More Properties of Limits: Order of Operations


 Irene Clemence Perkins
 2 years ago
 Views:
Transcription
1 math 30 day 5: calculating its 6 More Proerties of Limits: Order of Oerations THEOREM 45 (Order of Oerations, Continued) Assume that!a f () L and that m and n are ositive integers Then 5 (Power)!a [ f ()] n [!a f ()] n L n 6 (Fractional Power) Assume that m n is reduced Then h i n/m [ f!a ()]n/m f () L n/m,!a rovided that f () 0 for near a if m is even EXAMPLE 44 Determine!3 (4 ) 5 Indicate which it roerties were used at each ste SOLUTION (4!3 0)5 Powers [ 4 0] 5 Diff [ 4 0] 5!3!3!3 EXAMPLE 45 Determine! each ste Const Mult [4!3!3 0] 5 Thm 5,5 [4(3) 0] 5 () 5 3 Indicate which it roerties were used at SOLUTION Notice that ( ) / is a fractional ower function In n the language of Theorem 45, m is reduced and m is even Near, f () is ositive So Theorem 45 alies and we may calculate the it as! / Frac Pow Poly! (3) / 3 EXAMPLE 46 Determine! 3 ( + ) 4/3 Indicate which it roerties were used at each ste SOLUTION ( + ) 4/3 is a fractional ower function with m n 4 3 which is reduced and m 3 is odd Near, f () + is ositive So Theorem 45 alies and we may calculate the it as! 3 / Frac Pow Sum, Prod +! 3 + (8 3 + ) / 4 EXAMPLE 47 Determine! 3 ( 5) 3/4 SOLUTION ( 5) 3/4 is a fractional ower function In the language of Theorem 45, m n 3 4 is reduced and m 4 is even Near 3, 5 is negative Since ( 5) 3/4 is not even defined near 3, this it does not eist We now look at some secial cases of its with familiar functions THEOREM 46 (Secial Functions) Let n be a ositive integer and c be any constant 7 (Monomials)!a c n ca n 8 (Polynomials) If () c n n + c n n + + c + c 0 is a degree n olynomial, then () (a)!a
2 math 30 day 5: calculating its 7 9 (Rational Functions) If r() () is a rational function, then for any oint a in the q() domain of r() r() r(a)!a Theorem 46 says that the it of olynomial or rational function as! a is the same as the value of the function at a This is not true of all its For sin eamle, we saw that, yet we can t even ut 0 into this function!!0 Those secial or nice functions where!a f () f (a) are called continuous at a We will eamine them in deth in a few days For the moment we can say that olynomials are continuous everywhere and rational functions are continuous at every oint in their domains Proof Let s see how it roerties 7 through 9 follow from the revious roerties of its To rove the monomial roerty, use!a cn Const Mult c[ n ] Powers n Thm 5 c[ ] ca n!a!a To rove the olynomial roerty, since () c n n + c n n + + c + c 0 is a degree n olynomial, then () [c n n + c n n + + c + c 0 ]!a!a Sum!a c n n +!a c n n + +!a c +!a c 0 ] Monomial, Thm 5 c n a n + c n a n + + c a + c 0 (a) The rational function result is simler, still If r() () is a rational function, q() then () and q() are olynomials So for any oint a in the domain of r() (ie, q(a) 6 0), () Quotient r()!a () Polynomial (a)!a!a q()!a q() q(a) r(a) EXAMPLE 48 To see how these last results greatly simlify certain it calculations, 4 + let s determine! 3 + SOLUTION Since we have a rational function and the denominator is not 0 at, we see that 4 + Rational 4() + ()! 3 + 3()+ 7 That was easy! Several Cautions Most of the its we will encounter this term will not be so easy to determine While we will use the roerties we ve develoed and others below, most its will start off in the indeterminate form 0 Tyically we will need to 0 carry out some sort of algebraic maniulation to get the it in a form where the basic roerties aly For eamle, while!5 5 5
3 math 30 day 5: calculating its 8 is a rational function, roerty 8 above does not aly to the calculation of the it since 5 (the number is aroaching) is not in the domain of the function Consequently, some algebraic maniulation (in this case factoring) is required 5!5 5 ( 5)( + 5) + 5 Poly 0!5 5!5 There are two additional things to notice The first is mathematical grammar We continue to use the it symbol u until the actual numerical evaluation takes lace Writing something such as the following is simly wrong:!5 ((((((((((((((((((((h hhhhhhhhhhhhhhhhhhh 5 ( 5)( + 5) Among other things, the function + 5 is not the same as the constant 0 An even worse calculation to write is X XX X5 X XX! X X or even ((((((((((((((h!5 hhhhhhhhhhhhh Undefined The eression 0 0 is indeed not defined (and is certainly not equal to ) However, the it is indeterminate Near (but not equal to) 