7.6 Complex Fractions


 Alban Baldwin
 2 years ago
 Views:
Transcription
1 Section 7.6 Comple Fractions Comple Fractions In tis section we learn ow to simplify wat are called comple fractions, an eample of wic follows Note tat bot te numerator and denominator are fraction problems in teir own rigt, lending credence to wy we refer to suc a structure as a comple fraction. Tere are two very different tecniques we can use to simplify te comple fraction (. Te first tecnique is a natural coice. ( Simplifying Comple Fractions First Tecnique. To simplify a comple fraction, proceed as follows:. Simplify te numerator. 2. Simplify te denominator. 3. Simplify te division problem tat remains. Let s follow tis outline to simplify te comple fraction (. First, add te fractions in te numerator as follows Secondly, add te fractions in te denominator as follows Substitute te results from (2 and (3 into te numerator and denominator of (, respectively Te rigtand side of (4 is equivalent to Tis is a division problem, so invert and multiply, factor, ten cancel common factors (2 (3 (4 Copyrigted material. See: ttp://msenu.redwoods.edu/intalgtet/
2 696 Capter 7 Rational Functions Here is an arrangement of te work, from start to finis, presented witout comment. Tis is a good template to emulate wen doing your omework Now, let s look at a second approac to te problem. We saw tat simplifying te numerator in (2 required a common denominator of 6. Simplifying te denominator in (3 required a common denominator of 2. So, let s coose anoter common denominator, tis one a common denominator for bot numerator and denominator, namely, 2. Now, multiply top and bottom (numerator and denominator of te comple fraction ( by 2, as follows ( ( 2 2 Distribute te 2 in bot numerator and denominator and simplify. (5
3 Section 7.6 Comple Fractions 697 ( ( 2 2 ( 2 ( ( 3 ( Let s summarize tis second tecnique. Simplifying Comple Fractions Second Tecnique. To simplify a comple fraction, proceed as follows:. Find a common denominator for bot numerator and denominator. 2. Clear fractions from te numerator and denomaintor by multiplying eac by te common denominator found in te first step. Note tat for tis particular problem, te second metod is muc more efficient. It saves bot space and time and is more aestetically pleasing. It is te tecnique tat we will favor in te rest of tis section. Let s look at anoter eample. Eample 6. Use bot te First and Second Tecniques to simplify te epression 2. (7 State all restrictions. Let s use te first tecnique, simplifying numerator and denominator separately before dividing. First, make equivalent fractions wit a common denominator for te subtraction problem in te numerator of (7 and simplify. Do te same for te denominator. 2 Net, invert and multiply, ten factor ( + ( Let s invoke te sign cange rule and negate two parts of te fraction ( /, numerator and fraction bar, ten cancel te common factors.
4 698 Capter 7 Rational Functions 2 2 ( + ( ( + ( Hence, +. 2 Now, let s try te problem a second time, multiplying numerator and denominator by 2 to clear fractions from bot te numerator and denominator. ( ( 2 2 ( 2 2 ( 2 ( ( Te order in te numerator of te last fraction intimates tat a sign cange would be elpful. Negate te numerator and fraction bar, factor, ten cancel common factors ( ( ( + ( ( + ( + Tis is precisely te same answer found wit te first tecnique. To list te restrictions, we must make sure tat no values of make any denominator equal to zero, at te beginning of te problem, in te body of our work, or in te final answer. In te original problem, if 0, ten bot / and / 2 are undefined, so 0 is a restriction. In te body of our work, te factors + and found in various denominators make and restrictions. No oter denominators supply restrictions tat ave not already been listed. Hence, for all oter tan, 0, and, te leftand side of + 2 is identical to te rigtand side. Again, te calculator s table utility provides ample evidence of tis fact in te screensots sown in Figure. Note te ERR (error messages at eac of te restricted values of, but also note te perfect agreement of Y and Y2 at all oter values of. (8 Let s look at anoter eample, an important eample involving function notation.
