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

Save this PDF as:

Size: px
Start display at page:

Download "Precalculus MATH Sections 3.1, 3.2, 3.3. Exponential, Logistic and Logarithmic Functions" ## Transcription

1 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 of fuctios i terms of their applicatio. The mai reaso for this is because their rate of chage is proportioal to their amout at ay time. Form: Let f( ) = ab. Fid ad simplify f( + 1). How is this ew fuctio related to f? b is called the growth factor Domai Rage Itercepts Asymptotes Graphs b > 0, 0< b < 1 Epoetial Growth or Decay. Let a =1. The, f( ) = b

2 Liear Data vs Epoetial Eample: Determie if the followig data defies a liear or epoetial fuctio model. (a) y (b) y Applicatio: Compoud Iterest Formula: Derived from the simple iterest formula I = Pr t Eample: Suppose \$15,000 is ivested at a iterest rate of 7% compouded aually. Calculate the accout balace after 10 years. Eample: Now suppose the pricipal was compouded mothly. Calculate the accout balace for the same time period.

3 The Natural Epoetial Fuctio Cotiuous Compoudig Suppose \$1 was ivested ito a accout payig 100% aual iterest. Fill i the table below where is the umber of compoudig periods i oe year ad A is the amout i the accout after 1 year This ca be epressed i the followig way: as, 1+ e Or i limit otatio as: lim 1+ = e From this limit, the cotiuous compoudig of iterest formula ca be derived Where A is the amout at ay time t, P is the pricipal or iitial deposit or amout ivested r is the aual rate of iterest ad t is the time i years Eample: Suppose you ivest \$6,000 ito a accout that ears 8% iterest compouded cotiuously. How much will you have i the accout thirty years from ow?

4 Natural Epoetial Growth ad Decay Models: Applicatios: Eample: A job offer at a recet college graduates job fair offers a startig salary of \$44,000 with a guarateed raise of 6% each year. Fid a fuctio f that computes the salary durig the th year. Eample: A bacteria culture iitially cotais 10,000 bacteria ad is foud to double i size every 4 hours. Fid a epoetial model of the form f ( ) = Ca that represets the amout of bacterial i the culture at ay time hours. Eample: Bacterial Growth p 263 # 37 Eample: Radioactive Decay p 271 # 33 Eample: Half Life p 272 # 40 Note: the 3 rd problem above ca be modeled with a cotiuous decay model. I order to do this we eed to be able to solve epoetial equatios of the form e = b, where b is a positive costat. We could approimate solutios usig a calculator, however, there is a better way

5 Logistic Growth Aimal populatios are ot capable of urestricted growth because of limited habitat ad food supplies. Uder such coditios the populatio follows a logistic growth model: c c Pt () = or Pt () = kt 1 + ab 1 + ae kt where c, a, ad k are positive costats. For a certai fish populatio i a small pod c = 1200, k = 11, ad a = 0.2 ad t is measured i years. The fish were itroduced ito the pod at time t = 0. Usig the secod form for logistic growth (a) How may fish were origially put ito the pod? (b) Fid the populatio after 10, 20, ad 30 years. (c) Evaluate Pt () for large values of t. What does the populatio approach as t? (d) Approimate the value of t where the rate of chage of the populatio with respect to time is the greatest. Studets try P 263 # 56 P 272 # 45

6 Sectio 3.3 The Logarithmic Fuctio A logarithmic fuctio is best thought of as the iverse of a related epoetial fuctio. For eample, sketch a graph of the epoetial fuctio iverse of this fuctio. y = 2. Now, write a equatio for the Defiitio: The logarithmic fuctio with base b Domai *Iverse Properties Commo Logarithm Natural Logarithm Evaluatig Logarithms Eample: Evaluate the followig: (a) log381 (b) log4 64 Graphs ad Iverses (c) 0.67 l e Sketch the followig epoetial fuctios with their iverse o the same set of aes. Write a logarithmic equatio for each iverse. (a) f( ) = 2, (b) 1 f( ) = 2, (c) f( ) = 10, ad (d) f( ) = e.

