Operations with Polynomials


 Bethanie Tucker
 3 years ago
 Views:
Transcription
1 38 Chpter P Prerequisites P.4 Opertions with Polynomils Wht you should lern: How to identify the leding coefficients nd degrees of polynomils How to dd nd subtrct polynomils How to multiply polynomils How to find specil products How to use opertions with polynomils in ppliction problems Why you should lern it: Opertions with polynomils enble you to model vrious spects of the physicl world, such s the position of freeflling object, s shown in Eercises on pge 50. Bsic Definitions An lgebric epression contining only terms of the form k, where is ny rel number nd k is nonnegtive integer, is clled polynomil in one vrible or simply polynomil. Here re some emples of polynomils in one vrible. 3 8, , In the term k, is clled the coefficient, nd k the degree, of the term. Note tht the degree of the term is 1, nd the degree of constnt term is 0. Becuse polynomil is n lgebric sum, the coefficients tke on the signs between the terms. For instnce, hs coefficients 1, 4, 0, nd 3. Polynomils re usully written in order of descending powers of the vrible. This is referred to s stndrd form. For emple, the stndrd form of is Stndrd form The degree of polynomil is defined s the degree of the term with the highest power, nd the coefficient of this term is clled the leding coefficient of the polynomil. For instnce, the polynomil , is of fourth degree nd its leding coefficient is 3. nd 9 5 Definition of Polynomil in Let 1, 2, 3,..., n be rel numbers nd let n be nonnegtive integer. A polynomil in is n epression of the form n n n 1 n where n 0. The polynomil is of degree n, nd the number n is clled the leding coefficient. The number is clled the constnt term. 0 The following re not polynomils, for the resons stted. The epression is not polynomil becuse the eponent in The epression is not polynomil becuse the eponent in is negtive. is not n integer.
2 Section P.4 Opertions with Polynomils 39 Emple 1 Identifying Leding Coefficients nd Degrees Write the polynomil in stndrd form nd identify the degree nd leding coefficient of the polynomil. () (b) (c) Leding Polynomil Stndrd Form Degree Coefficient () (b) (c) A polynomil with only one term is monomil. Polynomils with two unlike terms re binomils, nd those with three unlike terms re trinomils. Here re some emples. Monomil: 5 3 Binomil: 4 3 Trinomil: The prefi mono mens one, the prefi bi mens two, nd the prefi tri mens three. Emple 2 Evluting Polynomil Find the vlue of when 4. When 4, the vlue of is Substitute 4 for. Evlute terms. Simplify. Adding nd Subtrcting Polynomils To dd two polynomils, simply combine like terms. This cn be done in either horizontl or verticl formt, s shown in Emples 3 nd 4. Emple 3 Adding Polynomils Horizontlly Use horizontl formt to dd nd Write originl polynomils Group like terms
3 40 Chpter P Prerequisites To use verticl formt to dd polynomils, lign the terms of the polynomils by their degrees, s shown in the following emple. Emple 4 Using Verticl Formt to Add Polynomils Use verticl formt to dd , , nd To subtrct one polynomil from nother, dd the opposite. You cn do this by chnging the sign of ech term of the polynomil tht is being subtrcted nd then dding the resulting like terms. Emple 5 Subtrcting Polynomils Horizontlly Use horizontl formt to subtrct from Write originl 4 polynomils Add the opposite Group like terms Study Tip The common error illustrted to the right is forgetting to chnge two of the signs in the polynomil tht is being subtrcted. When subtrcting polynomils, remember to dd the opposite of every term of the subtrcted polynomil. Be especilly creful to get the correct signs when you re subtrcting one polynomil from nother. One of the most common mistkes in lgebr is to forget to chnge signs correctly when subtrcting one epression from nother. Here is n emple. Wrong sign Wrong sign Emple 6 Using Verticl Formt to Subtrct Polynomils Use verticl formt to subtrct from Common error
4 Multiplying Polynomils Section P.4 Opertions with Polynomils 41 The simplest type of polynomil multipliction involves monomil multiplier. The product is obtined by direct ppliction of the. For instnce, to multiply the monomil 3 by the polynomil , multiply ech term of the polynomil by Emple 7 Finding Products with Monomil Multipliers Multiply the polynomil by the monomil. () (b) () Properties of eponents (b) Properties of eponents Outer First Inner Lst FOIL Digrm To multiply two binomils, you cn use both (left nd right) forms of the. For emple, if you tret the binomil 2 7 s single quntity, you cn multiply 3 2 by 2 7 s follows Product of First terms Product of Outer terms Product of Inner terms Product of Lst terms The four products in the boes bove suggest tht you cn put the product of two binomils in the FOIL form in just one step. This is clled the FOIL Method. Note tht the words first, outer, inner, nd lst refer to the positions of the terms in the originl product. Emple 8 Multiplying Binomils () Use the to multiply 2 by
5 42 Chpter P Prerequisites Emple 9 Multiplying Binomils (FOIL Method) Use the FOIL method to multiply the binomils. () 3 9 (b) F O I L () F O I L (b) To multiply two polynomils tht hve three or more terms, you cn use the sme bsic principle tht you use when multiplying monomils nd binomils. Tht is, ech term of one polynomil must be multiplied by ech term of the other polynomil. This cn be done using either horizontl or verticl formt. Emple 10 Multiplying Polynomils (Horizontl Formt) When multiplying two polynomils, it is best to write ech in stndrd form before using either the horizontl or verticl formt. This is illustrted in the net emple. Emple 11 Multiplying Polynomils (Verticl Formt) Write the polynomils in stndrd form nd use verticl formt to multiply With verticl formt, line up like terms in the sme verticl columns, much s you lign digits in wholenumber multipliction Stndrd form Stndrd form
6 Section P.4 Opertions with Polynomils 43 EXPLORATION Use the FOIL Method to find the product of where is constnt. Wht do you notice bout the number of terms in your product? Wht degree re the terms in your product? Polynomils re often written with eponents. As shown in the net emple, the properties of lgebr re used to simplify these epressions. Emple 12 Multiplying Polynomils Epnd Write ech fctor. Associtive Property of Multipliction Find 4 4. Emple 13 An Are Model for Multiplying Polynomils Show tht An pproprite re model to demonstrte the multipliction of two binomils would be A lw, the re formul for rectngle. Think of rectngle whose sides re 2 nd 2 1. The re of this rectngle is Are width length Another wy to find the re is to dd the res of the rectngulr prts, s shown in Figure P.11. There re two squres whose sides re, five rectngles whose sides re nd 1, nd two squres whose sides re 1. The totl re of these nine rectngles is Are sum of rectngulr res Becuse ech method must produce the sme re, you cn conclude tht Figure P Specil Products Some binomil products hve specil forms tht occur frequently in lgebr. For instnce, the product 3 3 is clled the product of the sum nd difference of two terms. With such products, the two middle terms cncel, s follows Sum nd difference of two terms Product hs no middle term.
