Rational Functions. Rational Function. Example. The degree of q x. If n d then f x is an improper rational function. x x. We have three forms

Size: px
Start display at page:

Download "Rational Functions. Rational Function. Example. The degree of q x. If n d then f x is an improper rational function. x x. We have three forms"

Transcription

1 Ratioal Fuctios We have three forms R Ratioal Fuctio a a1 a p p 0 b b1 b0 q q p a a1 a0 q b b1 b0 The egree of p The egree of q is is If the f is a improper ratioal fuctio Compare forms Epae a a a b b1 b0 1 0 R Factore p R q p Simplifie R q

2 To Fi the Domai Give R p q Solve q 0 Domai The omai of a ratioal fuctio is all real values ecept where the eomiator, q() = Domai: all reals ecept Domai: all reals ecept 1 a 1 1 Domai: all reals ecept 1 Poits Not i The Domai Poits ot i the omais of a ratioal fuctio are either holes or vertical asymptotes A poit is a vertical asymptote if it caot ot be compeletly factore out If R the zeroes of q p p q q after factorig, the 0 are vertical asymptotes (VA) the other zeroes of q 0 are the holes

3 Holes a Vertical Asymptotes 56 D f, is a VA is a hole s R 5 H Fi holes a vertical asymptotes is a VA is a hole a is a VA F a 1 are VA Horizotal a Oblique Asymptotes horizotal asymptotes (HA) oblique (slat) asymptotes (OA)

4 E Behavior p a a1 a0 R q b b b There is a oblique (slat) asymptote a y b is the horizotal asymptote y 0 is the horizotal asymptote A fuctio will ot have both a oblique a a horizotal asymptote Horizotal Asymptote A horizotal lie is a asymptote oly to the far left a the far right of the graph. "Far" left or "far" right is efie as aythig past the vertical asymptotes or -itercepts. Horizotal asymptotes are ot asymptotic i the mile. It is okay to cross a horizotal asymptote i the mile y 0 is HA y is HA 4 1 There is a OA There is o OA or HA The ability to aalyze these is importat i Math 15 4

5 Oblique Asymptotes Whe the egree of the umerator is eactly oe more tha the egree of the eomiator, the graph of the ratioal fuctio will have a oblique asymptote. Aother ame for a oblique asymptote is a slat asymptote. To fi the equatio of the oblique asymptote, perform log ivisio (sythetic if it will work) by iviig the eomiator ito the umerator a iscarig the remaier Fiig Oblique Asymptotes p a a a a R q b b1 b0 Divie p by q a obtai R f where f y f is a proper ratioal fuctio is the oblique asymptote r q Discar the remaier a the OA is y 4 5

6 Fi OA oblique asymptote y Fi OA 1 Usig sythetic ivisio R1 oblique asymptote y 6

7 Our ratioal fuctio Our ratioal fuctio a OA Fi OA OA y5 Our ratioal fuctio Our ratioal fuctio a OA 7

8 R Review a a1 a p p 0 b b1 b0 q q To fi the omai solve q 0 a To fi horizotal asymptotes eamie b To fi oblique asymptotes ivie a iscar remaier p q To fi the vertical asymptotes solve q 0 Remarks I busiess we frequetly ecouter problems ivolvig averages. For eample Average Cost is Cost ivie by AC C C Whe C() is a polyomial AC is a ratioal fuctio (a) Fi average cost (b) Fi the umber of uits to miimize average cost C C AC Note: I Math 15 we ofte fi it easier to use 1 AC

9 AC (a) Fi average cost (b) Fi the umber of uits to miimize average cost C AC (a) Fi average cost (b) Fi the umber of uits to miimize average cost C AC (a) Fi average cost (b) Fi the umber of uits to miimize average cost C = 100 uits 9

10 Solvig Ratioal Equatios To solve aalytically, multiply each term by the Least Commo Deomiator to obtai a polyomial equatio Remember, we ca solve ay equatio by graphig usig either the Zero Metho or the Itercept Metho (a) combie terms to form ratioal fuctio (b) fi umber of hours that will give a average sales of $60,000 The commo eomiator is 4 The average mothly sales (i thousas of ollars) epes o the umber of hours of traiig 0 S 40, S The average mothly sales (i thousas of ollars) epes o the umber of hours of traiig 0 S 40, Discar 1.5, because less tha 4 1.5, hours of traiig 10

11 The average mothly sales (i thousas of ollars) epes o 0 the umber of hours of traiig S 40 4 Solve by graphig Itercept Metho 0 S 40 4 Choose wiow The average mothly sales (i thousas of ollars) epes o the umber of hours of traiig 0 S 40 4 Itercept Metho The average mothly sales (i thousas of ollars) epes o the umber of hours of traiig appro 78.5 hours of traiig 11

Rational Function. To Find the Domain. ( x) ( ) q( x) ( ) ( ) ( ) , 0. where p x and are polynomial functions. The degree of q x

Rational Function. To Find the Domain. ( x) ( ) q( x) ( ) ( ) ( ) , 0. where p x and are polynomial functions. The degree of q x Graphig Ratioal Fuctios R Ratioal Fuctio p a + + a+ a 0 q q b + + b + b0 q, 0 where p a are polyomial fuctios p a + + a+ a0 q b + + b + b0 The egree of p The egree of q is is If > the f is a improper ratioal

More information

Curve Sketching Handout #5 Topic Interpretation Rational Functions

Curve Sketching Handout #5 Topic Interpretation Rational Functions Curve Sketchig Hadout #5 Topic Iterpretatio Ratioal Fuctios A ratioal fuctio is a fuctio f that is a quotiet of two polyomials. I other words, p ( ) ( ) f is a ratioal fuctio if p ( ) ad q ( ) are polyomials

More information

MA Lesson 26 Notes Graphs of Rational Functions (Asymptotes) Limits at infinity

MA Lesson 26 Notes Graphs of Rational Functions (Asymptotes) Limits at infinity MA 1910 Lesso 6 Notes Graphs of Ratioal Fuctios (Asymptotes) Limits at ifiity Defiitio of a Ratioal Fuctio: If P() ad Q() are both polyomial fuctios, Q() 0, the the fuctio f below is called a Ratioal Fuctio.