5, the fraction is not yet 0 0 You need to do more work to determine the it The work may involve factoring or other algebraic methods to simlify the eression so that we can more easily see what it is aroaching Another thing to notice is that 5 5 and + 5 are the same function as long as 6 5 where the first function is not defined but the second is However, we are interested in a it as! 5 so remember that this involves being close to, but 5 not equal to, 5 Consequently and + 5 are indeed the same!!5 5!5 43 Onesided Limits We have now stated a number of roerties for its All of these roerties also hold for onesided its, as well, with a slight modification for fractional owers THEOREM 47 (Onesided Limit Proerties) Limit roerties through 9 (the constant multile, sum, difference, roduct, quotient, integer ower, olynomial, and rational function rules) continue to hold for onesided its with the following modification for fractional owers Assume that m and n are ositive integers and that m n is reduced Then ale n/m (a)!a +[ f ()]n/m f () rovided that f () 0 for near a with > a if m!a + is even ale n/m (b) [ f!a ()]n/m f () rovided that f () 0 for near a with > a if m!a is even The net few eamles illustrate the use of it roerties with iecewise functions ( 3 +, if < EXAMPLE 49 Let f () Determine the following its if they if eist (a)! f () (b) f () (c) f () (d) f ()! +!!0
4 math 30 day 5: calculating its 9 SOLUTION We must be careful to use the correct definition of f for each it (a) As! from the left, is less than so f () 3 + there Thus! f () <! 3 + Poly 3( ) + 3 (b) As! from the right, is greater than so f () Thus f () > Root ! +! + (c) To determine f () we comare the one sided its Since f () 6!! + f (), we conclude that f () DNE!! (d) To determine f () we see that the values of near 0 are less than So!0 f () 3 + there So f () < 3 + Poly!0!0 We don t need to use the other definition for f since it does not aly to values of near 0 8 >< 3, if ale EXAMPLE 40 Let f () +, if < ale 5 Determine the following its if >: + if > 5 they eist (a) (d)!!5 f () (b) f ()! + (c)! f () (e) f ()!5 + (f )!5 SOLUTION We must be careful to use the correct definition of f for each it Note how we choose the function! (a) (b)! f () <! 3 Poly < ale 5 f ()! +! + Poly + (c) Since f ()! +! (d) (e)!5 f () < ale 5!5 + Poly 6 f () > 5!5 +!5 + + (f ) Since f () 6!5 +!5 f (), we conclude that! f () 5 6 f (), we conclude that!5 f () DNE 44 Most Limits Are Not Simle Let s return to the original motivation for calculating its We were interested in finding the sloe of a curve and this led to looking at its that have the form!a f () f (a) a Assuming that f is continuous, this it cannot be evaluated by any of the basic it roerties since the denominator is aroaching 0 More secifically, as! a, this difference quotient has the indeterminate form 0 0 To evaluate this it we must do more work Let s look at an EXAMPLE 4 Let f () Determine the sloe of this curve right at
5 math 30 day 5: calculating its 0 SOLUTION To find the sloe of a curve we must evaluate the difference quotient f () f (4) ( 3 + ) 5!4 4!4 4 Though this is a rational function, the it roerties do not aly since the denominator is 0 at 4, and so is the numerator (check it!) Instead, we must do more work f () f (4) 3 4 ( 4)( + ) + Poly 5!4 4!4 4!4 4!4 Only at the very last ste were we able to use a it roerty The Indeterminate Form 0 0 Many of the most imortant its we will see in the course have the indeterminate form 0 0 as in the revious eamle To evaluate such its, if they eist, requires more work tyically of the following tye factoring using conjugates simlifying making use of known its Let s look at some eamles of each Recall that if a > 0, then a + b and a b are called conjugates Notice that ( a + b)( a b) a b There is no middle term EXAMPLE 4 (Factoring) Factoring is one of the most critical tools in evaluating the sorts of its that arise in elementary calculus Evaluate! + 8 SOLUTION Notice that this it has the indeterminate form 0 0 Factoring is the key %0 ( )( )! + 8 &0! ( + 4)( ) ( ) Rational! Only at the very last ste were we able to use a it roerty EXAMPLE 43 (Factoring) Evaluate! 3 + SOLUTION This it has the indeterminate form 0 0 Factoring is the key %0! 3 + & 0 ( + 6)( + ) ( + 6)! ( + )! 4 4 Only at the very last ste were we able to use a it roerty EXAMPLE 44 (Conjugates) Evaluate!4 8) SOLUTION Notice that this it has the indeterminate form 0 0 Let s see how conjugates hel!4 % 0 ( 4) &0! !4 ( 8)( + )!4 4 (( 4)( + )!4 ( + ) EXAMPLE 45 (Conjugates) Here s another: Evaluate! Root