7 Solvig Epoetial ad Logarithmic Equatios Eample: Usig the iverse property, covertig to logarithmic form, or equivalet base property, solve each of the followig epoetial equatios (a) 1 2 = (b) 2e = 7 (c) = Eample: Covertig to epoetial form or usig the iverse property, solve each of the followig logarithmic equatios (a) log6 = 3 (b) log3 = 4 (c) l = 6.2 (d) 6log2 3 = 12 Chage of Base Formula: log a log = log b b a

### Pre-calculus Guided Notes: Chapter 11 Exponential and Logarithmic Functions Name: Pre-calculus Guided Notes: Chapter 11 Epoetial ad Logarithmic Fuctios Sectio 2 Epoetial Fuctios Paret Fuctio: y = > 1 0 < < 1 Domai Rage y-itercept ehavior Horizotal asymptote Vertical asymptote

### 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

### 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.

### 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

### Section 6.4: Series. Section 6.4 Series 413 ectio 64 eries 4 ectio 64: eries A couple decides to start a college fud for their daughter They pla to ivest \$50 i the fud each moth The fud pays 6% aual iterest, compouded mothly How much moey will they

### 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

### UNIT #5. Lesson #2 Arithmetic and Geometric Sequences. Lesson #3 Summation Notation. Lesson #4 Arithmetic Series. Lesson #5 Geometric Series UNIT #5 SEQUENCES AND SERIES Lesso # Sequeces Lesso # Arithmetic ad Geometric Sequeces Lesso #3 Summatio Notatio Lesso #4 Arithmetic Series Lesso #5 Geometric Series Lesso #6 Mortgage Paymets COMMON CORE

### End of year exam. Final Exam Review. 1.What is the inverse of the function Which transformations of the graph of. x will produce the graph of Name Date lass 1.What is the iverse of the fuctio f ( )? f 1 ( ) f 1 ( ) f ( ) ( ) 1 1. What trasformatios o the graph of f ( ) result i the graph of g( )? Traslate right by uits ad dow by uits. Stretch

### 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

### 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

### Castiel, Supernatural, Season 6, Episode 18 13 Differetial Equatios the aswer to your questio ca best be epressed as a series of partial differetial equatios... Castiel, Superatural, Seaso 6, Episode 18 A differetial equatio is a mathematical equatio

### The picture in figure 1.1 helps us to see that the area represents the distance traveled. Figure 1: Area represents distance travelled 1 Lecture : Area Area ad distace traveled Approximatig area by rectagles Summatio The area uder a parabola 1.1 Area ad distace Suppose we have the followig iformatio about the velocity of a particle, how

### ( ) 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

### 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

### In exercises 1 and 2, (a) write the repeating decimal as a geometric series and (b) write its sum as the ratio of two integers _ Chapter 9 Curve I eercises ad, (a) write the repeatig decimal as a geometric series ad (b) write its sum as the ratio of two itegers _.9.976 Distace A ball is dropped from a height of 8 meters. Each time

### 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

### Math 1030Q Spring Exam #2 Review. (9.2) 8x = log (9.2) 8x = log.32 8x log 9.2 log log log 9.2. x =.06418 Math 1030Q Sprig 2013 Exam #2 Review 1. Solve for x: 2 + 9.2 8x = 2.32 Solutio: 2 + 9.2 8x = 2.32 9.2 8x = 2.32 2 log 9.2 8x = log.32 8x log 9.2 = log.32 8x log 9.2 log.32 = 8 log 9.2 8 log 9.2 x =.06418

### Algebra 1 Hour Final Exam Review Days Semester Fial Eam Review Packet Name Algebra 1 Hour Fial Eam Review Days Assiged o Assigmet 6/6 Fial Eam Review Chapters 11 ad 1 Problems 54 7 6/7 Fial Eam Review Chapters 10 Problems 44 5 6/8 Fial Eam