5 Section 7.6 Comple Fractions 699 (a (b (c Figure. Using te table feature of te graping calculator to ceck te identity in (8. Eample 9. Given tat f(, simplify te epression List all restrictions. f( f(2. 2 Remember, f(2 means substitute 2 for. f(2 /2, so f( f(2 2 Because f( /, we know tat 2 2. To clear te fractions from te numerator, we d use a common denominator of 2. Tere are no fractions in te denominator tat need clearing, so te common denominator for numerator and denominator is 2. Multiply numerator and denominator by 2. ( ( ( f( f(2 2 ( 22 ( 22 Negate te numerator and fraction bar, ten cancel common factors. f( f( ( 2 2 2( ( 2 In te original problem, we ave a denominator of 2, so 2 is a restriction. If te body of our work, tere is a fraction /, wic is undefined wen 0, so 0 is also a restriction. Te remaining denominators provide no oter restrictions. Hence, for all values of ecept 0 and 2, te leftand side of is identical to te rigtand side. f( f(2 2 2
6 700 Capter 7 Rational Functions Let s look at anoter eample involving function notation. Eample 0. Given simplify te epression List all restrictions. f( 2, f( + f(. ( Te function notation f( + is asking us to replace eac instance of in te formula / 2 wit +. Tus, f( + /( + 2. Here is anoter way to tink of tis substitution. Suppose tat we remove te from so tat it reads f( 2, f( ( 2. (2 Now, if you want to compute f(2, simply insert a 2 in te blank area between parenteses. In our case, we want to compute f( +, so we insert an + in te blank space between parenteses in (2 to get f( + ( + 2. Wit tese preliminary remarks in mind, let s return to te problem. interpret te function notation as in our preliminary remarks and write f( + f( ( First, we Te common denominator for te numerator is found by listing eac factor to te igest power tat it occurs. Hence, te common denominator is 2 ( + 2. Te denominator as no fractions to be cleared, so it suffices to multiply bot numerator and denominator by 2 ( + 2. f( + f( ( ( ( ( ( ( ( ( ( ( + 2 ( 2 2 ( + 2
7 Section 7.6 Comple Fractions 70 We will now epand te numerator. Don t forget to use parenteses and distribute tat minus sign. f( + f( 2 ( ( ( ( + 2 Finally, factor a out of te numerator in opes of finding a common factor to cancel. f( + f( (2 + 2 ( + 2 (2 + 2 ( + 2 (2 + 2 ( + 2 We must now discuss te restrictions. In te original question (, te in te denominator must not equal zero. Hence, 0 is a restriction. In te final simplified form, te factor of 2 in te denominator is undefined if 0. Hence, 0 is a restriction. Finally, te factor of ( + 2 in te final denominator is undefined if + 0, so is a restriction. Te remaining denominators provide no additional restrictions. Hence, provided 0, 0, and, for all oter combinations of and, te leftand side of is identical to te rigtand side. f( + f( (2 + 2 ( + 2 Let s look at one final eample using function notation. Eample 3. If f( + (4 simplify f(f(. We first evaluate f at, ten evaluate f at te result of te first computation. Tus, we work te inner function first to obtain ( f(f( f. +
8 702 Capter 7 Rational Functions Te notation f(/( + is asking us to replace eac occurrence of in te formula /( + wit te epression /( +. Confusing? Here is an easy way to tink of tis substitution. Suppose tat we remove from f( +, replacing eac occurrence of wit empty parenteses, wic will produce te template f( ( ( +. (5 Now, if asked to compute f(3, simply insert 3 into te blank areas between parenteses. In tis case, we want to compute f(/(+, so we insert /(+ in te blank space between eac set of parenteses in (5 to obtain f ( We now ave a comple fraction. Te common denominator for bot top and bottom of tis comple fraction is +. Tus, we multiply bot numerator and denominator of our comple fraction by + and use te distributive property as follows ( + ( + + ( + ( + ( + ( ( + + ( + + (( + Cancel and simplify. ( ( + + ( ( + + (( ( In te final denominator, te value /2 makes te denominator 2 + equal to zero. Hence, /2 is a restriction. In te body of our work, several fractions ave denominators of + and are terefore undefined at. Tus, is a restriction. No oter denominators add additional restrictions. Hence, for all values of, ecept /2 and, te leftand side of f(f( 2 + is identical to te rigtand side.
P.4 Rational Expressions
7_0P04.qp /7/06 9:4 AM Page 7 Section P.4 Rational Epressions 7 P.4 Rational Epressions Domain of an Algebraic Epression Te set of real numbers for wic an algebraic epression is defined is te domain of
More informationA.4. Rational Expressions. Domain of an Algebraic Expression. What you should learn. Why you should learn it
A6 Appendi A Review of Fundamental Concepts of Algebra A.4 Rational Epressions Wat you sould learn Find domains of algebraic epressions. Simplify rational epressions. Add, subtract, multiply, and divide
More informationInstantaneous Rate of Change:
Instantaneous Rate of Cange: Last section we discovered tat te average rate of cange in F(x) can also be interpreted as te slope of a scant line. Te average rate of cange involves te cange in F(x) over
More informationUnderstanding the Derivative Backward and Forward by Dave Slomer
Understanding te Derivative Backward and Forward by Dave Slomer Slopes of lines are important, giving average rates of cange. Slopes of curves are even more important, giving instantaneous rates of cange.