6 math 30 day 5: calculating its SOLUTION Notice that this it has the indeterminate form 0 0 Use conjugates again! % 0 &0! ( + 4) 6! ( )( )! ( )( )! Root 6 EXAMPLE 46 (Conjugates) Evaluate! + 3 SOLUTION This it has the indeterminate form 0 0! % &0! ) ( )(! ( + 3) 4 ( )( + )( )!! ( + )( ) Prod, Root ( + ) 8 EXAMPLE 47 (Simlification) Sometimes its, like this net one, involve comound + fractions One method of attack is to carefully simlify them Evaluate! SOLUTION Notice that this it has the indeterminate form 0 0 Use common denominators to simlify! + % 0 &0! EXAMPLE 48 (Simlification) Evaluate! ( ) (+)( ) 4 %0! ( + )( )( ) &0 Rational! ( + )( ) 3 + SOLUTION Notice that this it has the indeterminate form 0 0 Use common denominators to simlify! + % 0 &0! (+) (+ %0! ( + )( ) &0! ( + ) EXAMPLE 49 (Simlification) Evaluate h!0 +h h SOLUTION Notice that this it has the indeterminate form 0 0 h!0 +h h &0 % 0 h!0 (+h) (+h)() h h h!0 ( + h)()(h) h!0 ( + h)()
7 math 30 day 5: calculating its EXAMPLE 40 (Simlification) Evaluate! 4 +7 SOLUTION Notice that this it has the indeterminate form 0 0 Use common denominators to simlify! 4 +7! 8 ( +7) ( +7)! ( +7)! ( )( + ) ( + 7)( ) ( + )! ( + 7) Practice Problems EXAMPLE 4 (Simlification) Evaluate!4 4 SOLUTION Notice that this it has the indeterminate form 0 0 Use factoring to simlify this rational function ( 4)( ) Linear!4 4!4 4!4 + EXAMPLE 4 (Simlification) Evaluate! 4 SOLUTION Notice that this it has the indeterminate form 0 0 Use factoring to simlify this rational function! + 4! + ( )( + )! 3 0 EXAMPLE 43 (Simlification) Evaluate!5 5 4 SOLUTION Notice that this it has the indeterminate form 0 0 Use factoring to simlify this rational function 3 0 ( + )( 5)!5 5!5 ( 5)( + 5) + 7! EXAMPLE 44 (Simlification) Evaluate! SOLUTION Notice that this it has the indeterminate form 0 0 Use factoring to simlify this rational function! 3 5 6! EXAMPLE 45 (Simlification) Evaluate!0 ( ) ( + )( 6) ( )( + )! ( + )( 6)! ( ) 6 SOLUTION Notice that this it has the indeterminate form 0 0 Use common denominators to simlify! !0 3 3(+) !0 6!0 ( + ) 6 6! +
8 math 30 day 5: calculating its 3 EXAMPLE 46 (Simlification) Evaluate! 4 SOLUTION Notice that this it has the indeterminate form 0 0 Use common denominators to simlify! 4! 4 4! EXAMPLE 47 (Simlification) Evaluate! 4 (4 )( ) ( )( + ) +! (4 )( )! SOLUTION Notice that this it has the indeterminate form 0 0 Use common denominators to simlify! + ( +)!! ( +)! EXAMPLE 48 (Simlification) Evaluate! ( + )( ) ( )( + )! ( + )( ) ( + )! ( + ) 4 SOLUTION Notice that this it has the indeterminate form 0 0 Use conjugates to simlify!! + +! ( )( + )! 3 EXAMPLE 49 (Simlification) Evaluate!3 + + Root SOLUTION Notice that this it has the indeterminate form 0 0 Use conjugates to simlify! ! EXAMPLE 430 (Simlification) Evaluate!0 4 ( 3)( + + )! ) ( 3)(!3 3 Root + + 4!3 SOLUTION Notice that this it has the indeterminate form 0 0 Use conjugates to simlify (4 ) 4!0!0 4 +!0 ( )( 4 + )!0 ( )( 4 + )!0 ( )( 4 + )!0 ( )( 4 + ) Root, Prod 4 4
1. The algebra of exponents 1.1. Natural Number Powers. It is easy to say what is meant by a n a (raised to) to the (power) n if n N.
CHAPTER 3: EXPONENTS AND POWER FUNCTIONS 1. The algebra of exponents 1.1. Natural Number Powers. It is easy to say what is meant by a n a (raised to) to the (power) n if n N. For example: In general, if
More informationThe Cubic Formula. The quadratic formula tells us the roots of a quadratic polynomial, a polynomial of the form ax 2 + bx + c. The roots (if b 2 b+
The Cubic Formula The quadratic formula tells us the roots of a quadratic olynomial, a olynomial of the form ax + bx + c. The roots (if b b+ 4ac 0) are b 4ac a and b b 4ac a. The cubic formula tells us
More informationAs we have seen, there is a close connection between Legendre symbols of the form
Gauss Sums As we have seen, there is a close connection between Legendre symbols of the form 3 and cube roots of unity. Secifically, if is a rimitive cube root of unity, then 2 ± i 3 and hence 2 2 3 In
More informationComplex Conjugation and Polynomial Factorization
Comlex Conjugation and Polynomial Factorization Dave L. Renfro Summer 2004 Central Michigan University I. The Remainder Theorem Let P (x) be a olynomial with comlex coe cients 1 and r be a comlex number.