### (c) Write, but do not evaluate, an integral expression for the volume of the solid generated when R is Calculus BC Fial Review Name: Revised 7 EXAM Date: Tuesday, May 9 Remiders:. Put ew batteries i your calculator. Make sure your calculator is i RADIAN mode.. Get a good ight s sleep. Eat breakfast. Brig:

### MATH CALCULUS II Objectives and Notes for Test 4 MATH 44 - CALCULUS II Objectives ad Notes for Test 4 To do well o this test, ou should be able to work the followig tpes of problems. Fid a power series represetatio for a fuctio ad determie the radius

### HONORS ALGEBRA 2 FINAL REVIEW Chapters 6, 7, 8, and 10 Name Date Sectios ad Scorig HONORS ALGEBRA FINAL REVIEW 0-8 Chapters 6,, 8, ad 0 Your fial eam will test your kowledge of the topics we studied i the secod half of the school year. There will be two sectios

### Lesson 1.1 Recursively Defined Sequences Lesso 1.1 Recursively Defied Sequeces 1. Tell whether each sequece is arithmetic, geometric, or either. a. 1,, 9, 13,... b. 2, 6, 18, 4,... c. 1, 1, 2, 3,, 8,... d. 16, 4, 1,.2,... e. 1, 1, 1, 1,... f..6,

### 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

### 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=

### 3.1 & 3.2 SEQUENCES. Definition 3.1: A sequence is a function whose domain is the positive integers (=Z ++ ) 3. & 3. SEQUENCES Defiitio 3.: A sequece is a fuctio whose domai is the positive itegers (=Z ++ ) Examples:. f() = for Z ++ or, 4, 6, 8, 0,. a = +/ or, ½, / 3, ¼, 3. b = /² or, ¼, / 9, 4. c = ( ) + or

### Section 7 Fundamentals of Sequences and Series ectio Fudametals of equeces ad eries. Defiitio ad examples of sequeces A sequece ca be thought of as a ifiite list of umbers. 0, -, -0, -, -0...,,,,,,. (iii),,,,... Defiitio: A sequece is a fuctio which

### ARITHMETIC PROGRESSIONS CHAPTER 5 ARITHMETIC PROGRESSIONS (A) Mai Cocepts ad Results A arithmetic progressio (AP) is a list of umbers i which each term is obtaied by addig a fixed umber d to the precedig term, except the first

### 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

### 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

### Math 2412 Review 3(answers) kt Math 4 Review 3(aswers) kt A t A e. If the half-life of radium is 690 years, ad you have 0 grams ow, how much will be preset i 50 years (rouded to three decimal places)?. The decay of radium is modeled

### 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

### 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 +... +

### Lecture 3. Digital Signal Processing. Chapter 3. z-transforms. Mikael Swartling Nedelko Grbic Bengt Mandersson. rev. 2016 Lecture 3 Digital Sigal Processig Chapter 3 z-trasforms Mikael Swartlig Nedelko Grbic Begt Madersso rev. 06 Departmet of Electrical ad Iformatio Techology Lud Uiversity z-trasforms We defie the z-trasform

### 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

### 6.3 Testing Series With Positive Terms 6.3. TESTING SERIES WITH POSITIVE TERMS 307 6.3 Testig Series With Positive Terms 6.3. Review of what is kow up to ow I theory, testig a series a i for covergece amouts to fidig the i= sequece of partial

### Compound Interest. S.Y.Tan. Compound Interest Compoud Iterest S.Y.Ta Compoud Iterest The yield of simple iterest is costat all throughout the ivestmet or loa term. =2000 r = 0% = 0. t = year =? I =? = 2000 (+ (0.)()) = 3200 I = - = 3200-2000 = 200