7 44 Chpter P Prerequisites Another common type of product is the squre of binomil. With this type of product, the middle term is lwys twice the product of the terms in the binomil Squre of binomil Outer nd inner terms re equl. Middle term is twice the product of the terms in the binomil. Specil Products Let u nd v be rel numbers, vribles, or lgebric epressions. Then the following formuls re true. Sum nd Difference of Sme Terms Emple u v u v u 2 v Squre of Binomil Emple u v 2 u 2 2uv v u v 2 u 2 2uv v b b The squre of binomil cn lso be demonstrted geometriclly. Consider squre, ech of whose sides re of length b. (See Figure P.12). The totl re includes one squre of re 2, two rectngles of re b ech, nd one squre of re b 2. So, the totl re is 2 2b b 2. + b Emple 14 Finding Specil Products b b + b Figure P.12 b 2 Multiply the polynomils. () (b) (c) 2 b 2 () Specil product Simplify. (b) Specil product Simplify. (c) 2 b b b 2 Specil product b 4b b 2 Simplify.
8 Applictions Section P.4 Opertions with Polynomils 45 There re mny pplictions tht require the evlution of polynomils. One commonly used seconddegree polynomil is clled position polynomil. This polynomil hs the form 16t 2 v 0 t s 0 Position polynomil where t is the time, mesured in seconds. The vlue of this polynomil gives the height (in feet) of freeflling object bove the ground, ssuming no ir resistnce. The coefficient of t, v 0, is clled the initil velocity of the object, nd the constnt term, s 0, is clled the initil height of the object. If the initil velocity is positive, the object ws projected upwrd (t t 0), nd if the initil velocity is negtive, the object ws projected downwrd. t = 0 t = ft t = 2 t = 3 Figure P.13 Emple 15 Finding the Height of FreeFlling Object An object is thrown downwrd from the top of 200foot building. The initil velocity is 10 feet per second. Use the position polynomil 16t 2 10t 200 to find the height of the object when t 1, t 2, nd t 3 (see Figure P.13). When t 1, the height of the object is Height feet. When t 2, the height of the object is Height feet. When t 3, the height of the object is Height feet. In Emple 15, the initil velocity is 10 feet per second. The vlue is negtive becuse the object ws thrown downwrd. If it hd been thrown upwrd, the initil velocity would hve been positive. If it hd been dropped, the initil velocity would hve been zero. Use your clcultor to determine the height of the object in Emple 15 when t Wht cn you conclude?
9 46 Chpter P Prerequisites Emple 16 Using Polynomil Models The numbers of pounds of poultry P nd of beef B consumed per person in the United Sttes from 1985 to 1998 cn be modeled by P 0.084t t 30.5, 5 t 18 Poultry (pounds per person) B 0.143t t 90.4, 5 t 18 Beef (pounds per person) where t 5 represents Find model tht represents the totl mount T of poultry nd beef consumed from 1985 to Estimte the totl mount T consumed in (Source: U.S. Deprtment of Agriculture) The sum of the two polynomil models would be P B 0.084t t t t 90.4 The model for the totl consumption of poultry nd beef is T P B 0.059t t t t Totl (pounds per person) Using this model, nd substituting t 16, you cn estimte the 1996 consumption to be T pounds per person. Emple 17 Geometry: Finding the Are of Shded Region Find n epression for the re of the shded portion of the figure First find the re of the lrge rectngle A 1 nd the re of the smll rectngle A 2. A nd A Then to find the re A of the shded portion, subtrct A 2 from A 1. A A 1 A 2 Write formul Substitute. Use FOIL Method nd specil product formul..