More informationClassical Fourier Series Introduction: Real Fourier Series
Math 344 May 1, 1 Classical Fourier Series Introduction: Real Fourier Series The orthogonality roerties of the sine and cosine functions make them good candidates as basis functions when orthogonal eansions
More informationMTH 3005  Calculus I Week 8: Limits at Infinity and Curve Sketching
MTH 35  Calculus I Week 8: Limits at Infinity and Curve Sketching Adam Gilbert Northeastern University January 2, 24 Objectives. After reviewing these notes the successful student will be prepared to
More information6.042/18.062J Mathematics for Computer Science December 12, 2006 Tom Leighton and Ronitt Rubinfeld. Random Walks
6.042/8.062J Mathematics for Comuter Science December 2, 2006 Tom Leighton and Ronitt Rubinfeld Lecture Notes Random Walks Gambler s Ruin Today we re going to talk about onedimensional random walks. In
More informationChapter 4. Polynomial and Rational Functions. 4.1 Polynomial Functions and Their Graphs
Chapter 4. Polynomial and Rational Functions 4.1 Polynomial Functions and Their Graphs A polynomial function of degree n is a function of the form P = a n n + a n 1 n 1 + + a 2 2 + a 1 + a 0 Where a s
More informationZeros of a Polynomial Function
Zeros of a Polynomial Function An important consequence of the Factor Theorem is that finding the zeros of a polynomial is really the same thing as factoring it into linear factors. In this section we
More informationMathematics 31 Precalculus and Limits
Mathematics 31 Precalculus and Limits Overview After completing this section, students will be epected to have acquired reliability and fluency in the algebraic skills of factoring, operations with radicals
More information3.1. RATIONAL EXPRESSIONS
3.1. RATIONAL EXPRESSIONS RATIONAL NUMBERS In previous courses you have learned how to operate (do addition, subtraction, multiplication, and division) on rational numbers (fractions). Rational numbers
More informationLIMITS AND CONTINUITY
LIMITS AND CONTINUITY 1 The concept of it Eample 11 Let f() = 2 4 Eamine the behavior of f() as approaches 2 2 Solution Let us compute some values of f() for close to 2, as in the tables below We see from
More informationPythagorean Triples and Rational Points on the Unit Circle
Pythagorean Triles and Rational Points on the Unit Circle Solutions Below are samle solutions to the roblems osed. You may find that your solutions are different in form and you may have found atterns
More informationSIMPLIFYING ALGEBRAIC FRACTIONS
Tallahassee Community College 5 SIMPLIFYING ALGEBRAIC FRACTIONS In arithmetic, you learned that a fraction is in simplest form if the Greatest Common Factor (GCF) of the numerator and the denominator is
More informationChapter 5 Review  Part I
Math 17 Chate Review Pat I Page 1 Chate Review  Pat I I. Tyes of Polynomials A. Basic Definitions 1. In the tem b m, b is called the coefficient, is called the vaiable, and m is called the eonent on the
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 information1.6 The Order of Operations
1.6 The Order of Operations Contents: Operations Grouping Symbols The Order of Operations Exponents and Negative Numbers Negative Square Roots Square Root of a Negative Number Order of Operations and Negative
More informationPrecalculus Prerequisites a.k.a. Chapter 0. August 16, 2013
Precalculus Prerequisites a.k.a. Chater 0 by Carl Stitz, Ph.D. Lakeland Community College Jeff Zeager, Ph.D. Lorain County Community College August 6, 0 Table of Contents 0 Prerequisites 0. Basic Set
More informationDiscrete Math I Practice Problems for Exam I
Discrete Math I Practice Problems for Exam I The ucoming exam on Thursday, January 12 will cover the material in Sections 1 through 6 of Chater 1. There may also be one question from Section 7. If there
More informationlecture 25: Gaussian quadrature: nodes, weights; examples; extensions
38 lecture 25: Gaussian quadrature: nodes, weights; examles; extensions 3.5 Comuting Gaussian quadrature nodes and weights When first aroaching Gaussian quadrature, the comlicated characterization of the
More informationQuadratic Equations and Inequalities
MA 134 Lecture Notes August 20, 2012 Introduction The purpose of this lecture is to... Introduction The purpose of this lecture is to... Learn about different types of equations Introduction The purpose
More informationRational Functions ( )
Rational Functions A rational function is a function of the form r P Q where P and Q are polynomials. We assume that P() and Q() have no factors in common, and Q() is not the zero polynomial. The domain
More informationAPPLICATIONS OF DIFFERENTIATION
4 APPLICATIONS OF DIFFERENTIATION APPLICATIONS OF DIFFERENTIATION So far, we have been concerned with some particular aspects of curve sketching: Domain, range, and symmetry (Chapter 1) Limits, continuity,
More information36 CHAPTER 1. LIMITS AND CONTINUITY. Figure 1.17: At which points is f not continuous?