### 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

### MIXED REVIEW of Problem Solving MIXED REVIEW of Problem Solvig STATE TEST PRACTICE classzoe.com Lessos 2.4 2.. MULTI-STEP PROBLEM A ball is dropped from a height of 2 feet. Each time the ball hits the groud, it bouces to 70% of its previous

### Sect 5.3 Proportions Sect 5. Proportios Objective a: Defiitio of a Proportio. A proportio is a statemet that two ratios or two rates are equal. A proportio takes a form that is similar to a aalogy i Eglish class. For example,

### MAT 271 Project: Partial Fractions for certain rational functions MAT 7 Project: Partial Fractios for certai ratioal fuctios Prerequisite kowledge: partial fractios from MAT 7, a very good commad of factorig ad complex umbers from Precalculus. To complete this project,

### Example 2. Find the upper bound for the remainder for the approximation from Example 1. Lesso 8- Error Approimatios 0 Alteratig Series Remaider: For a coverget alteratig series whe approimatig the sum of a series by usig oly the first terms, the error will be less tha or equal to the absolute

### Appendix: The Laplace Transform Appedix: The Laplace Trasform The Laplace trasform is a powerful method that ca be used to solve differetial equatio, ad other mathematical problems. Its stregth lies i the fact that it allows the trasformatio

### Section 1 of Unit 03 (Pure Mathematics 3) Algebra Sectio 1 of Uit 0 (Pure Mathematics ) Algebra Recommeded Prior Kowledge Studets should have studied the algebraic techiques i Pure Mathematics 1. Cotet This Sectio should be studied early i the course

### MATHEMATICS. 61. The differential equation representing the family of curves where c is a positive parameter, is of MATHEMATICS 6 The differetial equatio represetig the family of curves where c is a positive parameter, is of Order Order Degree (d) Degree (a,c) Give curve is y c ( c) Differetiate wrt, y c c y Hece differetial

### Section 13.3 Area and the Definite Integral Sectio 3.3 Area ad the Defiite Itegral We ca easily fid areas of certai geometric figures usig well-kow formulas: However, it is t easy to fid the area of a regio with curved sides: METHOD: To evaluate

### Recursive Algorithms. Recurrences. Recursive Algorithms Analysis Recursive Algorithms Recurreces Computer Sciece & Egieerig 35: Discrete Mathematics Christopher M Bourke cbourke@cseuledu A recursive algorithm is oe i which objects are defied i terms of other objects

### Subject: Differential Equations & Mathematical Modeling -III. Lesson: Power series solutions of Differential Equations. about ordinary points Power series solutio of Differetial equatios about ordiary poits Subject: Differetial Equatios & Mathematical Modelig -III Lesso: Power series solutios of Differetial Equatios about ordiary poits Lesso

### 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

### Most text will write ordinary derivatives using either Leibniz notation 2 3. y + 5y= e and y y. xx tt t Itroductio to Differetial Equatios Defiitios ad Termiolog Differetial Equatio: A equatio cotaiig the derivatives of oe or more depedet variables, with respect to oe or more idepedet variables, is said

### 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.

### 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.

### John Riley 30 August 2016 Joh Riley 3 August 6 Basic mathematics of ecoomic models Fuctios ad derivatives Limit of a fuctio Cotiuity 3 Level ad superlevel sets 3 4 Cost fuctio ad margial cost 4 5 Derivative of a fuctio 5 6 Higher

### 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

### Comparison Study of Series Approximation. and Convergence between Chebyshev. and Legendre Series Applied Mathematical Scieces, Vol. 7, 03, o. 6, 3-337 HIKARI Ltd, www.m-hikari.com http://d.doi.org/0.988/ams.03.3430 Compariso Study of Series Approimatio ad Covergece betwee Chebyshev ad Legedre Series

### COMPUTING SUMS AND THE AVERAGE VALUE OF THE DIVISOR FUNCTION (x 1) + x = n = n. COMPUTING SUMS AND THE AVERAGE VALUE OF THE DIVISOR FUNCTION Abstract. We itroduce a method for computig sums of the form f( where f( is ice. We apply this method to study the average value of d(, where