10 Section P.4 Opertions with Polynomils 47 P.4 Eercises In Eercises 1 12, write the polynomil in stndrd form, nd find its degree nd leding coefficient y z 16z t 16t t 4t 5 t z 2 8z 4z In Eercises 13 18, determine whether the polynomil is monomil, binomil, or trinomil y t u 7 9u z 2 In Eercises 19 26, give n emple of polynomil in one vrible stisfying the conditions. (Note: There is more thn one correct nswer.) 19. A monomil of degree A trinomil of degree A trinomil of degree 4 nd leding coefficient A binomil of degree 2 nd leding coefficient A monomil of degree 1 nd leding coefficient A binomil of degree 5 nd leding coefficient A monomil of degree A monomil of degree 2 nd leding coefficient 9 In Eercises 27 30, find the vlues of the polynomil t the given vlues of the vrible () 2 (b) () 2 (b) () 1 (b) t 4 4t 3 () t 1 (b) t 2 3 In Eercises 31 34, perform the ddition using horizontl formt y 6 4y 2 6y 3 In Eercises 35 38, perform the ddition using verticl formt b 3 b 2 2b 7 b v 2 v 3 4v 1 2v 2 3v In Eercises 39 42, perform the subtrction using horizontl formt y 4 2 3y y 2 3y 4 y 4 y 2 In Eercises 43 46, perform the subtrction using verticl formt z 2 z z 3 2z 2 z t 4 5t 2 t 4 0.3t In Eercises 47 64, perform the opertions s 12s s 2 6s y 4 18y 18 11y 4 8y s 6s 30s y 2 2y y 2 y 3y 2 6y
11 48 Chpter P Prerequisites y 2 3y 9 3 4y y 2 2y t t t v 2 8 v 1 3 v z 2 z 11 3 z 2 4z 5 2 2z 2 5z t 4 2t 2 t 5t 4 9t 2 4t 3 8t 2 5t t 3 2t 2 t 8 3t 3 t 2 4t 2 4 2t 2 3t 1 t 3 1 In Eercises 65 68, use clcultor to perform the opertions y y y k k 14.61k k In Eercises 69 96, perform the multipliction nd simplify n 3n y 5 y 72. 5z 2z y 2 3y 2 7y y 2 y y 4 y 81. 2t 1 t z 5 2z b 5 1 3b y 3 2y 2 y 3 2y 5 2 3y 2 y 3 2b b 5 4y y 9 5t 3 4 2t t 3t t t 1 t 1 3 2t 5 5 8y 3 2y 1 y 7 In Eercises , perform the multipliction using horizontl formt t 3 t 2 5t u 5 2u 2 3u In Eercises , perform the multipliction using verticl formt s 2 5s 6 3s 4 In Eercises , perform the multipliction y 7 y c 6c n m 8n m t 9 2t z 1 5z t t b 4 0.1b b y y 7 4z t 2 5t 1 2t 2 5 2z 2 3z 7 3z y y t 2 2t 7 2t 2 8t y y z y u v 3 2 In Eercises , perform the opertions nd simplify k 8 k 8 k 2 k t 3 2 t
12 Section P.4 Opertions with Polynomils 49 Geometry In Eercises , find n epression for the perimeter or circumference of the figure y y y y 5 Geometry In Eercises , find n epression for the re of the shded portion of the figure y Geometric Modeling In Eercises , () perform the multipliction lgebriclly nd (b) use geometric re model to verify your solution to prt () y y t 3 t z 5 z 1 Geometric Modeling In Eercises 157 nd 158, use the re model to write two different epressions for the totl re. Then equte the two epressions nd nme the lgebric property tht is illustrted b b t 7t + 4 5t 2 6t 159. Geometry The length of rectngle is times its width w. Find epressions for () the perimeter nd (b) the re of the rectngle Geometry The bse of tringle is 3 nd its height is 5. Find n epression for the re A of the tringle Personl Finnce After 2 yers, n investment of $1000 compounded nnully t n interest rte of r will yield n mount r 2. Find this product.
13 50 Chpter P Prerequisites 162. Personl Finnce After 2 yers, n investment of $1000 compounded nnully t n interest rte of 9.5% will yield n mount Find this product. FreeFlling Object In Eercises , use the position polynomil to determine whether the freeflling object ws dropped, thrown upwrd, or thrown downwrd. Also determine the height of the object t time t t t 2 50t t 2 24t t 2 32t FreeFlling Object An object is thrown upwrd from the top of 200foot building (see figure). The initil velocity is 40 feet per second. Use the position polynomil 16t 2 40t 200 to find the height of the object when t 1, t 2, nd t 3. (b) During the given period, the per cpit consumption of whole milk ws decresing nd the per cpit consumption of lowft milk ws incresing (see figure). Ws the combined per cpit consumption of milk incresing or decresing? Milk (in gllons per cpit) y Yer (0 1990) Whole milk Lowft milk t Synthesis 200 ft 250 ft Figure for 167 Figure for FreeFlling Object An object is thrown downwrd from the top of 250foot building (see figure). The initil velocity is 25 feet per second. Use the position polynomil 16t 2 25t 250 to find the height of the object when t 1, t 2, nd t The per cpit consumption (verge consumption per person) of whole milk W nd lowft milk L in the United Sttes between 1990 nd 1998 cn be pproimted by these two polynomil models. W 0.024t t t 8 Whole milk L 0.016t t t 8 Lowft milk In these models, W nd L represent the verge consumption per person in gllons nd t represents the yer, with t 0 corresponding to (Source: U.S. Deprtment of Agriculture) () Find polynomil model tht represents the per cpit consumption of milk (of both types) during this time period. Use this model to find the per cpit consumption of milk in 1993 nd in Eplin why y 2 is not equl to 2 y Is every trinomil seconddegree polynomil? Eplin Cn two thirddegree polynomils be dded to produce seconddegree polynomil? If so, give n emple Perform the multiplictions. () 1 1 (b) (c) From the pttern formed by these products, cn you predict the result of ? 174. Eplin why 2 3 is not polynomil.
ARITHMETIC OPERATIONS. The real numbers have the following properties: a b c ab ac
REVIEW OF ALGEBRA Here we review the bsic rules nd procedures of lgebr tht you need to know in order to be successful in clculus. ARITHMETIC OPERATIONS The rel numbers hve the following properties: b b
More informationSummary Information and Formulae MTH109 College Algebra
Generl Formuls Summry Informtion nd Formule MTH109 College Algebr Temperture: F = 9 5 C + 32 nd C = 5 ( 9 F 32 ) F = degrees Fhrenheit C = degrees Celsius Simple Interest: I = Pr t I = Interest erned (chrged)
More informationALevel Mathematics Transition Task (compulsory for all maths students and all further maths student)
ALevel Mthemtics Trnsition Tsk (compulsory for ll mths students nd ll further mths student) Due: st Lesson of the yer. Length:  hours work (depending on prior knowledge) This trnsition tsk provides revision
More informationMATHEMATICS AND STATISTICS 1.2
MATHEMATICS AND STATISTICS. Apply lgebric procedures in solving problems Eternlly ssessed 4 credits Electronic technology, such s clcultors or computers, re not permitted in the ssessment of this stndr
More informationExponents and Polynomials
C H A P T E R 5 Eponents nd Polynomils ne sttistic tht cn be used to mesure the generl helth of ntion or group within ntion is life epectncy. This dt is considered more ccurte thn mny other sttistics becuse
More information5.2 Exponent Properties Involving Quotients
5. Eponent Properties Involving Quotients Lerning Objectives Use the quotient of powers property. Use the power of quotient property. Simplify epressions involving quotient properties of eponents. Use
More informationfractions Let s Learn to
5 simple lgebric frctions corne lens pupil retin Norml vision light focused on the retin concve lens Shortsightedness (myopi) light focused in front of the retin Corrected myopi light focused on the retin
More informationBefore we can begin Ch. 3 on Radicals, we need to be familiar with perfect squares, cubes, etc. Try and do as many as you can without a calculator!!!