More information2.0 5Minute Review: Polynomial Functions
mat 3 day 3: intro to limits 5Minute Review: Polynomial Functions You sould be familiar wit polynomials Tey are among te simplest of functions DEFINITION A polynomial is a function of te form y = p(x)
More informationDerivatives Math 120 Calculus I D Joyce, Fall 2013
Derivatives Mat 20 Calculus I D Joyce, Fall 203 Since we ave a good understanding of its, we can develop derivatives very quickly. Recall tat we defined te derivative f x of a function f at x to be te
More informationFinite Difference Approximations
Capter Finite Difference Approximations Our goal is to approximate solutions to differential equations, i.e., to find a function (or some discrete approximation to tis function) tat satisfies a given relationsip
More informationTangent Lines and Rates of Change
Tangent Lines and Rates of Cange 922005 Given a function y = f(x), ow do you find te slope of te tangent line to te grap at te point P(a, f(a))? (I m tinking of te tangent line as a line tat just skims
More information11.2 Instantaneous Rate of Change
11. Instantaneous Rate of Cange Question 1: How do you estimate te instantaneous rate of cange? Question : How do you compute te instantaneous rate of cange using a limit? Te average rate of cange is useful
More information1.6. Analyse Optimum Volume and Surface Area. Maximum Volume for a Given Surface Area. Example 1. Solution
1.6 Analyse Optimum Volume and Surface Area Estimation and oter informal metods of optimizing measures suc as surface area and volume often lead to reasonable solutions suc as te design of te tent in tis
More informationMath 113 HW #5 Solutions
Mat 3 HW #5 Solutions. Exercise.5.6. Suppose f is continuous on [, 5] and te only solutions of te equation f(x) = 6 are x = and x =. If f() = 8, explain wy f(3) > 6. Answer: Suppose we ad tat f(3) 6. Ten
More informationLecture 10: What is a Function, definition, piecewise defined functions, difference quotient, domain of a function
Lecture 10: Wat is a Function, definition, piecewise defined functions, difference quotient, domain of a function A function arises wen one quantity depends on anoter. Many everyday relationsips between
More informationACT Math Facts & Formulas
Numbers, Sequences, Factors Integers:..., 3, 2, 1, 0, 1, 2, 3,... Rationals: fractions, tat is, anyting expressable as a ratio of integers Reals: integers plus rationals plus special numbers suc as
More information2 Limits and Derivatives
2 Limits and Derivatives 2.7 Tangent Lines, Velocity, and Derivatives A tangent line to a circle is a line tat intersects te circle at exactly one point. We would like to take tis idea of tangent line
More informationTrapezoid Rule. y 2. y L
Trapezoid Rule and Simpson s Rule c 2002, 2008, 200 Donald Kreider and Dwigt Lar Trapezoid Rule Many applications of calculus involve definite integrals. If we can find an antiderivative for te integrand,
More informationACTIVITY: Deriving the Area Formula of a Trapezoid
4.3 Areas of Trapezoids a trapezoid? How can you derive a formula for te area of ACTIVITY: Deriving te Area Formula of a Trapezoid Work wit a partner. Use a piece of centimeter grid paper. a. Draw any
More informationOptimal Pricing Strategy for Second Degree Price Discrimination
Optimal Pricing Strategy for Second Degree Price Discrimination Alex O Brien May 5, 2005 Abstract Second Degree price discrimination is a coupon strategy tat allows all consumers access to te coupon. Purcases
More informationExam 2 Review. . You need to be able to interpret what you get to answer various questions.