36 CHAPTER 1. LIMITS AND CONTINUITY 1.3 Continuity Before Calculus became clearly de ned, continuity meant that one could draw the graph of a function without having to lift the pen and pencil. While this
More informationPre Cal 2 1 Lesson with notes 1st.notebook. January 22, Operations with Complex Numbers
0 2 Operations with Complex Numbers Objectives: To perform operations with pure imaginary numbers and complex numbers To use complex conjugates to write quotients of complex numbers in standard form Complex
More informationSolving Rational Equations
Lesson M Lesson : Student Outcomes Students solve rational equations, monitoring for the creation of extraneous solutions. Lesson Notes In the preceding lessons, students learned to add, subtract, multiply,
More informationTRANSCENDENTAL NUMBERS
TRANSCENDENTAL NUMBERS JEREMY BOOHER. Introduction The Greeks tried unsuccessfully to square the circle with a comass and straightedge. In the 9th century, Lindemann showed that this is imossible by demonstrating
More informationTaylor Polynomials and Taylor Series Math 126
Taylor Polynomials and Taylor Series Math 26 In many problems in science and engineering we have a function f(x) which is too complicated to answer the questions we d like to ask. In this chapter, we will
More informationMath 2250 Exam #1 Practice Problem Solutions. g(x) = x., h(x) =
Math 50 Eam # Practice Problem Solutions. Find the vertical asymptotes (if any) of the functions g() = + 4, h() = 4. Answer: The only number not in the domain of g is = 0, so the only place where g could
More informationSolutions to SelfTest for Chapter 4 c4sts  p1
Solutions to SelfTest for Chapter 4 c4sts  p1 1. Graph a polynomial function. Label all intercepts and describe the end behavior. a. P(x) = x 4 2x 3 15x 2. (1) Domain = R, of course (since this is a
More informationPOLYNOMIAL FUNCTIONS
POLYNOMIAL FUNCTIONS Polynomial Division.. 314 The Rational Zero Test.....317 Descarte s Rule of Signs... 319 The Remainder Theorem.....31 Finding all Zeros of a Polynomial Function.......33 Writing a
More informationCBus Voltage Calculation
D E S I G N E R N O T E S CBus Voltage Calculation Designer note number: 3121256 Designer: Darren Snodgrass Contact Person: Darren Snodgrass Aroved: Date: Synosis: The guidelines used by installers
More informationMain page. Given f ( x, y) = c we differentiate with respect to x so that
Further Calculus Implicit differentiation Parametric differentiation Related rates of change Small variations and linear approximations Stationary points Curve sketching  asymptotes Curve sketching the
More informationSTUDY GUIDE FOR SOME BASIC INTERMEDIATE ALGEBRA SKILLS
STUDY GUIDE FOR SOME BASIC INTERMEDIATE ALGEBRA SKILLS The intermediate algebra skills illustrated here will be used extensively and regularly throughout the semester Thus, mastering these skills is an
More information3.4 Complex Zeros and the Fundamental Theorem of Algebra
86 Polynomial Functions.4 Complex Zeros and the Fundamental Theorem of Algebra In Section., we were focused on finding the real zeros of a polynomial function. In this section, we expand our horizons and
More informationZeros of Polynomial Functions
Review: Synthetic Division Find (x 25x  5x 3 + x 4 ) (5 + x). Factor Theorem Solve 2x 35x 2 + x + 2 =0 given that 2 is a zero of f(x) = 2x 35x 2 + x + 2. Zeros of Polynomial Functions Introduction
More informationMath 5330 Spring Notes Prime Numbers
Math 5330 Sring 206 Notes Prime Numbers The study of rime numbers is as old as mathematics itself. This set of notes has a bunch of facts about rimes, or related to rimes. Much of this stuff is old dating
More information1.3 Algebraic Expressions
1.3 Algebraic Expressions A polynomial is an expression of the form: a n x n + a n 1 x n 1 +... + a 2 x 2 + a 1 x + a 0 The numbers a 1, a 2,..., a n are called coefficients. Each of the separate parts,
More informationSummer Mathematics Packet Say Hello to Algebra 2. For Students Entering Algebra 2
Summer Math Packet Student Name: Say Hello to Algebra 2 For Students Entering Algebra 2 This summer math booklet was developed to provide students in middle school an opportunity to review grade level
More information2.1 Simple & Compound Propositions
2.1 Simle & Comound Proositions 1 2.1 Simle & Comound Proositions Proositional Logic can be used to analyse, simlify and establish the equivalence of statements. A knowledge of logic is essential to the
More informationLecture 5 : Continuous Functions Definition 1 We say the function f is continuous at a number a if