More information

CALCULUS BASIC SUMMER REVIEW

CALCULUS BASIC SUMMER REVIEW CALCULUS BASIC SUMMER REVIEW NAME rise y y y Slope of a o vertical lie: m ru Poit Slope Equatio: y y m( ) The slope is m ad a poit o your lie is, ). ( y Slope-Itercept Equatio: y m b slope= m y-itercept=

More information

Unit 4: Polynomial and Rational Functions

Unit 4: Polynomial and Rational Functions 48 Uit 4: Polyomial ad Ratioal Fuctios Polyomial Fuctios A polyomial fuctio y px ( ) is a fuctio of the form p( x) ax + a x + a x +... + ax + ax+ a 1 1 1 0 where a, a 1,..., a, a1, a0are real costats ad

More information

Northwest High School s Algebra 2/Honors Algebra 2 Summer Review Packet

Northwest High School s Algebra 2/Honors Algebra 2 Summer Review Packet Northwest High School s Algebra /Hoors Algebra Summer Review Packet This packet is optioal! It will NOT be collected for a grade et school year! This packet has bee desiged to help you review various mathematical

More information

(Figure 2.9), we observe x. and we write. (b) as x, x 1. and we write. We say that the line y 0 is a horizontal asymptote of the graph of f.

(Figure 2.9), we observe x. and we write. (b) as x, x 1. and we write. We say that the line y 0 is a horizontal asymptote of the graph of f. The symbol for ifiity ( ) does ot represet a real umber. We use to describe the behavior of a fuctio whe the values i its domai or rage outgrow all fiite bouds. For eample, whe we say the limit of f as

More information

Continuous Functions

Continuous Functions Cotiuous Fuctios Q What does it mea for a fuctio to be cotiuous at a poit? Aswer- I mathematics, we have a defiitio that cosists of three cocepts that are liked i a special way Cosider the followig defiitio

More information

A.1 Algebra Review: Polynomials/Rationals. Definitions:

A.1 Algebra Review: Polynomials/Rationals. Definitions: MATH 040 Notes: Uit 0 Page 1 A.1 Algera Review: Polyomials/Ratioals Defiitios: A polyomial is a sum of polyomial terms. Polyomial terms are epressios formed y products of costats ad variales with whole

More information

( ) 2 + k The vertex is ( h, k) ( )( x q) The x-intercepts are x = p and x = q.

( ) 2 + k The vertex is ( h, k) ( )( x q) The x-intercepts are x = p and x = q. A Referece Sheet Number Sets Quadratic Fuctios Forms Form Equatio Stadard Form Vertex Form Itercept Form y ax + bx + c The x-coordiate of the vertex is x b a y a x h The axis of symmetry is x b a + k The

More information

Algebra II Notes Unit Seven: Powers, Roots, and Radicals

Algebra II Notes Unit Seven: Powers, Roots, and Radicals Syllabus Objectives: 7. The studets will use properties of ratioal epoets to simplify ad evaluate epressios. 7.8 The studet will solve equatios cotaiig radicals or ratioal epoets. b a, the b is the radical.

More information

Lyman Memorial High School. Honors Pre-Calculus Prerequisite Packet. Name:

Lyman Memorial High School. Honors Pre-Calculus Prerequisite Packet. Name: Lyma Memorial High School Hoors Pre-Calculus Prerequisite Packet 2018 Name: Dear Hoors Pre-Calculus Studet, Withi this packet you will fid mathematical cocepts ad skills covered i Algebra I, II ad Geometry.

More information

Mathematics 1 Outcome 1a. Pascall s Triangle and the Binomial Theorem (8 pers) Cumulative total = 8 periods. Lesson, Outline, Approach etc.

Mathematics 1 Outcome 1a. Pascall s Triangle and the Binomial Theorem (8 pers) Cumulative total = 8 periods. Lesson, Outline, Approach etc. prouce for by Tom Strag Pascall s Triagle a the Biomial Theorem (8 pers) Mathematics 1 Outcome 1a Lesso, Outlie, Approach etc. Nelso MIA - AH M1 1 Itrouctio to Pascal s Triagle via routes alog a set of

More information

k=1 s k (x) (3) and that the corresponding infinite series may also converge; moreover, if it converges, then it defines a function S through its sum

k=1 s k (x) (3) and that the corresponding infinite series may also converge; moreover, if it converges, then it defines a function S through its sum 0. L Hôpital s rule You alreay kow from Lecture 0 that ay sequece {s k } iuces a sequece of fiite sums {S } through S = s k, a that if s k 0 as k the {S } may coverge to the it k= S = s s s 3 s 4 = s k.

More information

Polynomial and Rational Functions. Polynomial functions and Their Graphs. Polynomial functions and Their Graphs. Examples

Polynomial and Rational Functions. Polynomial functions and Their Graphs. Polynomial functions and Their Graphs. Examples Polomial ad Ratioal Fuctios Polomial fuctios ad Their Graphs Math 44 Precalculus Polomial ad Ratioal Fuctios Polomial Fuctios ad Their Graphs Polomial fuctios ad Their Graphs A Polomial of degree is a

More information

Representing Functions as Power Series. 3 n ...