Exam Review Exam covers 1.6,.1.3, 1.5, 4.14., and 5.15.3. You sould know ow to do all te omework problems from tese sections and you sould practice your understanding on several old exams in te exam
More informationArea of Trapezoids. Find the area of the trapezoid. 7 m. 11 m. 2 Use the Area of a Trapezoid. Find the value of b 2
Page 1 of. Area of Trapezoids Goal Find te area of trapezoids. Recall tat te parallel sides of a trapezoid are called te bases of te trapezoid, wit lengts denoted by and. base, eigt Key Words trapezoid
More informationThe EOQ Inventory Formula
Te EOQ Inventory Formula James M. Cargal Matematics Department Troy University Montgomery Campus A basic problem for businesses and manufacturers is, wen ordering supplies, to determine wat quantity of
More informationThe Derivative. Not for Sale
3 Te Te Derivative 3. Limits 3. Continuity 3.3 Rates of Cange 3. Definition of te Derivative 3.5 Grapical Differentiation Capter 3 Review Etended Application: A Model for Drugs Administered Intravenously
More informationSurface Areas of Prisms and Cylinders
12.2 TEXAS ESSENTIAL KNOWLEDGE AND SKILLS G.10.B G.11.C Surface Areas of Prisms and Cylinders Essential Question How can you find te surface area of a prism or a cylinder? Recall tat te surface area of
More informationSection 3.3. Differentiation of Polynomials and Rational Functions. Difference Equations to Differential Equations
Difference Equations to Differential Equations Section 3.3 Differentiation of Polynomials an Rational Functions In tis section we begin te task of iscovering rules for ifferentiating various classes of
More informationVerifying Numerical Convergence Rates
1 Order of accuracy Verifying Numerical Convergence Rates We consider a numerical approximation of an exact value u. Te approximation depends on a small parameter, suc as te grid size or time step, and
More informationMath Test Sections. The College Board: Expanding College Opportunity
Taking te SAT I: Reasoning Test Mat Test Sections Te materials in tese files are intended for individual use by students getting ready to take an SAT Program test; permission for any oter use must be sougt
More informationAverage and Instantaneous Rates of Change: The Derivative
9.3 verage and Instantaneous Rates of Cange: Te Derivative 609 OBJECTIVES 9.3 To define and find average rates of cange To define te derivative as a rate of cange To use te definition of derivative to
More informationLecture 10. Limits (cont d) Onesided limits. (Relevant section from Stewart, Seventh Edition: Section 2.4, pp. 113.)
Lecture 10 Limits (cont d) Onesided its (Relevant section from Stewart, Sevent Edition: Section 2.4, pp. 113.) As you may recall from your earlier course in Calculus, we may define onesided its, were
More informationGeometric Stratification of Accounting Data
Stratification of Accounting Data Patricia Gunning * Jane Mary Horgan ** William Yancey *** Abstract: We suggest a new procedure for defining te boundaries of te strata in igly skewed populations, usual
More informationSAT Subject Math Level 1 Facts & Formulas
Numbers, Sequences, Factors Integers:..., 3, 2, 1, 0, 1, 2, 3,... Reals: integers plus fractions, decimals, and irrationals ( 2, 3, π, etc.) Order Of Operations: Aritmetic Sequences: PEMDAS (Parenteses
More informationME422 Mechanical Control Systems Modeling Fluid Systems
Cal Poly San Luis Obispo Mecanical Engineering ME422 Mecanical Control Systems Modeling Fluid Systems Owen/Ridgely, last update Mar 2003 Te dynamic euations for fluid flow are very similar to te dynamic
More informationChapter 11. Limits and an Introduction to Calculus. Selected Applications
Capter Limits and an Introduction to Calculus. Introduction to Limits. Tecniques for Evaluating Limits. Te Tangent Line Problem. Limits at Infinit and Limits of Sequences.5 Te Area Problem Selected Applications
More informationDifferentiable Functions
Capter 8 Differentiable Functions A differentiable function is a function tat can be approximated locally by a linear function. 8.. Te derivative Definition 8.. Suppose tat f : (a, b) R and a < c < b.
More information1 Derivatives of Piecewise Defined Functions
MATH 1010E University Matematics Lecture Notes (week 4) Martin Li 1 Derivatives of Piecewise Define Functions For piecewise efine functions, we often ave to be very careful in computing te erivatives.
More informationArea of a Parallelogram
Area of a Parallelogram Focus on After tis lesson, you will be able to... φ develop te φ formula for te area of a parallelogram calculate te area of a parallelogram One of te sapes a marcing band can make
More informationCHAPTER 8: DIFFERENTIAL CALCULUS
CHAPTER 8: DIFFERENTIAL CALCULUS 1. Rules of Differentiation As we ave seen, calculating erivatives from first principles can be laborious an ifficult even for some relatively simple functions. It is clearly
More informationDetermine the perimeter of a triangle using algebra Find the area of a triangle using the formula
Student Name: Date: Contact Person Name: Pone Number: Lesson 0 Perimeter, Area, and Similarity of Triangles Objectives Determine te perimeter of a triangle using algebra Find te area of a triangle using
More informationThe differential amplifier
DiffAmp.doc 1 Te differential amplifier Te emitter coupled differential amplifier output is V o = A d V d + A c V C Were V d = V 1 V 2 and V C = (V 1 + V 2 ) / 2 In te ideal differential amplifier A c
More informationNew Vocabulary volume
. Plan Objectives To find te volume of a prism To find te volume of a cylinder Examples Finding Volume of a Rectangular Prism Finding Volume of a Triangular Prism 3 Finding Volume of a Cylinder Finding
More informationCollege Planning Using Cash Value Life Insurance
College Planning Using Cas Value Life Insurance CAUTION: Te advisor is urged to be extremely cautious of anoter college funding veicle wic provides a guaranteed return of premium immediately if funded
More informationChapter 10: Refrigeration Cycles
Capter 10: efrigeration Cycles Te vapor compression refrigeration cycle is a common metod for transferring eat from a low temperature to a ig temperature. Te above figure sows te objectives of refrigerators
More informationLet's Learn About Notes
2002 Product of Australia Contents Let's Learn About Notes by Beatrice Wilder Seet Seet 2 Seet 3 Seet 4 Seet 5 Seet 6 Seet 7 Seet 8 Seet 9 Seet 0 Seet Seet 2 Seet 3 Basic Information About Notes Lines
More informationThis supplement is meant to be read after Venema s Section 9.2. Throughout this section, we assume all nine axioms of Euclidean geometry.