### 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

### 9.3 The INTEGRAL TEST; p-series Lecture 9.3 & 9.4 Math 0B Nguye of 6 Istructor s Versio 9.3 The INTEGRAL TEST; p-series I this ad the followig sectio, you will study several covergece tests that apply to series with positive terms. Note

### Once we have a sequence of numbers, the next thing to do is to sum them up. Given a sequence (a n ) n=1 . Ifiite Series Oce we have a sequece of umbers, the ext thig to do is to sum them up. Give a sequece a be a sequece: ca we give a sesible meaig to the followig expressio? a = a a a a While summig ifiitely

### 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

### 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,

### Estimation for Complete Data Estimatio for Complete Data complete data: there is o loss of iformatio durig study. complete idividual complete data= grouped data A complete idividual data is the oe i which the complete iformatio of

### 6.1. Sequences as Discrete Functions. Investigate 6.1 Sequeces as Discrete Fuctios The word sequece is used i everyday laguage. I a sequece, the order i which evets occur is importat. For example, builders must complete work i the proper sequece to costruct

### WORKING WITH NUMBERS 1 WORKING WITH NUMBERS WHAT YOU NEED TO KNOW The defiitio of the differet umber sets: is the set of atural umbers {0, 1,, 3, }. is the set of itegers {, 3,, 1, 0, 1,, 3, }; + is the set of positive itegers;

### 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

### 62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + 62. Power series Defiitio 16. (Power series) Give a sequece {c }, the series c x = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + is called a power series i the variable x. The umbers c are called the coefficiets of

### 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

### multiplies all measures of center and the standard deviation and range by k, while the variance is multiplied by k 2. Lesso 3- Lesso 3- Scale Chages of Data Vocabulary scale chage of a data set scale factor scale image BIG IDEA Multiplyig every umber i a data set by k multiplies all measures of ceter ad the stadard deviatio

### Math 113 Exam 3 Practice Math Exam Practice Exam will cover.-.9. This sheet has three sectios. The first sectio will remid you about techiques ad formulas that you should kow. The secod gives a umber of practice questios for you

### 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

### Exponents. Learning Objectives. Pre-Activity Sectio. Pre-Activity Preparatio Epoets A Chai Letter Chai letters are geerated every day. If you sed a chai letter to three frieds ad they each sed it o to three frieds, who each sed it o to three frieds,

### GRADE 12 SEPTEMBER 2015 MATHEMATICS P1 NATIONAL SENIOR CERTIFICATE GRADE 1 SEPTEMBER 015 MATHEMATICS P1 MARKS: 150 TIME: 3 hours *MATHE1* This questio paper cosists of 10 pages, icludig a iformatio sheet. MATHEMATICS P1 (EC/SEPTEMBER 015) INSTRUCTIONS

### You may work in pairs or purely individually for this assignment. CS 04 Problem Solvig i Computer Sciece OOC Assigmet 6: Recurreces You may work i pairs or purely idividually for this assigmet. Prepare your aswers to the followig questios i a plai ASCII text file or

### a is some real number (called the coefficient) other Precalculus Notes for Sectio.1 Liear/Quadratic Fuctios ad Modelig http://www.schooltube.com/video/77e0a939a3344194bb4f Defiitios: A moomial is a term of the form tha zero ad is a oegative iteger. a where

### CHAPTER I: Vector Spaces CHAPTER I: Vector Spaces Sectio 1: Itroductio ad Examples This first chapter is largely a review of topics you probably saw i your liear algebra course. So why cover it? (1) Not everyoe remembers everythig

### Objective Mathematics 6. If si () + cos () =, the is equal to :. If <

### 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)

### Recurrence Relations Recurrece Relatios Aalysis of recursive algorithms, such as: it factorial (it ) { if (==0) retur ; else retur ( * factorial(-)); } Let t be the umber of multiplicatios eeded to calculate factorial(). The