Representing Functions as Power Series. 3 n ... Math Fall 7 Lab Represetig Fuctios as Power Series I. Itrouctio I sectio.8 we leare the series c c c c c... () is calle a power series. It is a uctio o whose omai is the set o all or which it coverges.

More information

P.3 Polynomials and Special products

P.3 Polynomials and Special products Precalc Fall 2016 Sectios P.3, 1.2, 1.3, P.4, 1.4, P.2 (radicals/ratioal expoets), 1.5, 1.6, 1.7, 1.8, 1.1, 2.1, 2.2 I Polyomial defiitio (p. 28) a x + a x +... + a x + a x 1 1 0 1 1 0 a x + a x +... +

More information

September 2012 C1 Note. C1 Notes (Edexcel) Copyright - For AS, A2 notes and IGCSE / GCSE worksheets 1

September 2012 C1 Note. C1 Notes (Edexcel) Copyright   - For AS, A2 notes and IGCSE / GCSE worksheets 1 September 0 s (Edecel) Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets September 0 Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets September 0 Copyright

More information

RADICAL EXPRESSION. If a and x are real numbers and n is a positive integer, then x is an. n th root theorems: Example 1 Simplify

RADICAL EXPRESSION. If a and x are real numbers and n is a positive integer, then x is an. n th root theorems: Example 1 Simplify Example 1 Simplify 1.2A Radical Operatios a) 4 2 b) 16 1 2 c) 16 d) 2 e) 8 1 f) 8 What is the relatioship betwee a, b, c? What is the relatioship betwee d, e, f? If x = a, the x = = th root theorems: RADICAL

More information

Properties and Tests of Zeros of Polynomial Functions

Properties and Tests of Zeros of Polynomial Functions Properties ad Tests of Zeros of Polyomial Fuctios The Remaider ad Factor Theorems: Sythetic divisio ca be used to fid the values of polyomials i a sometimes easier way tha substitutio. This is show by

More information

In algebra one spends much time finding common denominators and thus simplifying rational expressions. For example:

In algebra one spends much time finding common denominators and thus simplifying rational expressions. For example: 74 The Method of Partial Fractios I algebra oe speds much time fidig commo deomiators ad thus simplifyig ratioal epressios For eample: + + + 6 5 + = + = = + + + + + ( )( ) 5 It may the seem odd to be watig

More information

3.2 Properties of Division 3.3 Zeros of Polynomials 3.4 Complex and Rational Zeros of Polynomials

3.2 Properties of Division 3.3 Zeros of Polynomials 3.4 Complex and Rational Zeros of Polynomials Math 60 www.timetodare.com 3. Properties of Divisio 3.3 Zeros of Polyomials 3.4 Complex ad Ratioal Zeros of Polyomials I these sectios we will study polyomials algebraically. Most of our work will be cocered

More information

Mini Lecture 10.1 Radical Expressions and Functions. 81x d. x 4x 4

Mini Lecture 10.1 Radical Expressions and Functions. 81x d. x 4x 4 Mii Lecture 0. Radical Expressios ad Fuctios Learig Objectives:. Evaluate square roots.. Evaluate square root fuctios.. Fid the domai of square root fuctios.. Use models that are square root fuctios. 5.

More information

7.1 Finding Rational Solutions of Polynomial Equations

7.1 Finding Rational Solutions of Polynomial Equations Name Class Date 7.1 Fidig Ratioal Solutios of Polyomial Equatios Essetial Questio: How do you fid the ratioal roots of a polyomial equatio? Resource Locker Explore Relatig Zeros ad Coefficiets of Polyomial

More information

Series: Infinite Sums

Series: Infinite Sums Series: Ifiite Sums Series are a way to mae sese of certai types of ifiitely log sums. We will eed to be able to do this if we are to attai our goal of approximatig trascedetal fuctios by usig ifiite degree

More information

TEACHER CERTIFICATION STUDY GUIDE

TEACHER CERTIFICATION STUDY GUIDE COMPETENCY 1. ALGEBRA SKILL 1.1 1.1a. ALGEBRAIC STRUCTURES Kow why the real ad complex umbers are each a field, ad that particular rigs are ot fields (e.g., itegers, polyomial rigs, matrix rigs) Algebra

More information

Analytic Number Theory Solutions

Analytic Number Theory Solutions Aalytic Number Theory Solutios Sea Li Corell Uiversity sl6@corell.eu Ja. 03 Itrouctio This ocumet is a work-i-progress solutio maual for Tom Apostol s Itrouctio to Aalytic Number Theory. The solutios were

More information

The type of limit that is used to find TANGENTS and VELOCITIES gives rise to the central idea in DIFFERENTIAL CALCULUS, the DERIVATIVE.

The type of limit that is used to find TANGENTS and VELOCITIES gives rise to the central idea in DIFFERENTIAL CALCULUS, the DERIVATIVE. NOTES : LIMITS AND DERIVATIVES Name: Date: Period: Iitial: LESSON.1 THE TANGENT AND VELOCITY PROBLEMS Pre-Calculus Mathematics Limit Process Calculus The type of it that is used to fid TANGENTS ad VELOCITIES

More information

MATH 10550, EXAM 3 SOLUTIONS

MATH 10550, EXAM 3 SOLUTIONS MATH 155, EXAM 3 SOLUTIONS 1. I fidig a approximate solutio to the equatio x 3 +x 4 = usig Newto s method with iitial approximatio x 1 = 1, what is x? Solutio. Recall that x +1 = x f(x ) f (x ). Hece,

More information

CAMI Education linked to CAPS: Mathematics. Grade The main topics in the FET Mathematics Curriculum NUMBER