Mat 444/445 Geometry for Teacers Summer 2008 Supplement : Similar Triangles Tis supplement is meant to be read after Venema s Section 9.2. Trougout tis section, we assume all nine axioms of uclidean geometry.
More informationM(0) = 1 M(1) = 2 M(h) = M(h 1) + M(h 2) + 1 (h > 1)
Insertion and Deletion in VL Trees Submitted in Partial Fulfillment of te Requirements for Dr. Eric Kaltofen s 66621: nalysis of lgoritms by Robert McCloskey December 14, 1984 1 ackground ccording to Knut
More informationFINITE DIFFERENCE METHODS
FINITE DIFFERENCE METHODS LONG CHEN Te best known metods, finite difference, consists of replacing eac derivative by a difference quotient in te classic formulation. It is simple to code and economic to
More informationFinite Volume Discretization of the Heat Equation
Lecture Notes 3 Finite Volume Discretization of te Heat Equation We consider finite volume discretizations of te onedimensional variable coefficient eat equation, wit Neumann boundary conditions u t x
More informationDifferential Calculus: Differentiation (First Principles, Rules) and Sketching Graphs (Grade 12)
OpenStaxCNX moule: m39313 1 Differential Calculus: Differentiation (First Principles, Rules) an Sketcing Graps (Grae 12) Free Hig Scool Science Texts Project Tis work is prouce by OpenStaxCNX an license
More informationChapter 1. Rates of Change. By the end of this chapter, you will
Capter 1 Rates of Cange Our world is in a constant state of cange. Understanding te nature of cange and te rate at wic it takes place enables us to make important predictions and decisions. For eample,
More informationSimilar interpretations can be made for total revenue and total profit functions.
EXERCISE 37 Tings to remember: 1. MARGINAL COST, REVENUE, AND PROFIT If is te number of units of a product produced in some time interval, ten: Total Cost C() Marginal Cost C'() Total Revenue R() Marginal
More informationHardness Measurement of Metals Static Methods
Hardness Testing Principles and Applications Copyrigt 2011 ASM International Konrad Herrmann, editor All rigts reserved. www.asminternational.org Capter 2 Hardness Measurement of Metals Static Metods T.
More informationCan a LumpSum Transfer Make Everyone Enjoy the Gains. from Free Trade?
Can a LumpSum Transfer Make Everyone Enjoy te Gains from Free Trade? Yasukazu Icino Department of Economics, Konan University June 30, 2010 Abstract I examine lumpsum transfer rules to redistribute te
More informationSections 3.1/3.2: Introducing the Derivative/Rules of Differentiation
Sections 3.1/3.2: Introucing te Derivative/Rules of Differentiation 1 Tangent Line Before looking at te erivative, refer back to Section 2.1, looking at average velocity an instantaneous velocity. Here
More informationf(a + h) f(a) f (a) = lim
Lecture 7 : Derivative AS a Function In te previous section we defined te derivative of a function f at a number a (wen te function f is defined in an open interval containing a) to be f (a) 0 f(a + )
More information 1  Handout #22 May 23, 2012 Huffman Encoding and Data Compression. CS106B Spring 2012. Handout by Julie Zelenski with minor edits by Keith Schwarz
CS106B Spring 01 Handout # May 3, 01 Huffman Encoding and Data Compression Handout by Julie Zelenski wit minor edits by Keit Scwarz In te early 1980s, personal computers ad ard disks tat were no larger
More informationNotes: Most of the material in this chapter is taken from Young and Freedman, Chap. 12.