Nme: Algebr II Honors PreChpter Homework Before we cn begin Ch on Rdicls, we need to be fmilir with perfect squres, cubes, etc Try nd do s mny s you cn without clcultor!!! n The nth root of n n Be ble
More informationSUMMER KNOWHOW STUDY AND LEARNING CENTRE
SUMMER KNOWHOW STUDY AND LEARNING CENTRE Indices & Logrithms 2 Contents Indices.2 Frctionl Indices.4 Logrithms 6 Exponentil equtions. Simplifying Surds 13 Opertions on Surds..16 Scientific Nottion..18
More informationAdding and Subtracting Rational Expressions
6.4 Adding nd Subtrcting Rtionl Epressions Essentil Question How cn you determine the domin of the sum or difference of two rtionl epressions? You cn dd nd subtrct rtionl epressions in much the sme wy
More informationMathcad Lecture #1 Inclass Worksheet Mathcad Basics
Mthcd Lecture #1 Inclss Worksheet Mthcd Bsics At the end of this lecture, you will be ble to: Evlute mthemticl epression numericlly Assign vrible nd use them in subsequent clcultions Distinguish between
More informationAdvanced Algebra & Trigonometry Midterm Review Packet
Nme Dte Advnced Alger & Trigonometry Midterm Review Pcket The Advnced Alger & Trigonometry midterm em will test your generl knowledge of the mteril we hve covered since the eginning of the school yer.
More informationUnit 1 Exponentials and Logarithms
HARTFIELD PRECALCULUS UNIT 1 NOTES PAGE 1 Unit 1 Eponentils nd Logrithms (2) Eponentil Functions (3) The number e (4) Logrithms (5) Specil Logrithms (7) Chnge of Bse Formul (8) Logrithmic Functions (10)
More informationapproaches as n becomes larger and larger. Since e > 1, the graph of the natural exponential function is as below
. Eponentil nd rithmic functions.1 Eponentil Functions A function of the form f() =, > 0, 1 is clled n eponentil function. Its domin is the set of ll rel f ( 1) numbers. For n eponentil function f we hve.
More informationMatrices and Determinants
Nme Chpter 8 Mtrices nd Determinnts Section 8.1 Mtrices nd Systems of Equtions Objective: In this lesson you lerned how to use mtrices, Gussin elimintion, nd GussJordn elimintion to solve systems of liner
More informationMATHS NOTES. SUBJECT: Maths LEVEL: Higher TEACHER: Aidan Roantree. The Institute of Education Topics Covered: Powers and Logs
MATHS NOTES The Institute of Eduction 06 SUBJECT: Mths LEVEL: Higher TEACHER: Aidn Rontree Topics Covered: Powers nd Logs About Aidn: Aidn is our senior Mths techer t the Institute, where he hs been teching
More information3.1 Exponential Functions and Their Graphs
. Eponentil Functions nd Their Grphs Sllbus Objective: 9. The student will sketch the grph of eponentil, logistic, or logrithmic function. 9. The student will evlute eponentil or logrithmic epressions.
More informationQuotient Rule: am a n = am n (a 0) Negative Exponents: a n = 1 (a 0) an Power Rules: (a m ) n = a m n (ab) m = a m b m
Formuls nd Concepts MAT 099: Intermedite Algebr repring for Tests: The formuls nd concepts here m not be inclusive. You should first tke our prctice test with no notes or help to see wht mteril ou re comfortble
More informationPreSession Review. Part 1: Basic Algebra; Linear Functions and Graphs
PreSession Review Prt 1: Bsic Algebr; Liner Functions nd Grphs A. Generl Review nd Introduction to Algebr Hierrchy of Arithmetic Opertions Opertions in ny expression re performed in the following order:
More informationTHE DISCRIMINANT & ITS APPLICATIONS
THE DISCRIMINANT & ITS APPLICATIONS The discriminnt ( Δ ) is the epression tht is locted under the squre root sign in the qudrtic formul i.e. Δ b c. For emple: Given +, Δ () ( )() The discriminnt is used
More informationLesson 1: Quadratic Equations
Lesson 1: Qudrtic Equtions Qudrtic Eqution: The qudrtic eqution in form is. In this section, we will review 4 methods of qudrtic equtions, nd when it is most to use ech method. 1. 3.. 4. Method 1: Fctoring
More informationFaith Scholarship Service Friendship
Immcult Mthemtics Summer Assignment The purpose of summer ssignment is to help you keep previously lerned fcts fresh in your mind for use in your net course. Ecessive time spent reviewing t the beginning
More informationEquations and Inequalities
Equtions nd Inequlities Equtions nd Inequlities Curriculum Redy ACMNA: 4, 5, 6, 7, 40 www.mthletics.com Equtions EQUATIONS & Inequlities & INEQUALITIES Sometimes just writing vribles or pronumerls in
More informationand that at t = 0 the object is at position 5. Find the position of the object at t = 2.