CAMI Education linked to CAPS: Mathematics. Grade The main topics in the FET Mathematics Curriculum NUMBER - 1 - CAMI Eucatio like to CAPS: Grae 1 The mai topics i the FET Curriculum NUMBER TOPIC 1 Fuctios Number patters, sequeces a series 3 Fiace, growth a ecay 4 Algebra 5 Differetial Calculus 6 Probability

More information

Polynomial Functions and Their Graphs

Polynomial Functions and Their Graphs Polyomial Fuctios ad Their Graphs I this sectio we begi the study of fuctios defied by polyomial expressios. Polyomial ad ratioal fuctios are the most commo fuctios used to model data, ad are used extesively

More information

LESSON 2: SIMPLIFYING RADICALS

LESSON 2: SIMPLIFYING RADICALS High School: Workig with Epressios LESSON : SIMPLIFYING RADICALS N.RN.. C N.RN.. B 5 5 C t t t t t E a b a a b N.RN.. 4 6 N.RN. 4. N.RN. 5. N.RN. 6. 7 8 N.RN. 7. A 7 N.RN. 8. 6 80 448 4 5 6 48 00 6 6 6

More information

First, note that the LS residuals are orthogonal to the regressors. X Xb X y = 0 ( normal equations ; (k 1) ) So,

First, note that the LS residuals are orthogonal to the regressors. X Xb X y = 0 ( normal equations ; (k 1) ) So, 0 2. OLS Part II The OLS residuals are orthogoal to the regressors. If the model icludes a itercept, the orthogoality of the residuals ad regressors gives rise to three results, which have limited practical

More information

Sail into Summer with Math!

Sail into Summer with Math! Sail ito Summer with Math! For Studets Eterig Hoors Geometry This summer math booklet was developed to provide studets i kidergarte through the eighth grade a opportuity to review grade level math objectives

More information

n m CHAPTER 3 RATIONAL EXPONENTS AND RADICAL FUNCTIONS 3-1 Evaluate n th Roots and Use Rational Exponents Real nth Roots of a n th Root of a

n m CHAPTER 3 RATIONAL EXPONENTS AND RADICAL FUNCTIONS 3-1 Evaluate n th Roots and Use Rational Exponents Real nth Roots of a n th Root of a CHAPTER RATIONAL EXPONENTS AND RADICAL FUNCTIONS Big IDEAS: 1) Usig ratioal expoets ) Performig fuctio operatios ad fidig iverse fuctios ) Graphig radical fuctios ad solvig radical equatios Sectio: Essetial

More information

G r a d e 1 1 P r e - C a l c u l u s M a t h e m a t i c s ( 3 0 S )

G r a d e 1 1 P r e - C a l c u l u s M a t h e m a t i c s ( 3 0 S ) G r a d e 1 1 P r e - C a l c u l u s M a t h e m a t i c s ( 3 0 S ) Grade 11 Pre-Calculus Mathematics (30S) is desiged for studets who ited to study calculus ad related mathematics as part of post-secodary

More information

We will conclude the chapter with the study a few methods and techniques which are useful

We will conclude the chapter with the study a few methods and techniques which are useful Chapter : Coordiate geometry: I this chapter we will lear about the mai priciples of graphig i a dimesioal (D) Cartesia system of coordiates. We will focus o drawig lies ad the characteristics of the graphs

More information

AP Calculus BC Review Applications of Derivatives (Chapter 4) and f,

AP Calculus BC Review Applications of Derivatives (Chapter 4) and f, AP alculus B Review Applicatios of Derivatives (hapter ) Thigs to Kow ad Be Able to Do Defiitios of the followig i terms of derivatives, ad how to fid them: critical poit, global miima/maima, local (relative)

More information

GRAPHING LINEAR EQUATIONS. Linear Equations ( 3,1 ) _x-axis. Origin ( 0, 0 ) Slope = change in y change in x. Equation for l 1.

GRAPHING LINEAR EQUATIONS. Linear Equations ( 3,1 ) _x-axis. Origin ( 0, 0 ) Slope = change in y change in x. Equation for l 1. GRAPHING LINEAR EQUATIONS Quadrat II Quadrat I ORDERED PAIR: The first umer i the ordered pair is the -coordiate ad the secod umer i the ordered pair is the y-coordiate. (,1 ) Origi ( 0, 0 ) _-ais Liear

More information

1 Generating functions for balls in boxes

1 Generating functions for balls in boxes Math 566 Fall 05 Some otes o geeratig fuctios Give a sequece a 0, a, a,..., a,..., a geeratig fuctio some way of represetig the sequece as a fuctio. There are may ways to do this, with the most commo ways

More information

Exponential and Trigonometric Functions Lesson #1

Exponential and Trigonometric Functions Lesson #1 Epoetial ad Trigoometric Fuctios Lesso # Itroductio To Epoetial Fuctios Cosider a populatio of 00 mice which is growig uder plague coditios. If the mouse populatio doubles each week we ca costruct a table

More information

Order doesn t matter. There exists a number (zero) whose sum with any number is the number.

Order doesn t matter. There exists a number (zero) whose sum with any number is the number. P. Real Numbers ad Their Properties Natural Numbers 1,,3. Whole Numbers 0, 1,,... Itegers..., -1, 0, 1,... Real Numbers Ratioal umbers (p/q) Where p & q are itegers, q 0 Irratioal umbers o-termiatig ad

More information

Chapter 4. Fourier Series

Chapter 4. Fourier Series Chapter 4. Fourier Series At this poit we are ready to ow cosider the caoical equatios. Cosider, for eample the heat equatio u t = u, < (4.) subject to u(, ) = si, u(, t) = u(, t) =. (4.) Here,

More information

The z-transform. 7.1 Introduction. 7.2 The z-transform Derivation of the z-transform: x[n] = z n LTI system, h[n] z = re j

The z-transform. 7.1 Introduction. 7.2 The z-transform Derivation of the z-transform: x[n] = z n LTI system, h[n] z = re j The -Trasform 7. Itroductio Geeralie the complex siusoidal represetatio offered by DTFT to a represetatio of complex expoetial sigals. Obtai more geeral characteristics for discrete-time LTI systems. 7.