Capter 6. Fluid Mecanics Notes: Most of te material in tis capter is taken from Young and Freedman, Cap. 12. 6.1 Fluid Statics Fluids, i.e., substances tat can flow, are te subjects of tis capter. But
More informationWriting Mathematics Papers
Writing Matematics Papers Tis essay is intended to elp your senior conference paper. It is a somewat astily produced amalgam of advice I ave given to students in my PDCs (Mat 4 and Mat 9), so it s not
More informationThe modelling of business rules for dashboard reporting using mutual information
8 t World IMACS / MODSIM Congress, Cairns, Australia 37 July 2009 ttp://mssanz.org.au/modsim09 Te modelling of business rules for dasboard reporting using mutual information Gregory Calbert Command, Control,
More informationSolution Derivations for Capa #7
Solution Derivations for Capa #7 1) Consider te beavior of te circuit, wen various values increase or decrease. (Select Iincreases, Ddecreases, If te first is I and te rest D, enter IDDDD). A) If R1
More information1 Density functions, cummulative density functions, measures of central tendency, and measures of dispersion
Density functions, cummulative density functions, measures of central tendency, and measures of dispersion densityfunctionsintro.tex October, 9 Note tat tis section of notes is limitied to te consideration
More informationProof of the Power Rule for Positive Integer Powers
Te Power Rule A function of te form f (x) = x r, were r is any real number, is a power function. From our previous work we know tat x x 2 x x x x 3 3 x x In te first two cases, te power r is a positive
More informationNote nine: Linear programming CSE 101. 1 Linear constraints and objective functions. 1.1 Introductory example. Copyright c Sanjoy Dasgupta 1
Copyrigt c Sanjoy Dasgupta Figure. (a) Te feasible region for a linear program wit two variables (see tet for details). (b) Contour lines of te objective function: for different values of (profit). Te
More informationConservation of Energy 1 of 8
Conservation of Energy 1 of 8 Conservation of Energy Te important conclusions of tis capter are: If a system is isolated and tere is no friction (no nonconservative forces), ten KE + PE = constant (Our
More informationWeek #15  Word Problems & Differential Equations Section 8.2
Week #1  Word Problems & Differential Equations Section 8. From Calculus, Single Variable by HugesHallett, Gleason, McCallum et. al. Copyrigt 00 by Jon Wiley & Sons, Inc. Tis material is used by permission
More informationMath WarmUp for Exam 1 Name: Solutions
Disclaimer: Tese review problems do not represent te exact questions tat will appear te exam. Tis is just a warmup to elp you begin studying. It is your responsibility to review te omework problems, webwork
More informationAn Interest Rate Model
An Interest Rate Model Concepts and Buzzwords Building Price Tree from Rate Tree Lognormal Interest Rate Model Nonnegativity Volatility and te Level Effect Readings Tuckman, capters 11 and 12. Lognormal
More informationGeoActivity. 1 Use a straightedge to draw a line through one of the vertices of an index card. height is perpendicular to the bases.
Page of 7 8. Area of Parallelograms Goal Find te area of parallelograms. Key Words ase of a parallelogram eigt of a parallelogram parallelogram p. 0 romus p. GeoActivity Use a straigtedge to draw a line
More informationA LowTemperature Creep Experiment Using Common Solder
A LowTemperature Creep Experiment Using Common Solder L. Roy Bunnell, Materials Science Teacer Soutridge Hig Scool Kennewick, WA 99338 roy.bunnell@ksd.org Copyrigt: Edmonds Community College 2009 Abstract:
More informationChapter 7 Numerical Differentiation and Integration
45 We ave a abit in writing articles publised in scientiþc journals to make te work as Þnised as possible, to cover up all te tracks, to not worry about te blind alleys or describe ow you ad te wrong idea
More informationSAT Math MustKnow Facts & Formulas
SAT Mat MustKnow Facts & Formuas Numbers, Sequences, Factors Integers:..., 3, 2, 1, 0, 1, 2, 3,... Rationas: fractions, tat is, anyting expressabe as a ratio of integers Reas: integers pus rationas
More informationA strong credit score can help you score a lower rate on a mortgage
NET GAIN Scoring points for your financial future AS SEEN IN USA TODAY S MONEY SECTION, JULY 3, 2007 A strong credit score can elp you score a lower rate on a mortgage By Sandra Block Sales of existing