### 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.

### Student s Printed Name: Studet s Prited Name: Istructor: XID: C Sectio: No questios will be aswered durig this eam. If you cosider a questio to be ambiguous, state your assumptios i the margi ad do the best you ca to provide

### 10.1 Sequences. n term. We will deal a. a n or a n n. ( 1) n ( 1) n 1 2 ( 1) a =, 0 0,,,,, ln n. n an 2. n term. 0. Sequeces A sequece is a list of umbers writte i a defiite order: a, a,, a, a is called the first term, a is the secod term, ad i geeral eclusively with ifiite sequeces ad so each term Notatio: the sequece

### MATHEMATICS (Three hours and a quarter) MATHEMATICS (Three hours ad a quarter) (The first fiftee miutes of the eamiatio are for readig the paper oly. Cadidates must NOT start writig durig this time.) Aswer Questio from Sectio A ad questios from

### 1 6 = 1 6 = + Factorials and Euler s Gamma function Royal Holloway Uiversity of Lodo Departmet of Physics Factorials ad Euler s Gamma fuctio Itroductio The is a self-cotaied part of the course dealig, essetially, with the factorial fuctio ad its geeralizatio

### 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

### Sigma notation. 2.1 Introduction Sigma otatio. Itroductio We use sigma otatio to idicate the summatio process whe we have several (or ifiitely may) terms to add up. You may have see sigma otatio i earlier courses. It is used to idicate

### is also known as the general term of the sequence Lesso : Sequeces ad Series Outlie Objectives: I ca determie whether a sequece has a patter. I ca determie whether a sequece ca be geeralized to fid a formula for the geeral term i the sequece. I ca determie

### Mathematics Extension 1 016 Bored of Studies Trial Eamiatios Mathematics Etesio 1 3 rd ctober 016 Geeral Istructios Total Marks 70 Readig time 5 miutes Workig time hours Write usig black or blue pe Black pe is preferred Board-approved

### 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

### 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

### Time-Domain Representations of LTI Systems 2.1 Itroductio Objectives: 1. Impulse resposes of LTI systems 2. Liear costat-coefficiets differetial or differece equatios of LTI systems 3. Bloc diagram represetatios of LTI systems 4. State-variable

### μ are complex parameters. Other A New Numerical Itegrator for the Solutio of Iitial Value Problems i Ordiary Differetial Equatios. J. Suday * ad M.R. Odekule Departmet of Mathematical Scieces, Adamawa State Uiversity, Mubi, Nigeria.

### NATIONAL SENIOR CERTIFICATE GRADE 12 NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P FEBRUARY/MARCH 04 MARKS: 50 TIME: 3 hours This questio paper cosists of 9 pages, diagram sheet ad iformatio sheet. Please tur over Mathematics/P DBE/Feb.

### A sequence of numbers is a function whose domain is the positive integers. We can see that the sequence Sequeces A sequece of umbers is a fuctio whose domai is the positive itegers. We ca see that the sequece,, 2, 2, 3, 3,... is a fuctio from the positive itegers whe we write the first sequece elemet as

### MATH 1A FINAL (7:00 PM VERSION) SOLUTION. (Last edited December 25, 2013 at 9:14pm.) MATH A FINAL (7: PM VERSION) SOLUTION (Last edited December 5, 3 at 9:4pm.) Problem. (i) Give the precise defiitio of the defiite itegral usig Riema sums. (ii) Write a epressio for the defiite itegral

### 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 Pre-alculus Practice Eam MULTIPLE-HOIE (alculator permitted ). Solve cos = si, 0 0.9 0.40,.5 c. 0.79 d. 0.79,.8. Determie the equatio of a circle with cetre ( 0,0) passig through the poit P (,5) + = c. Mathematics 6 HWK Solutios 8. p580 A abbreviatio: iff is a abbreviatio for if ad oly if. Geometric Series: Several of these problems use what we worked out i class cocerig the geometric series, which I