More information

REVISION SHEET FP1 (MEI) ALGEBRA. Identities In mathematics, an identity is a statement which is true for all values of the variables it contains.

REVISION SHEET FP1 (MEI) ALGEBRA. Identities In mathematics, an identity is a statement which is true for all values of the variables it contains. The mai ideas are: Idetities REVISION SHEET FP (MEI) ALGEBRA Before the exam you should kow: If a expressio is a idetity the it is true for all values of the variable it cotais The relatioships betwee

More information

FUNCTIONS (11 UNIVERSITY)

FUNCTIONS (11 UNIVERSITY) FINAL EXAM REVIEW FOR MCR U FUNCTIONS ( UNIVERSITY) Overall Remiders: To study for your eam your should redo all your past tests ad quizzes Write out all the formulas i the course to help you remember

More information

THE LEGENDRE POLYNOMIALS AND THEIR PROPERTIES. r If one now thinks of obtaining the potential of a distributed mass, the solution becomes-

THE LEGENDRE POLYNOMIALS AND THEIR PROPERTIES. r If one now thinks of obtaining the potential of a distributed mass, the solution becomes- THE LEGENDRE OLYNOMIALS AND THEIR ROERTIES The gravitatioal potetial ψ at a poit A at istace r from a poit mass locate at B ca be represete by the solutio of the Laplace equatio i spherical cooriates.

More information

x c the remainder is Pc ().

x c the remainder is Pc (). Algebra, Polyomial ad Ratioal Fuctios Page 1 K.Paulk Notes Chapter 3, Sectio 3.1 to 3.4 Summary Sectio Theorem Notes 3.1 Zeros of a Fuctio Set the fuctio to zero ad solve for x. The fuctio is zero at these

More information

Zeros of Polynomials

Zeros of Polynomials Math 160 www.timetodare.com 4.5 4.6 Zeros of Polyomials I these sectios we will study polyomials algebraically. Most of our work will be cocered with fidig the solutios of polyomial equatios of ay degree

More information

Precalculus MATH Sections 3.1, 3.2, 3.3. Exponential, Logistic and Logarithmic Functions

Precalculus MATH Sections 3.1, 3.2, 3.3. Exponential, Logistic and Logarithmic Functions Precalculus MATH 2412 Sectios 3.1, 3.2, 3.3 Epoetial, Logistic ad Logarithmic Fuctios Epoetial fuctios are used i umerous applicatios coverig may fields of study. They are probably the most importat group

More information

Mth 95 Notes Module 1 Spring Section 4.1- Solving Systems of Linear Equations in Two Variables by Graphing, Substitution, and Elimination

Mth 95 Notes Module 1 Spring Section 4.1- Solving Systems of Linear Equations in Two Variables by Graphing, Substitution, and Elimination Mth 9 Notes Module Sprig 4 Sectio 4.- Solvig Sstems of Liear Equatios i Two Variales Graphig, Sustitutio, ad Elimiatio A Solutio to a Sstem of Two (or more) Liear Equatios is the commo poit(s) of itersectio

More information

KNOWLEDGE OF NUMBER SENSE, CONCEPTS, AND OPERATIONS

KNOWLEDGE OF NUMBER SENSE, CONCEPTS, AND OPERATIONS DOMAIN I. COMPETENCY.0 MATHEMATICS KNOWLEDGE OF NUMBER SENSE, CONCEPTS, AND OPERATIONS Skill. Apply ratio ad proportio to solve real-world problems. A ratio is a compariso of umbers. If a class had boys

More information

FLC Ch 8 & 9. Evaluate. Check work. a) b) c) d) e) f) g) h) i) j) k) l) m) n) o) 3. p) q) r) s) t) 3.

FLC Ch 8 & 9. Evaluate. Check work. a) b) c) d) e) f) g) h) i) j) k) l) m) n) o) 3. p) q) r) s) t) 3. Math 100 Elemetary Algebra Sec 8.1: Radical Expressios List perfect squares ad evaluate their square root. Kow these perfect squares for test. Def The positive (pricipal) square root of x, writte x, is

More information

APPENDIX F Complex Numbers

APPENDIX F Complex Numbers APPENDIX F Complex Numbers Operatios with Complex Numbers Complex Solutios of Quadratic Equatios Polar Form of a Complex Number Powers ad Roots of Complex Numbers Operatios with Complex Numbers Some equatios

More information

2.3 Warmup. Graph the derivative of the following functions. Where necessary, approximate the derivative.

2.3 Warmup. Graph the derivative of the following functions. Where necessary, approximate the derivative. . Warmup Grap te erivative of te followig fuctios. Were ecessar, approimate te erivative. Differetiabilit Must a fuctio ave a erivative at eac poit were te fuctio is efie? Or If f a is efie, must f ( a)

More information

as best you can in any three (3) of a f. [15 = 3 5 each] e. y = sec 2 (arctan(x)) f. y = sin (e x )

as best you can in any three (3) of a f. [15 = 3 5 each] e. y = sec 2 (arctan(x)) f. y = sin (e x ) Mathematics Y Calculus I: Calculus of oe variable Tret Uiversity, Summer Solutios to the Fial Examiatio Time: 9: :, o Weesay, August,. Brought to you by Stefa. Istructios: Show all your work a justify

More information

Mathematics: Paper 1

Mathematics: Paper 1 GRADE 1 EXAMINATION JULY 013 Mathematics: Paper 1 EXAMINER: Combied Paper MODERATORS: JE; RN; SS; AVDB TIME: 3 Hours TOTAL: 150 PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY 1. This questio paper cosists

More information

3. If x and y are real numbers, what is the simplified radical form

3. If x and y are real numbers, what is the simplified radical form lgebra II Practice Test Objective:.a. Which is equivalet to 98 94 4 49?. Which epressio is aother way to write 5 4? 5 5 4 4 4 5 4 5. If ad y are real umbers, what is the simplified radical form of 5 y

More information

( a) ( ) 1 ( ) 2 ( ) ( ) 3 3 ( ) =!