7.2 The Fundmentl Theorem of Clculus 49 re mny, mny problems tht pper much different on the surfce but tht turn out to be the sme s these problems, in the sense tht when we try to pproimte solutions we
More informationProperties of Integrals, Indefinite Integrals. Goals: Definition of the Definite Integral Integral Calculations using Antiderivatives
Block #6: Properties of Integrls, Indefinite Integrls Gols: Definition of the Definite Integrl Integrl Clcultions using Antiderivtives Properties of Integrls The Indefinite Integrl 1 Riemnn Sums  1 Riemnn
More informationChapters Five Notes SN AA U1C5
Chpters Five Notes SN AA U1C5 Nme Period Section 5: Fctoring Qudrtic Epressions When you took lger, you lerned tht the first thing involved in fctoring is to mke sure to fctor out ny numers or vriles
More informationReview Factoring Polynomials:
Chpter 4 Mth 0 Review Fctoring Polynomils:. GCF e. A) 5 5 A) 4 + 9. Difference of Squres b = ( + b)( b) e. A) 9 6 B) C) 98y. Trinomils e. A) + 5 4 B) + C) + 5 + Solving Polynomils:. A) ( 5)( ) = 0 B) 4
More informationSample pages. 9:04 Equations with grouping symbols
Equtions 9 Contents I know the nswer is here somewhere! 9:01 Inverse opertions 9:0 Solving equtions Fun spot 9:0 Why did the tooth get dressed up? 9:0 Equtions with pronumerls on both sides GeoGebr ctivity
More informationSTRAND B: NUMBER THEORY
Mthemtics SKE, Strnd B UNIT B Indices nd Fctors: Tet STRAND B: NUMBER THEORY B Indices nd Fctors Tet Contents Section B. Squres, Cubes, Squre Roots nd Cube Roots B. Inde Nottion B. Fctors B. Prime Fctors,
More informationReview of basic calculus
Review of bsic clculus This brief review reclls some of the most importnt concepts, definitions, nd theorems from bsic clculus. It is not intended to tech bsic clculus from scrtch. If ny of the items below
More informationChapter 3 Exponential and Logarithmic Functions Section 3.1
Chpter 3 Eponentil nd Logrithmic Functions Section 3. EXPONENTIAL FUNCTIONS AND THEIR GRAPHS Eponentil Functions Eponentil functions re nonlgebric functions. The re clled trnscendentl functions. The eponentil
More information0.1 THE REAL NUMBER LINE AND ORDER
6000_000.qd //0 :6 AM Pge 00 CHAPTER 0 A Preclculus Review 0. THE REAL NUMBER LINE AND ORDER Represent, clssify, nd order rel numers. Use inequlities to represent sets of rel numers. Solve inequlities.
More informationSections 1.3, 7.1, and 9.2: Properties of Exponents and Radical Notation
Sections., 7., nd 9.: Properties of Eponents nd Rdicl Nottion Let p nd q be rtionl numbers. For ll rel numbers nd b for which the epressions re rel numbers, the following properties hold. i = + p q p q
More informationThe First Fundamental Theorem of Calculus. If f(x) is continuous on [a, b] and F (x) is any antiderivative. f(x) dx = F (b) F (a).
The Fundmentl Theorems of Clculus Mth 4, Section 0, Spring 009 We now know enough bout definite integrls to give precise formultions of the Fundmentl Theorems of Clculus. We will lso look t some bsic emples
More informationLogarithms. Logarithm is another word for an index or power. POWER. 2 is the power to which the base 10 must be raised to give 100.
Logrithms. Logrithm is nother word for n inde or power. THIS IS A POWER STATEMENT BASE POWER FOR EXAMPLE : We lred know tht; = NUMBER 10² = 100 This is the POWER Sttement OR 2 is the power to which the
More informationExponents and Logarithms Exam Questions
Eponents nd Logrithms Em Questions Nme: ANSWERS Multiple Choice 1. If 4, then is equl to:. 5 b. 8 c. 16 d.. Identify the vlue of the intercept of the function ln y.. 1 b. 0 c. d.. Which eqution is represented
More informationThe Fundamental Theorem of Calculus. The Total Change Theorem and the Area Under a Curve.
Clculus Li Vs The Fundmentl Theorem of Clculus. The Totl Chnge Theorem nd the Are Under Curve. Recll the following fct from Clculus course. If continuous function f(x) represents the rte of chnge of F
More informationConsolidation Worksheet
Cmbridge Essentils Mthemtics Core 8 NConsolidtion Worksheet N Consolidtion Worksheet Work these out. 8 b 7 + 0 c 6 + 7 5 Use the number line to help. 2 Remember + 2 2 +2 2 2 + 2 Adding negtive number is
More informationSESSION 2 Exponential and Logarithmic Functions. Math 301 R 3. (Revisit, Review and Revive)
Mth 01 R (Revisit, Review nd Revive) SESSION Eponentil nd Logrithmic Functions 1 Eponentil nd Logrithmic Functions Key Concepts The Eponent Lws m n 1 n n m m n m n m mn m m m m mn m m m b n b b b Simplify
More informationAdd and Subtract Rational Expressions. You multiplied and divided rational expressions. You will add and subtract rational expressions.
TEKS 8. A..A, A.0.F Add nd Subtrct Rtionl Epressions Before Now You multiplied nd divided rtionl epressions. You will dd nd subtrct rtionl epressions. Why? So you cn determine monthly cr lon pyments, s
More information3 x x x 1 3 x a a a 2 7 a Ba 1 NOW TRY EXERCISES 89 AND a 2/ Evaluate each expression.