Lecture 5 : Continuous Functions Definition We say the function f is continuous at a number a if f(x) = f(a). (i.e. we can make the value of f(x) as close as we like to f(a) by taking x sufficiently close
More informationArithmetic Operations. The real numbers have the following properties: In particular, putting a 1 in the Distributive Law, we get
Review of Algebra REVIEW OF ALGEBRA Review of Algebra Here we review the basic rules and procedures of algebra that you need to know in order to be successful in calculus. Arithmetic Operations The real
More informationThe Method of Partial Fractions Math 121 Calculus II Spring 2015
Rational functions. as The Method of Partial Fractions Math 11 Calculus II Spring 015 Recall that a rational function is a quotient of two polynomials such f(x) g(x) = 3x5 + x 3 + 16x x 60. The method
More informationZeros of Polynomial Functions
Zeros of Polynomial Functions Objectives: 1.Use the Fundamental Theorem of Algebra to determine the number of zeros of polynomial functions 2.Find rational zeros of polynomial functions 3.Find conjugate
More informationSOME PROPERTIES OF EXTENSIONS OF SMALL DEGREE OVER Q. 1. Quadratic Extensions
SOME PROPERTIES OF EXTENSIONS OF SMALL DEGREE OVER Q TREVOR ARNOLD Abstract This aer demonstrates a few characteristics of finite extensions of small degree over the rational numbers Q It comrises attemts
More informationAssignment 9; Due Friday, March 17
Assignment 9; Due Friday, March 17 24.4b: A icture of this set is shown below. Note that the set only contains oints on the lines; internal oints are missing. Below are choices for U and V. Notice that
More informationPRIME NUMBERS AND THE RIEMANN HYPOTHESIS
PRIME NUMBERS AND THE RIEMANN HYPOTHESIS CARL ERICKSON This minicourse has two main goals. The first is to carefully define the Riemann zeta function and exlain how it is connected with the rime numbers.
More informationPlacement Test Review Materials for
Placement Test Review Materials for 1 To The Student This workbook will provide a review of some of the skills tested on the COMPASS placement test. Skills covered in this workbook will be used on the
More informationMath 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers.
Math 0980 Chapter Objectives Chapter 1: Introduction to Algebra: The Integers. 1. Identify the place value of a digit. 2. Write a number in words or digits. 3. Write positive and negative numbers used
More informationLecture 21 and 22: The Prime Number Theorem
Lecture and : The Prime Number Theorem (New lecture, not in Tet) The location of rime numbers is a central question in number theory. Around 88, Legendre offered eerimental evidence that the number π()
More informationSample Problems. 2. Rationalize the denominator in each of the following expressions Simplify each of the following expressions.
Lecture Notes Radical Exressions age Samle Problems. Simlify each of the following exressions. Assume that a reresents a ositive number. a) b) g) 7 + 7 c) h) 7 x y d) 0 + i) j) x x + e) k) ( x) ( + x)
More informationEquations Warmup: Rules for manipulating equations
Equations Warmu: Rules or maniulating equations earning objectives: 3.A.3. to be able to rearrange equations using the ollowing rules: add or subtract the same thing to both sides i a b then a + c b +
More informationInverse Trigonometric Functions
Precalculus  MAT 75 Page: Inverse Trigonometric Functions Inverse Sine Function Remember that to have an inverse of a function, the function must be :. The Sine function is obviousl NOT a : function over
More information3.5 Summary of Curve Sketching
3.5 Summary of Curve Sketching Follow these steps to sketch the curve. 1. Domain of f() 2. and y intercepts (a) intercepts occur when f() = 0 (b) yintercept occurs when = 0 3. Symmetry: Is it even or
More informationMATH 10550, EXAM 2 SOLUTIONS. x 2 + 2xy y 2 + x = 2
MATH 10550, EXAM SOLUTIONS (1) Find an equation for the tangent line to at the point (1, ). + y y + = Solution: The equation of a line requires a point and a slope. The problem gives us the point so we
More informationSUNY ECC. ACCUPLACER Preparation Workshop. Algebra Skills
SUNY ECC ACCUPLACER Preparation Workshop Algebra Skills Gail A. Butler Ph.D. Evaluating Algebraic Epressions Substitute the value (#) in place of the letter (variable). Follow order of operations!!! E)
More informationPolynomials can be added or subtracted simply by adding or subtracting the corresponding terms, e.g., if
1. Polynomials 1.1. Definitions A polynomial in x is an expression obtained by taking powers of x, multiplying them by constants, and adding them. It can be written in the form c 0 x n + c 1 x n 1 + c
More informationMATH Fundamental Mathematics II.