( a) ( ) 1 ( ) 2 ( ) ( ) 3 3 ( ) =! .8,.9: Taylor ad Maclauri Series.8. Although we were able to fid power series represetatios for a limited group of fuctios i the previous sectio, it is ot immediately obvious whether ay give fuctio has

More information

COMM 602: Digital Signal Processing

COMM 602: Digital Signal Processing COMM 60: Digital Sigal Processig Lecture 4 -Properties of LTIS Usig Z-Trasform -Iverse Z-Trasform Properties of LTIS Usig Z-Trasform Properties of LTIS Usig Z-Trasform -ve +ve Properties of LTIS Usig Z-Trasform

More information

We are mainly going to be concerned with power series in x, such as. (x)} converges - that is, lims N n

We are mainly going to be concerned with power series in x, such as. (x)} converges - that is, lims N n Review of Power Series, Power Series Solutios A power series i x - a is a ifiite series of the form c (x a) =c +c (x a)+(x a) +... We also call this a power series cetered at a. Ex. (x+) is cetered at

More information

ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER / Statistics

ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER / Statistics ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER 1 018/019 DR. ANTHONY BROWN 8. Statistics 8.1. Measures of Cetre: Mea, Media ad Mode. If we have a series of umbers the

More information

Name Date PRECALCULUS SUMMER PACKET

Name Date PRECALCULUS SUMMER PACKET Name Date PRECALCULUS SUMMER PACKET This packet covers some of the cocepts that you eed to e familiar with i order to e successful i Precalculus. This summer packet is due o the first day of school! Make

More information

Find a formula for the exponential function whose graph is given , 1 2,16 1, 6

Find a formula for the exponential function whose graph is given , 1 2,16 1, 6 Math 4 Activity (Due by EOC Apr. ) Graph the followig epoetial fuctios by modifyig the graph of f. Fid the rage of each fuctio.. g. g. g 4. g. g 6. g Fid a formula for the epoetial fuctio whose graph is

More information

CHAPTER 10 INFINITE SEQUENCES AND SERIES

CHAPTER 10 INFINITE SEQUENCES AND SERIES CHAPTER 10 INFINITE SEQUENCES AND SERIES 10.1 Sequeces 10.2 Ifiite Series 10.3 The Itegral Tests 10.4 Compariso Tests 10.5 The Ratio ad Root Tests 10.6 Alteratig Series: Absolute ad Coditioal Covergece

More information

SNAP Centre Workshop. Basic Algebraic Manipulation

SNAP Centre Workshop. Basic Algebraic Manipulation SNAP Cetre Workshop Basic Algebraic Maipulatio 8 Simplifyig Algebraic Expressios Whe a expressio is writte i the most compact maer possible, it is cosidered to be simplified. Not Simplified: x(x + 4x)

More information

Definition of z-transform.

Definition of z-transform. - Trasforms Frequecy domai represetatios of discretetime sigals ad LTI discrete-time systems are made possible with the use of DTFT. However ot all discrete-time sigals e.g. uit step sequece are guarateed

More information

REGRESSION (Physics 1210 Notes, Partial Modified Appendix A)

REGRESSION (Physics 1210 Notes, Partial Modified Appendix A) REGRESSION (Physics 0 Notes, Partial Modified Appedix A) HOW TO PERFORM A LINEAR REGRESSION Cosider the followig data poits ad their graph (Table I ad Figure ): X Y 0 3 5 3 7 4 9 5 Table : Example Data

More information

REVISION SHEET FP1 (MEI) ALGEBRA. Identities In mathematics, an identity is a statement which is true for all values of the variables it contains.

REVISION SHEET FP1 (MEI) ALGEBRA. Identities In mathematics, an identity is a statement which is true for all values of the variables it contains. the Further Mathematics etwork wwwfmetworkorguk V 07 The mai ideas are: Idetities REVISION SHEET FP (MEI) ALGEBRA Before the exam you should kow: If a expressio is a idetity the it is true for all values

More information

U8L1: Sec Equations of Lines in R 2

U8L1: Sec Equations of Lines in R 2 MCVU U8L: Sec. 8.9. Equatios of Lies i R Review of Equatios of a Straight Lie (-D) Cosider the lie passig through A (-,) with slope, as show i the diagram below. I poit slope form, the equatio of the lie

More information

4755 Mark Scheme June Question Answer Marks Guidance M1* Attempt to find M or 108M -1 M 108 M1 A1 [6] M1 A1

4755 Mark Scheme June Question Answer Marks Guidance M1* Attempt to find M or 108M -1 M 108 M1 A1 [6] M1 A1 4755 Mark Scheme Jue 05 * Attempt to fid M or 08M - M 08 8 4 * Divide by their determiat,, at some stage Correct determiat, (A0 for det M= 08 stated, all other OR 08 8 4 5 8 7 5 x, y,oe 8 7 4xy 8xy dep*

More information

Mth 138 College Algebra Review Guide for Exam III

Mth 138 College Algebra Review Guide for Exam III Mth 138 College Algebra Review Guide for Exam III Thomas W. Judso Stephe F. Austi State Uiversity Sprig 2018 Exam III Details Exam III will be o Thursday, April 19 ad will cover material up to Chapter