SECTION. Eponents nd Rdicls 7 B 7 7 7 7 7 7 7 NOW TRY EXERCISES 89 AND 9 7. EXERCISES CONCEPTS. () Using eponentil nottion, we cn write the product s. In the epression 3 4,the numer 3 is clled the, nd
More informationSection 6.3 The Fundamental Theorem, Part I
Section 6.3 The Funmentl Theorem, Prt I (3//8) Overview: The Funmentl Theorem of Clculus shows tht ifferentition n integrtion re, in sense, inverse opertions. It is presente in two prts. We previewe Prt
More informationINTRODUCTION TO LINEAR ALGEBRA
ME Applied Mthemtics for Mechnicl Engineers INTRODUCTION TO INEAR AGEBRA Mtrices nd Vectors Prof. Dr. Bülent E. Pltin Spring Sections & / ME Applied Mthemtics for Mechnicl Engineers INTRODUCTION TO INEAR
More informationCalculus 2: Integration. Differentiation. Integration
Clculus 2: Integrtion The reverse process to differentition is known s integrtion. Differentition f() f () Integrtion As it is the opposite of finding the derivtive, the function obtined b integrtion is
More informationChapter 1: Logarithmic functions and indices
Chpter : Logrithmic functions nd indices. You cn simplify epressions y using rules of indices m n m n m n m n ( m ) n mn m m m m n m m n Emple Simplify these epressions: 5 r r c 4 4 d 6 5 e ( ) f ( ) 4
More informationMATH 144: Business Calculus Final Review
MATH 144: Business Clculus Finl Review 1 Skills 1. Clculte severl limits. 2. Find verticl nd horizontl symptotes for given rtionl function. 3. Clculte derivtive by definition. 4. Clculte severl derivtives
More informationAQA Further Pure 1. Complex Numbers. Section 1: Introduction to Complex Numbers. The number system
Complex Numbers Section 1: Introduction to Complex Numbers Notes nd Exmples These notes contin subsections on The number system Adding nd subtrcting complex numbers Multiplying complex numbers Complex
More informationEXPONENT. Section 2.1. Do you see a pattern? Do you see a pattern? Try a) ( ) b) ( ) c) ( ) d)
Section. EXPONENT RULES Do ou see pttern? Do ou see pttern? Tr ) ( ) b) ( ) c) ( ) d) Eponent rules strt here:. Epnd the following s bove. ) b) 7 c) d) How n 's re ou ultipling in ech proble? ) b) c) d)
More informationMath 8 Winter 2015 Applications of Integration
Mth 8 Winter 205 Applictions of Integrtion Here re few importnt pplictions of integrtion. The pplictions you my see on n exm in this course include only the Net Chnge Theorem (which is relly just the Fundmentl
More informationSection 3.2: Negative Exponents
Section 3.2: Negtive Exponents Objective: Simplify expressions with negtive exponents using the properties of exponents. There re few specil exponent properties tht del with exponents tht re not positive.
More informationIdentify graphs of linear inequalities on a number line.
COMPETENCY 1.0 KNOWLEDGE OF ALGEBRA SKILL 1.1 Identify grphs of liner inequlities on number line.  When grphing firstdegree eqution, solve for the vrible. The grph of this solution will be single point
More informationThe graphs of Rational Functions
Lecture 4 5A: The its of Rtionl Functions s x nd s x + The grphs of Rtionl Functions The grphs of rtionl functions hve severl differences compred to power functions. One of the differences is the behvior
More informationPrecalculus Spring 2017
Preclculus Spring 2017 Exm 3 Summry (Section 4.1 through 5.2, nd 9.4) Section P.5 Find domins of lgebric expressions Simplify rtionl expressions Add, subtrct, multiply, & divide rtionl expressions Simplify
More information13.3. The Area Bounded by a Curve. Introduction. Prerequisites. Learning Outcomes
The Are Bounded b Curve 3.3 Introduction One of the importnt pplictions of integrtion is to find the re bounded b curve. Often such n re cn hve phsicl significnce like the work done b motor, or the distnce
More information4.4 Areas, Integrals and Antiderivatives
. res, integrls nd ntiderivtives 333. Ares, Integrls nd Antiderivtives This section explores properties of functions defined s res nd exmines some connections mong res, integrls nd ntiderivtives. In order
More informationSCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics. Basic Algebra
SCHOOL OF ENGINEERING & BUILT ENVIRONMENT Mthemtics Bsic Algebr Opertions nd Epressions Common Mistkes Division of Algebric Epressions Eponentil Functions nd Logrithms Opertions nd their Inverses Mnipulting
More informationThe Algebra (aljabr) of Matrices
Section : Mtri lgebr nd Clculus Wshkewicz College of Engineering he lgebr (ljbr) of Mtrices lgebr s brnch of mthemtics is much broder thn elementry lgebr ll of us studied in our high school dys. In sense
More informationElementary Mathematical Concepts and Operations
Elementry Mthemticl Concepts nd Opertions After studying this chpter you should be ble to: dd, subtrct, multiply nd divide positive nd negtive numbers understnd the concept of squre root expnd nd evlute
More informationInClass Problems 2 and 3: Projectile Motion Solutions. InClass Problem 2: Throwing a Stone Down a Hill
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Deprtment of Physics Physics 8T Fll Term 4 InClss Problems nd 3: Projectile Motion Solutions We would like ech group to pply the problem solving strtegy with the
More informationTheoretical foundations of Gaussian quadrature