More informationCatalogue no. 12001XIE. Survey Methodology. December 2004
Catalogue no. 1001XIE Survey Metodology December 004 How to obtain more information Specific inquiries about tis product and related statistics or services sould be directed to: Business Survey Metods
More information7.7 Solving Rational Equations
Section 7.7 Solving Rational Equations 7 7.7 Solving Rational Equations When simplifying comple fractions in the previous section, we saw that multiplying both numerator and denominator by the appropriate
More informationAreas and Centroids. Nothing. Straight Horizontal line. Straight Sloping Line. Parabola. Cubic
Constructing Sear and Moment Diagrams Areas and Centroids Curve Equation Sape Centroid (From Fat End of Figure) Area Noting Noting a x 0 Straigt Horizontal line /2 Straigt Sloping Line /3 /2 Paraola /4
More informationGEOMETRY. Grades 6 8 Part 1 of 2. Lessons & Worksheets to Build Skills in Measuring Perimeter, Area, Surface Area, and Volume. Poster/Teaching Guide
Grades 6 8 Part 1 of 2 Poster/Teacing Guide GEOMETRY Lessons & Workseets to Build Skills in Measuring Perimeter, Area, Surface Area, and Volume Aligned wit NCTM Standards Mat Grants Available Lesson 1
More informationThe Derivative as a Function
Section 2.2 Te Derivative as a Function 200 Kiryl Tsiscanka Te Derivative as a Function DEFINITION: Te derivative of a function f at a number a, denoted by f (a), is if tis limit exists. f (a) f(a+) f(a)
More informationIn other words the graph of the polynomial should pass through the points
Capter 3 Interpolation Interpolation is te problem of fitting a smoot curve troug a given set of points, generally as te grap of a function. It is useful at least in data analysis (interpolation is a form
More informationA New Hybrid Metaheuristic Approach for Stratified Sampling
A New Hybrid Metaeuristic Approac for Stratified Sampling Timur KESKİNTÜRK 1, Sultan KUZU 2, Baadır F. YILDIRIM 3 1 Scool of Business, İstanbul University, İstanbul, Turkey 2 Scool of Business, İstanbul
More informationModule 2. The Science of Surface and Ground Water. Version 2 CE IIT, Kharagpur
Module Te Science of Surface and Ground Water Version CE IIT, Karagpur Lesson 6 Principles of Ground Water Flow Version CE IIT, Karagpur Instructional Objectives On completion of te lesson, te student
More information2.28 EDGE Program. Introduction
Introduction Te Economic Diversification and Growt Enterprises Act became effective on 1 January 1995. Te creation of tis Act was to encourage new businesses to start or expand in Newfoundland and Labrador.
More informationModule 1: Introduction to Finite Element Analysis Lecture 1: Introduction
Module : Introduction to Finite Element Analysis Lecture : Introduction.. Introduction Te Finite Element Metod (FEM) is a numerical tecnique to find approximate solutions of partial differential equations.
More information4.1 Rightangled Triangles 2. 4.2 Trigonometric Functions 19. 4.3 Trigonometric Identities 36. 4.4 Applications of Trigonometry to Triangles 53
ontents 4 Trigonometry 4.1 Rigtangled Triangles 4. Trigonometric Functions 19 4.3 Trigonometric Identities 36 4.4 pplications of Trigonometry to Triangles 53 4.5 pplications of Trigonometry to Waves 65
More informationMATHEMATICS FOR ENGINEERING DIFFERENTIATION TUTORIAL 1  BASIC DIFFERENTIATION
MATHEMATICS FOR ENGINEERING DIFFERENTIATION TUTORIAL 1  BASIC DIFFERENTIATION Tis tutorial is essential prerequisite material for anyone stuing mecanical engineering. Tis tutorial uses te principle of
More informationInvestigating CostEfficient Ways of Running a Survey of Pacific Peoples. Undertaken for the Ministry of Pacific Island Affairs
Investigating CostEfficient Ways of Running a Survey of Pacific Peoples Undertaken for te Ministry of Pacific Island Affairs by Temaleti Tupou, assisted by Debra Taylor and Tracey Savage Survey Metods,
More informationSAT Math Facts & Formulas
Numbers, Sequences, Factors SAT Mat Facts & Formuas Integers:..., 3, 2, 1, 0, 1, 2, 3,... Reas: integers pus fractions, decimas, and irrationas ( 2, 3, π, etc.) Order Of Operations: Aritmetic Sequences:
More informationCHAPTER 7. Di erentiation
CHAPTER 7 Di erentiation 1. Te Derivative at a Point Definition 7.1. Let f be a function defined on a neigborood of x 0. f is di erentiable at x 0, if te following it exists: f 0 fx 0 + ) fx 0 ) x 0 )=.