More information

Appendix F: Complex Numbers

Appendix F: Complex Numbers Appedix F Complex Numbers F1 Appedix F: Complex Numbers Use the imagiary uit i to write complex umbers, ad to add, subtract, ad multiply complex umbers. Fid complex solutios of quadratic equatios. Write

More information

Define and Use Sequences and Series

Define and Use Sequences and Series . a., A..A; P..A, P..B TEKS Defie ad Use Sequeces ad Series Before You idetified ad wrote fuctios. Now You will recogize ad write rules for umber patters. Why? So you ca fid agle measures, as i Ex.. Key

More information

INTEGRATION BY PARTS (TABLE METHOD)

INTEGRATION BY PARTS (TABLE METHOD) INTEGRATION BY PARTS (TABLE METHOD) Suppose you wat to evaluate cos d usig itegratio by parts. Usig the u dv otatio, we get So, u dv d cos du d v si cos d si si d or si si d We see that it is ecessary

More information

PAijpam.eu ON DERIVATION OF RATIONAL SOLUTIONS OF BABBAGE S FUNCTIONAL EQUATION

PAijpam.eu ON DERIVATION OF RATIONAL SOLUTIONS OF BABBAGE S FUNCTIONAL EQUATION Iteratioal Joural of Pure ad Applied Mathematics Volume 94 No. 204, 9-20 ISSN: 3-8080 (prited versio); ISSN: 34-3395 (o-lie versio) url: http://www.ijpam.eu doi: http://dx.doi.org/0.2732/ijpam.v94i.2 PAijpam.eu

More information

Lesson 10: Limits and Continuity

Lesson 10: Limits and Continuity www.scimsacademy.com Lesso 10: Limits ad Cotiuity SCIMS Academy 1 Limit of a fuctio The cocept of limit of a fuctio is cetral to all other cocepts i calculus (like cotiuity, derivative, defiite itegrals

More information

3. Z Transform. Recall that the Fourier transform (FT) of a DT signal xn [ ] is ( ) [ ] = In order for the FT to exist in the finite magnitude sense,

3. Z Transform. Recall that the Fourier transform (FT) of a DT signal xn [ ] is ( ) [ ] = In order for the FT to exist in the finite magnitude sense, 3. Z Trasform Referece: Etire Chapter 3 of text. Recall that the Fourier trasform (FT) of a DT sigal x [ ] is ω ( ) [ ] X e = j jω k = xe I order for the FT to exist i the fiite magitude sese, S = x [

More information

a 3, a 4, ... are the terms of the sequence. The number a n is the nth term of the sequence, and the entire sequence is denoted by a n

a 3, a 4, ... are the terms of the sequence. The number a n is the nth term of the sequence, and the entire sequence is denoted by a n 60_090.qxd //0 : PM Page 59 59 CHAPTER 9 Ifiite Series Sectio 9. EXPLORATION Fidig Patters Describe a patter for each of the followig sequeces. The use your descriptio to write a formula for the th term

More information

Advanced Algebra SS Semester 2 Final Exam Study Guide Mrs. Dunphy

Advanced Algebra SS Semester 2 Final Exam Study Guide Mrs. Dunphy Advaced Algebra SS Semester 2 Fial Exam Study Guide Mrs. Duphy My fial is o at Iformatio about the Fial Exam The fial exam is cumulative, coverig previous mathematic coursework, especially Algebra I. All

More information

Honors Algebra 2 Summer Assignment

Honors Algebra 2 Summer Assignment Hoors Algera Summer Assigmet Dear Future Hoors Algera Studet, Cogratulatios o your erollmet i Hoors Algera! Below you will fid the summer assigmet questios. It is assumed that these cocepts, alog with

More information

Root Finding COS 323

Root Finding COS 323 Root Fidig COS 323 Remider Sig up for Piazza Assigmet 0 is posted, due Tue 9/25 Last time.. Floatig poit umbers ad precisio Machie epsilo Sources of error Sesitivity ad coditioig Stability ad accuracy

More information

Maximum and Minimum Values

Maximum and Minimum Values Sec 4.1 Maimum ad Miimum Values A. Absolute Maimum or Miimum / Etreme Values A fuctio Similarly, f has a Absolute Maimum at c if c f f has a Absolute Miimum at c if c f f for every poit i the domai. f

More information

THE KENNESAW STATE UNIVERSITY HIGH SCHOOL MATHEMATICS COMPETITION PART II Calculators are NOT permitted Time allowed: 2 hours

THE KENNESAW STATE UNIVERSITY HIGH SCHOOL MATHEMATICS COMPETITION PART II Calculators are NOT permitted Time allowed: 2 hours THE 06-07 KENNESAW STATE UNIVERSITY HIGH SCHOOL MATHEMATICS COMPETITION PART II Calculators are NOT permitted Time allowed: hours Let x, y, ad A all be positive itegers with x y a) Prove that there are

More information

Part I: Covers Sequence through Series Comparison Tests

Part I: Covers Sequence through Series Comparison Tests Part I: Covers Sequece through Series Compariso Tests. Give a example of each of the followig: (a) A geometric sequece: (b) A alteratig sequece: (c) A sequece that is bouded, but ot coverget: (d) A sequece

More information

3.3 Rules for Differentiation Calculus. Drum Roll please [In a Deep Announcer Voice] And now the moment YOU VE ALL been waiting for

3.3 Rules for Differentiation Calculus. Drum Roll please [In a Deep Announcer Voice] And now the moment YOU VE ALL been waiting for . Rules or Dieretiatio Calculus. RULES FOR DIFFERENTIATION Drum Roll please [I a Deep Aoucer Voice] A ow the momet YOU VE ALL bee waitig or Rule #1 Derivative o a Costat Fuctio I c is a costat value, the