Theoreticl foundtions of Gussin qudrture 1 Inner product vector spce Definition 1. A vector spce (or liner spce) is set V = {u, v, w,...} in which the following two opertions re defined: (A) Addition of
More informationMathematics Extension 1
04 Bored of Studies Tril Emintions Mthemtics Etension Written by Crrotsticks & Trebl. Generl Instructions Totl Mrks 70 Reding time 5 minutes. Working time hours. Write using blck or blue pen. Blck pen
More informationCHAPTER 9. Rational Numbers, Real Numbers, and Algebra
CHAPTER 9 Rtionl Numbers, Rel Numbers, nd Algebr Problem. A mn s boyhood lsted 1 6 of his life, he then plyed soccer for 1 12 of his life, nd he mrried fter 1 8 more of his life. A dughter ws born 9 yers
More informationLesson 2.4 Exercises, pages
Lesson. Exercises, pges A. Expnd nd simplify. ) + b) ( ) () 0  ( ) () 0 c) 7 + d) (7) ( ) 7  + 8 () ( 8). Expnd nd simplify. ) b)  7  + 7 7( ) ( ) ( ) 7( 7) 8 (7) P DO NOT COPY.. Multiplying nd Dividing
More informationREVIEW Chapter 1 The Real Number System
Mth 7 REVIEW Chpter The Rel Number System In clss work: Solve ll exercises. (Sections. &. Definition A set is collection of objects (elements. The Set of Nturl Numbers N N = {,,,, 5, } The Set of Whole
More informationExponentials & Logarithms Unit 8
U n i t 8 AdvF Dte: Nme: Eponentils & Logrithms Unit 8 Tenttive TEST dte Big ide/lerning Gols This unit begins with the review of eponent lws, solving eponentil equtions (by mtching bses method nd tril
More informationHIGHER SCHOOL CERTIFICATE EXAMINATION MATHEMATICS 3 UNIT (ADDITIONAL) AND 3/4 UNIT (COMMON) Time allowed Two hours (Plus 5 minutes reading time)
HIGHER SCHOOL CERTIFICATE EXAMINATION 998 MATHEMATICS 3 UNIT (ADDITIONAL) AND 3/4 UNIT (COMMON) Time llowed Two hours (Plus 5 minutes reding time) DIRECTIONS TO CANDIDATES Attempt ALL questions ALL questions
More information12.1 Introduction to Rational Expressions
. Introduction to Rtionl Epressions A rtionl epression is rtio of polynomils; tht is, frction tht hs polynomil s numertor nd/or denomintor. Smple rtionl epressions: 0 EVALUATING RATIONAL EXPRESSIONS To
More information38 Riemann sums and existence of the definite integral.
38 Riemnn sums nd existence of the definite integrl. In the clcultion of the re of the region X bounded by the grph of g(x) = x 2, the xxis nd 0 x b, two sums ppered: ( n (k 1) 2) b 3 n 3 re(x) ( n These
More informationDETERMINANTS. All Mathematical truths are relative and conditional. C.P. STEINMETZ
All Mthemticl truths re reltive nd conditionl. C.P. STEINMETZ 4. Introduction DETERMINANTS In the previous chpter, we hve studied bout mtrices nd lgebr of mtrices. We hve lso lernt tht system of lgebric
More information7.1 Integral as Net Change and 7.2 Areas in the Plane Calculus
7.1 Integrl s Net Chnge nd 7. Ares in the Plne Clculus 7.1 INTEGRAL AS NET CHANGE Notecrds from 7.1: Displcement vs Totl Distnce, Integrl s Net Chnge We hve lredy seen how the position of n oject cn e
More informationGoals: Determine how to calculate the area described by a function. Define the definite integral. Explore the relationship between the definite
Unit #8 : The Integrl Gols: Determine how to clculte the re described by function. Define the definite integrl. Eplore the reltionship between the definite integrl nd re. Eplore wys to estimte the definite
More information13.3. The Area Bounded by a Curve. Introduction. Prerequisites. Learning Outcomes
The Are Bounded b Curve 3.3 Introduction One of the importnt pplictions of integrtion is to find the re bounded b curve. Often such n re cn hve phsicl significnce like the work done b motor, or the distnce
More informationReview of Calculus, cont d
Jim Lmbers MAT 460 Fll Semester 200910 Lecture 3 Notes These notes correspond to Section 1.1 in the text. Review of Clculus, cont d Riemnn Sums nd the Definite Integrl There re mny cses in which some
More informationBridging the gap: GCSE AS Level
Bridging the gp: GCSE AS Level CONTENTS Chpter Removing rckets pge Chpter Liner equtions Chpter Simultneous equtions 8 Chpter Fctors 0 Chpter Chnge the suject of the formul Chpter 6 Solving qudrtic equtions
More informationObj: SWBAT Recall the many important types and properties of functions
Obj: SWBAT Recll the mny importnt types nd properties of functions Functions Domin nd Rnge Function Nottion Trnsformtion of Functions Combintions/Composition of Functions OnetoOne nd Inverse Functions
More informationPHYS Summer Professor Caillault Homework Solutions. Chapter 2
PHYS 1111  Summer 2007  Professor Cillult Homework Solutions Chpter 2 5. Picture the Problem: The runner moves long the ovl trck. Strtegy: The distnce is the totl length of trvel, nd the displcement
More informationMath 1431 Section M TH 4:00 PM 6:00 PM Susan Wheeler Office Hours: Wed 6:00 7:00 PM Online ***NOTE LABS ARE MON AND WED
Mth 43 Section 4839 M TH 4: PM 6: PM Susn Wheeler swheeler@mth.uh.edu Office Hours: Wed 6: 7: PM Online ***NOTE LABS ARE MON AND WED t :3 PM to 3: pm ONLINE Approimting the re under curve given the type
More informationPART 1 MULTIPLE CHOICE Circle the appropriate response to each of the questions below. Each question has a value of 1 point.