More informationComputer Vision System for Tracking Players in Sports Games
Computer Vision System for Tracking Players in Sports Games Abstract Janez Perš, Stanislav Kovacic Faculty of Electrical Engineering, University of Lublana Tržaška 5, 000 Lublana anez.pers@kiss.unil.si,
More informationResearch on the Antiperspective Correction Algorithm of QR Barcode
Researc on te Antiperspective Correction Algoritm of QR Barcode Jianua Li, YiWen Wang, YiJun Wang,Yi Cen, Guoceng Wang Key Laboratory of Electronic Tin Films and Integrated Devices University of Electronic
More informationAdvice for Undergraduates on Special Aspects of Writing Mathematics
Advice for Undergraduates on Special Aspects of Writing Matematics Abstract Tere are several guides to good matematical writing for professionals, but few for undergraduates. Yet undergraduates wo write
More informationDiscovering Area Formulas of Quadrilaterals by Using Composite Figures
Activity: Format: Ojectives: Related 009 SOL(s): Materials: Time Required: Directions: Discovering Area Formulas of Quadrilaterals y Using Composite Figures Small group or Large Group Participants will
More informationProjective Geometry. Projective Geometry
Euclidean versus Euclidean geometry describes sapes as tey are Properties of objects tat are uncanged by rigid motions» Lengts» Angles» Parallelism Projective geometry describes objects as tey appear Lengts,
More informationEC201 Intermediate Macroeconomics. EC201 Intermediate Macroeconomics Problem set 8 Solution
EC201 Intermediate Macroeconomics EC201 Intermediate Macroeconomics Prolem set 8 Solution 1) Suppose tat te stock of mone in a given econom is given te sum of currenc and demand for current accounts tat
More informationFrom Raids to Deportation: Current Immigration Law Enforcement
W W a t i s g o i n g o n? From Raids to Deportation: Current Immigration Law Enforcement O u r c u r r e n t i m m i g r a t i o n l a w s a r e b r o k e n a n d n e e d t o b e f i x e d. T e r e c
More informationAnswers to Basic Algebra Review
Answers to Basic Algebra Review 1. 1.1 Follow the sign rules when adding and subtracting: If the numbers have the same sign, add them together and keep the sign. If the numbers have different signs, subtract
More informationVOL. 6, NO. 9, SEPTEMBER 2011 ISSN ARPN Journal of Engineering and Applied Sciences
VOL. 6, NO. 9, SEPTEMBER 0 ISSN 896608 0060 Asian Researc Publising Network (ARPN). All rigts reserved. BIT ERROR RATE, PERFORMANCE ANALYSIS AND COMPARISION OF M x N EQUALIZER BASED MAXIMUM LIKELIHOOD
More information2.1: The Derivative and the Tangent Line Problem
.1.1.1: Te Derivative and te Tangent Line Problem Wat is te deinition o a tangent line to a curve? To answer te diiculty in writing a clear deinition o a tangent line, we can deine it as te iting position
More informationf(x + h) f(x) h as representing the slope of a secant line. As h goes to 0, the slope of the secant line approaches the slope of the tangent line.
Derivative of f(z) Dr. E. Jacobs Te erivative of a function is efine as a limit: f (x) 0 f(x + ) f(x) We can visualize te expression f(x+) f(x) as representing te slope of a secant line. As goes to 0,
More information4.4 The Derivative. 51. Disprove the claim: If lim f (x) = L, then either lim f (x) = L or. 52. If lim x a. f (x) = and lim x a. g(x) =, then lim x a
Capter 4 Real Analysis 281 51. Disprove te claim: If lim f () = L, ten eiter lim f () = L or a a lim f () = L. a 52. If lim a f () = an lim a g() =, ten lim a f + g =. 53. If lim f () = an lim g() = L
More informationRecall from last time: Events are recorded by local observers with synchronized clocks. Event 1 (firecracker explodes) occurs at x=x =0 and t=t =0
1/27 Day 5: Questions? Time Dilation engt Contraction PH3 Modern Pysics P11 I sometimes ask myself ow it came about tat I was te one to deelop te teory of relatiity. Te reason, I tink, is tat a normal
More informationSimplifying Exponential Expressions
Simplifying Eponential Epressions Eponential Notation Base Eponent Base raised to an eponent Eample: What is the base and eponent of the following epression? 7 is the base 7 is the eponent Goal To write
More informationTheoretical calculation of the heat capacity
eoretical calculation of te eat capacity Principle of equipartition of energy Heat capacity of ideal and real gases Heat capacity of solids: DulongPetit, Einstein, Debye models Heat capacity of metals
More information