More information

+ {JEE Advace 03} Sept 0 Name: Batch (Day) Phoe No. IT IS NOT ENOUGH TO HAVE A GOOD MIND, THE MAIN THING IS TO USE IT WELL Marks: 00. If A (α, β) = (a) A( α, β) = A( α, β) (c) Adj (A ( α, β)) = Sol : We

More information

Set Notation Review. N the set of positive integers (aka set of natural numbers) {1, 2, 3, }

Set Notation Review. N the set of positive integers (aka set of natural numbers) {1, 2, 3, } 11. Notes o Mathematical Iductio Before we delve ito the today s topic, let s review some basic set otatio Set Notatio Review N the set of positive itegers (aa set of atural umbers) {1,, 3, } Z the set

More information

Name: Math 10550, Final Exam: December 15, 2007

Name: Math 10550, Final Exam: December 15, 2007 Math 55, Fial Exam: December 5, 7 Name: Be sure that you have all pages of the test. No calculators are to be used. The exam lasts for two hours. Whe told to begi, remove this aswer sheet ad keep it uder

More information

COS 341 Discrete Mathematics. Exponential Generating Functions and Recurrence Relations

COS 341 Discrete Mathematics. Exponential Generating Functions and Recurrence Relations COS 341 Discrete Mathematics Epoetial Geeratig Fuctios ad Recurrece Relatios 1 Tetbook? 1 studets said they eeded the tetbook, but oly oe studet bought the tet from Triagle. If you do t have the book,

More information

Chapter 6: Numerical Series

Chapter 6: Numerical Series Chapter 6: Numerical Series 327 Chapter 6 Overview: Sequeces ad Numerical Series I most texts, the topic of sequeces ad series appears, at first, to be a side topic. There are almost o derivatives or itegrals

More information

THE SOLUTION OF NONLINEAR EQUATIONS f( x ) = 0.

THE SOLUTION OF NONLINEAR EQUATIONS f( x ) = 0. THE SOLUTION OF NONLINEAR EQUATIONS f( ) = 0. Noliear Equatio Solvers Bracketig. Graphical. Aalytical Ope Methods Bisectio False Positio (Regula-Falsi) Fied poit iteratio Newto Raphso Secat The root of

More information

, 4 is the second term U 2

, 4 is the second term U 2 Balliteer Istitute 995-00 wwwleavigcertsolutioscom Leavig Cert Higher Maths Sequeces ad Series A sequece is a array of elemets seperated by commas E,,7,0,, The elemets are called the terms of the sequece

More information

Infinite Sequences and Series

Infinite Sequences and Series Chapter 6 Ifiite Sequeces ad Series 6.1 Ifiite Sequeces 6.1.1 Elemetary Cocepts Simply speakig, a sequece is a ordered list of umbers writte: {a 1, a 2, a 3,...a, a +1,...} where the elemets a i represet

More information

II. Descriptive Statistics D. Linear Correlation and Regression. 1. Linear Correlation

II. Descriptive Statistics D. Linear Correlation and Regression. 1. Linear Correlation II. Descriptive Statistics D. Liear Correlatio ad Regressio I this sectio Liear Correlatio Cause ad Effect Liear Regressio 1. Liear Correlatio Quatifyig Liear Correlatio The Pearso product-momet correlatio

More information

6.451 Principles of Digital Communication II Wednesday, March 9, 2005 MIT, Spring 2005 Handout #12. Problem Set 5 Solutions

6.451 Principles of Digital Communication II Wednesday, March 9, 2005 MIT, Spring 2005 Handout #12. Problem Set 5 Solutions 6.51 Priciples of Digital Commuicatio II Weesay, March 9, 2005 MIT, Sprig 2005 Haout #12 Problem Set 5 Solutios Problem 5.1 (Eucliea ivisio algorithm). (a) For the set F[x] of polyomials over ay fiel F,

More information

MATH 1080: Calculus of One Variable II Fall 2017 Textbook: Single Variable Calculus: Early Transcendentals, 7e, by James Stewart.

MATH 1080: Calculus of One Variable II Fall 2017 Textbook: Single Variable Calculus: Early Transcendentals, 7e, by James Stewart. MATH 1080: Calculus of Oe Variable II Fall 2017 Textbook: Sigle Variable Calculus: Early Trascedetals, 7e, by James Stewart Uit 3 Skill Set Importat: Studets should expect test questios that require a

More information

[ 11 ] z of degree 2 as both degree 2 each. The degree of a polynomial in n variables is the maximum of the degrees of its terms.

[ 11 ] z of degree 2 as both degree 2 each. The degree of a polynomial in n variables is the maximum of the degrees of its terms. [ 11 ] 1 1.1 Polyomial Fuctios 1 Algebra Ay fuctio f ( x) ax a1x... a1x a0 is a polyomial fuctio if ai ( i 0,1,,,..., ) is a costat which belogs to the set of real umbers ad the idices,, 1,...,1 are atural

More information

Chapter 7: Numerical Series

Chapter 7: Numerical Series Chapter 7: Numerical Series Chapter 7 Overview: Sequeces ad Numerical Series I most texts, the topic of sequeces ad series appears, at first, to be a side topic. There are almost o derivatives or itegrals

More information

Paired Data and Linear Correlation

Paired Data and Linear Correlation Paired Data ad Liear Correlatio Example. A group of calculus studets has take two quizzes. These are their scores: Studet st Quiz Score ( data) d Quiz Score ( data) 7 5 5 0 3 0 3 4 0 5 5 5 5 6 0 8 7 0

More information