MATH 10032 Fundamental Mathematics II http://www.math.kent.edu/ebooks/10032/funmath2.pdf Department of Mathematical Sciences Kent State University December 29, 2008 2 Contents 1 Fundamental Mathematics
More informationCalculus Card Matching
Card Matching Card Matching A Game of Matching Functions Description Give each group of students a packet of cards. Students work as a group to match the cards, by thinking about their card and what information
More informationHFCC Math Lab Intermediate Algebra  17 DIVIDING RADICALS AND RATIONALIZING THE DENOMINATOR
HFCC Math Lab Intermediate Algebra  17 DIVIDING RADICALS AND RATIONALIZING THE DENOMINATOR Dividing Radicals: To divide radical expression we use Step 1: Simplify each radical Step 2: Apply the Quotient
More informationCLASS NOTES. We bring down (copy) the leading coefficient below the line in the same column.
SYNTHETIC DIVISION CLASS NOTES When factoring or evaluating polynomials we often find that it is convenient to divide a polynomial by a linear (first degree) binomial of the form x k where k is a real
More information1 Lecture: Integration of rational functions by decomposition
Lecture: Integration of rational functions by decomposition into partial fractions Recognize and integrate basic rational functions, except when the denominator is a power of an irreducible quadratic.
More informationEffect Sizes Based on Means
CHAPTER 4 Effect Sizes Based on Means Introduction Raw (unstardized) mean difference D Stardized mean difference, d g Resonse ratios INTRODUCTION When the studies reort means stard deviations, the referred
More informationZero: If P is a polynomial and if c is a number such that P (c) = 0 then c is a zero of P.
MATH 11011 FINDING REAL ZEROS KSU OF A POLYNOMIAL Definitions: Polynomial: is a function of the form P (x) = a n x n + a n 1 x n 1 + + a x + a 1 x + a 0. The numbers a n, a n 1,..., a 1, a 0 are called
More informationPythagorean Triples. Chapter 2. a 2 + b 2 = c 2
Chapter Pythagorean Triples The Pythagorean Theorem, that beloved formula of all high school geometry students, says that the sum of the squares of the sides of a right triangle equals the square of the
More informationFirst Degree Equations First degree equations contain variable terms to the first power and constants.
Section 4 7: Solving 2nd Degree Equations First Degree Equations First degree equations contain variable terms to the first power and constants. 2x 6 = 14 2x + 3 = 4x 15 First Degree Equations are solved
More informationLesson 9: Radicals and Conjugates
Student Outcomes Students understand that the sum of two square roots (or two cube roots) is not equal to the square root (or cube root) of their sum. Students convert expressions to simplest radical form.
More informationContents. 6 Graph Sketching 87. 6.1 Increasing Functions and Decreasing Functions... 87. 6.2 Intervals Monotonically Increasing or Decreasing...
Contents 6 Graph Sketching 87 6.1 Increasing Functions and Decreasing Functions.......................... 87 6.2 Intervals Monotonically Increasing or Decreasing....................... 88 6.3 Etrema Maima
More information( ) the longrun limit to the right and
(Section 2.3: Limits and Infinity I) 2.3. SECTION 2.3: LIMITS AND INFINITY I LEARNING OBJECTIVES Understand longrun its and relate them to horizontal asymptotes of graphs. Be able to evaluate longrun
More informationLIMIT COMPUTATION: FORMULAS AND TECHNIQUES, INCLUDING L HÔPITAL S RULE
LIMIT COMPUTATION: FORMULAS AND TECHNIQUES, INCLUDING L HÔPITAL S RULE MATH 153, SECTION 55 (VIPUL NAIK) Corresponding material in the book: Sections 11.4, 11.5, 11.6. What students should already know:
More informationMATH REVIEW KIT. Reproduced with permission of the Certified General Accountant Association of Canada.
MATH REVIEW KIT Reproduced with permission of the Certified General Accountant Association of Canada. Copyright 00 by the Certified General Accountant Association of Canada and the UBC Real Estate Division.
More informationAn Insight into Division Algorithm, Remainder and Factor Theorem
An Insight into Division Algorithm, Remainder and Factor Theorem Division Algorithm Recall division of a positive integer by another positive integer For eample, 78 7, we get and remainder We confine the
More informationZeros of Polynomial Functions
Zeros of Polynomial Functions The Rational Zero Theorem If f (x) = a n x n + a n1 x n1 + + a 1 x + a 0 has integer coefficients and p/q (where p/q is reduced) is a rational zero, then p is a factor of
More informationAlum Rock Elementary Union School District Algebra I Study Guide for Benchmark III
Alum Rock Elementary Union School District Algebra I Study Guide for Benchmark III Name Date Adding and Subtracting Polynomials Algebra Standard 10.0 A polynomial is a sum of one ore more monomials. Polynomial
More informationDirections Please read carefully!
