd N 2 be nilpotent mtrices over the field F. Prove tht N 1 nd N 2 re similr if nd only if they hve the sme miniml polynomil. Solution: If
More informationTallahassee Community College. Simplifying Radicals
Tllhssee Communit College Simplifing Rdils The squre root of n positive numer is the numer tht n e squred to get the numer whose squre root we re seeking. For emple, 1 euse if we squre we get 1, whih is
More information4 Geometry: Shapes. 4.1 Circumference and area of a circle. FM Functional Maths AU (AO2) Assessing Understanding PS (AO3) Problem Solving HOMEWORK 4A
Geometry: Shpes. Circumference nd re of circle HOMEWORK D C 3 5 6 7 8 9 0 3 U Find the circumference of ech of the following circles, round off your nswers to dp. Dimeter 3 cm Rdius c Rdius 8 m d Dimeter
More informationTwo hours UNIVERSITY OF MANCHESTER SCHOOL OF COMPUTER SCIENCE. Date: Friday 16 th May 2008. Time: 14:00 16:00
COMP20212 Two hours UNIVERSITY OF MANCHESTER SCHOOL OF COMPUTER SCIENCE Digitl Design Techniques Dte: Fridy 16 th My 2008 Time: 14:00 16:00 Plese nswer ny THREE Questions from the FOUR questions provided
More informationThree squares with sides 3, 4, and 5 units are used to form the right triangle shown. In a right triangle, the sides have special names.
1 The Pythgoren Theorem MAIN IDEA Find length using the Pythgoren Theorem. New Voulry leg hypotenuse Pythgoren Theorem Mth Online glenoe.om Extr Exmples Personl Tutor SelfChek Quiz Three squres with
More informationLearning Outcomes. Computer Systems  Architecture Lecture 4  Boolean Logic. What is Logic? Boolean Logic 10/28/2010
/28/2 Lerning Outcomes At the end of this lecture you should: Computer Systems  Architecture Lecture 4  Boolen Logic Eddie Edwrds eedwrds@doc.ic.c.uk http://www.doc.ic.c.uk/~eedwrds/compsys (Hevily sed
More informationIn the following there are presented four different kinds of simulation games for a given Büchi automaton A = :
Simultion Gmes Motivtion There re t lest two distinct purposes for which it is useful to compute simultion reltionships etween the sttes of utomt. Firstly, with the use of simultion reltions it is possile
More informationRotating DC Motors Part II
Rotting Motors rt II II.1 Motor Equivlent Circuit The next step in our consiertion of motors is to evelop n equivlent circuit which cn be use to better unerstn motor opertion. The rmtures in rel motors
More informationSmall Business Networking
Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd business. Introducing technology
More informationLecture 3 Gaussian Probability Distribution
Lecture 3 Gussin Probbility Distribution Introduction l Gussin probbility distribution is perhps the most used distribution in ll of science. u lso clled bell shped curve or norml distribution l Unlike
More informationVersion 001 CIRCUITS holland (1290) 1
Version CRCUTS hollnd (9) This printout should hve questions Multiplechoice questions my continue on the next column or pge find ll choices efore nswering AP M 99 MC points The power dissipted in wire
More informationUse Geometry Expressions to create a more complex locus of points. Find evidence for equivalence using Geometry Expressions.
Lerning Objectives Loci nd Conics Lesson 3: The Ellipse Level: Preclculus Time required: 120 minutes In this lesson, students will generlize their knowledge of the circle to the ellipse. The prmetric nd
More information50 MATHCOUNTS LECTURES (10) RATIOS, RATES, AND PROPORTIONS
0 MATHCOUNTS LECTURES (0) RATIOS, RATES, AND PROPORTIONS BASIC KNOWLEDGE () RATIOS: Rtios re use to ompre two or more numers For n two numers n ( 0), the rtio is written s : = / Emple : If 4 stuents in
More informationRegular Sets and Expressions
Regulr Sets nd Expressions Finite utomt re importnt in science, mthemtics, nd engineering. Engineers like them ecuse they re super models for circuits (And, since the dvent of VLSI systems sometimes finite
More information