PART MULTIPLE CHOICE Circle the pproprite response to ech of the questions below. Ech question hs vlue of point.. If in sequence the second level difference is constnt, thn the sequence is:. rithmetic
More informationa < a+ x < a+2 x < < a+n x = b, n A i n f(x i ) x. i=1 i=1
Mth 33 Volume Stewrt 5.2 Geometry of integrls. In this section, we will lern how to compute volumes using integrls defined by slice nlysis. First, we recll from Clculus I how to compute res. Given the
More informationMA Lesson 21 Notes
MA 000 Lesson 1 Notes ( 5) How would person solve n eqution with vrible in n eponent, such s 9? (We cnnot rewrite this eqution esil with the sme bse.) A nottion ws developed so tht equtions such s this
More information50. Use symmetry to evaluate xx D is the region bounded by the square with vertices 5, Prove Property 11. y y CAS
68 CHAPTE MULTIPLE INTEGALS 46. e da, 49. Evlute tn 3 4 da, where,. [Hint: Eploit the fct tht is the disk with center the origin nd rdius is smmetric with respect to both es.] 5. Use smmetr to evlute 3
More informationThe Trapezoidal Rule
_.qd // : PM Pge 9 SECTION. Numericl Integrtion 9 f Section. The re of the region cn e pproimted using four trpezoids. Figure. = f( ) f( ) n The re of the first trpezoid is f f n. Figure. = Numericl Integrtion
More informationMath Calculus with Analytic Geometry II
orem of definite Mth 5.0 with Anlytic Geometry II Jnury 4, 0 orem of definite If < b then b f (x) dx = ( under f bove xxis) ( bove f under xxis) Exmple 8 0 3 9 x dx = π 3 4 = 9π 4 orem of definite Problem
More information4.1 OnetoOne Functions; Inverse Functions. EX) Find the inverse of the following functions. State if the inverse also forms a function or not.
4.1 OnetoOne Functions; Inverse Functions Finding Inverses of Functions To find the inverse of function simply switch nd y vlues. Input becomes Output nd Output becomes Input. EX) Find the inverse of
More informationI do slope intercept form With my shades on MartinGay, Developmental Mathematics
AATA Dte: 1//1 SWBAT simplify rdicls. Do Now: ACT Prep HW Requests: Pg 49 #1745 odds Continue Vocb sheet In Clss: Complete Skills Prctice WS HW: Complete Worksheets For Wednesdy stmped pges Bring stmped
More informationRiemann Sums and Riemann Integrals
Riemnn Sums nd Riemnn Integrls Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University August 26, 2013 Outline 1 Riemnn Sums 2 Riemnn Integrls 3 Properties
More informationScientific notation is a way of expressing really big numbers or really small numbers.
Scientific Nottion (Stndrd form) Scientific nottion is wy of expressing relly big numbers or relly smll numbers. It is most often used in scientific clcultions where the nlysis must be very precise. Scientific
More informationHow do we solve these things, especially when they get complicated? How do we know when a system has a solution, and when is it unique?
XII. LINEAR ALGEBRA: SOLVING SYSTEMS OF EQUATIONS Tody we re going to tlk bout solving systems of liner equtions. These re problems tht give couple of equtions with couple of unknowns, like: 6 2 3 7 4
More informationPROPERTIES OF AREAS In general, and for an irregular shape, the definition of the centroid at position ( x, y) is given by
PROPERTES OF RES Centroid The concept of the centroid is prol lred fmilir to ou For plne shpe with n ovious geometric centre, (rectngle, circle) the centroid is t the centre f n re hs n is of smmetr, the
More informationGrade 10 Math Academic Levels (MPM2D) Unit 4 Quadratic Relations
Grde 10 Mth Acdemic Levels (MPMD) Unit Qudrtic Reltions Topics Homework Tet ook Worksheet D 1 Qudrtic Reltions in Verte Qudrtic Reltions in Verte Form (Trnsltions) Form (Trnsltions) D Qudrtic Reltions
More informationOptimization Lecture 1 Review of Differential Calculus for Functions of Single Variable.
Optimiztion Lecture 1 Review of Differentil Clculus for Functions of Single Vrible http://users.encs.concordi.c/~luisrod, Jnury 14 Outline Optimiztion Problems Rel Numbers nd Rel Vectors Open, Closed nd
More information13.4 Work done by Constant Forces
13.4 Work done by Constnt Forces We will begin our discussion of the concept of work by nlyzing the motion of n object in one dimension cted on by constnt forces. Let s consider the following exmple: push
More informationMath 1B, lecture 4: Error bounds for numerical methods
Mth B, lecture 4: Error bounds for numericl methods Nthn Pflueger 4 September 0 Introduction The five numericl methods descried in the previous lecture ll operte by the sme principle: they pproximte the
More informationPractice Final. Name: Problem 1. Show all of your work, label your answers clearly, and do not use a calculator.
Nme: MATH 2250 Clculus Eric Perkerson Dte: December 11, 2015 Prctice Finl Show ll of your work, lbel your nswers clerly, nd do not use clcultor. Problem 1 Compute the following limits, showing pproprite
More informationProblem Set 9. Figure 1: Diagram. This picture is a rough sketch of the 4 parabolas that give us the area that we need to find. The equations are:
(x + y ) = y + (x + y ) = x + Problem Set 9 Discussion: Nov., Nov. 8, Nov. (on probbility nd binomil coefficients) The nme fter the problem is the designted writer of the solution of tht problem. (No one
More informationRiemann Sums and Riemann Integrals
Riemnn Sums nd Riemnn Integrls Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University August 26, 203 Outline Riemnn Sums Riemnn Integrls Properties Abstrct
More informationThe Fundamental Theorem of Calculus
Chpter The Fundmentl Theorem of Clculus In this chpter we will formulte one of the most importnt results of clculus, the Fundmentl Theorem. This result will link together the notions of n integrl nd derivtive.
More informationChapter 9 Definite Integrals
Chpter 9 Definite Integrls In the previous chpter we found how to tke n ntiderivtive nd investigted the indefinite integrl. In this chpter the connection etween ntiderivtives nd definite integrls is estlished
More informationCalculus Module C21. Areas by Integration. Copyright This publication The Northern Alberta Institute of Technology All Rights Reserved.
Clculus Module C Ares Integrtion Copright This puliction The Northern Alert Institute of Technolog 7. All Rights Reserved. LAST REVISED Mrch, 9 Introduction to Ares Integrtion Sttement of Prerequisite
More informationTO: Next Year s AP Calculus Students
TO: Net Yer s AP Clculus Students As you probbly know, the students who tke AP Clculus AB nd pss the Advnced Plcement Test will plce out of one semester of college Clculus; those who tke AP Clculus BC
More information