Math Xa Algebra Practice Problems (Solutions) Fall 2008 Directions Please read carefully! You will not be allowed to use a calculator or any other aids on the Algebra PreTest or PostTest. Be sure to
More informationDraft Material. Determine the derivatives of polynomial functions by simplifying the algebraic expression lim h and then
CHAPTER : DERIVATIVES Specific Expectations Addressed in the Chapter Generate, through investigation using technology, a table of values showing the instantaneous rate of change of a polynomial function,
More informationFlorida Math 0028. Correlation of the ALEKS course Florida Math 0028 to the Florida Mathematics Competencies  Upper
Florida Math 0028 Correlation of the ALEKS course Florida Math 0028 to the Florida Mathematics Competencies  Upper Exponents & Polynomials MDECU1: Applies the order of operations to evaluate algebraic
More informationa > 0 parabola opens a < 0 parabola opens
Objective 8 Quadratic Functions The simplest quadratic function is f() = 2. Objective 8b Quadratic Functions in (h, k) form Appling all of Obj 4 (reflections and translations) to the function. f() = a(
More information3. Power of a Product: Separate letters, distribute to the exponents and the bases
Chapter 5 : Polynomials and Polynomial Functions 5.1 Properties of Exponents Rules: 1. Product of Powers: Add the exponents, base stays the same 2. Power of Power: Multiply exponents, bases stay the same
More informationStochastic Derivation of an Integral Equation for Probability Generating Functions
Journal of Informatics and Mathematical Sciences Volume 5 (2013), Number 3,. 157 163 RGN Publications htt://www.rgnublications.com Stochastic Derivation of an Integral Equation for Probability Generating
More informationHow to solve a Cubic Equation Part 3 General Depression and a New Covariant
ow to Solve a ubic Equation Part 3 ow to solve a ubic Equation Part 3 General Deression and a New ovariant James F. Blinn Microsoft Research blinn@microsoft.com Originally ublished in IEEE omuter Grahics
More informationReal Roots of Univariate Polynomials with Real Coefficients
Real Roots of Univariate Polynomials with Real Coefficients mostly written by Christina Hewitt March 22, 2012 1 Introduction Polynomial equations are used throughout mathematics. When solving polynomials
More informationVerifying Trigonometric Identities. Introduction. is true for all real numbers x. So, it is an identity. Verifying Trigonometric Identities
333202_0502.qxd 382 2/5/05 Chapter 5 5.2 9:0 AM Page 382 Analytic Trigonometry Verifying Trigonometric Identities What you should learn Verify trigonometric identities. Why you should learn it You can
More information0.8 Rational Expressions and Equations
96 Prerequisites 0.8 Rational Expressions and Equations We now turn our attention to rational expressions  that is, algebraic fractions  and equations which contain them. The reader is encouraged to
More informationwhere a, b, c, and d are constants with a 0, and x is measured in radians. (π radians =
Introduction to Modeling 3.61 3.6 Sine and Cosine Functions The general form of a sine or cosine function is given by: f (x) = asin (bx + c) + d and f(x) = acos(bx + c) + d where a, b, c, and d are constants
More informationDiscrete Mathematics: Homework 7 solution. Due: 2011.6.03
EE 2060 Discrete Mathematics spring 2011 Discrete Mathematics: Homework 7 solution Due: 2011.6.03 1. Let a n = 2 n + 5 3 n for n = 0, 1, 2,... (a) (2%) Find a 0, a 1, a 2, a 3 and a 4. (b) (2%) Show that
More informationConfidence Intervals for CaptureRecapture Data With Matching
Confidence Intervals for CatureRecature Data With Matching Executive summary Caturerecature data is often used to estimate oulations The classical alication for animal oulations is to take two samles
More informationMTH124: Honors Algebra I
MTH124: Honors Algebra I This course prepares students for more advanced courses while they develop algebraic fluency, learn the skills needed to solve equations, and perform manipulations with numbers,
More information6 EXTENDING ALGEBRA. 6.0 Introduction. 6.1 The cubic equation. Objectives
6 EXTENDING ALGEBRA Chapter 6 Extending Algebra Objectives After studying this chapter you should understand techniques whereby equations of cubic degree and higher can be solved; be able to factorise
More informationAlgebraic Expressions and Equations: Classification of Expressions and Equations
OpenStaxCNX module: m21848 1 Algebraic Expressions and Equations: Classification of Expressions and Equations Wade Ellis Denny Burzynski This work is produced by OpenStaxCNX and licensed under the Creative
More informationPYTHAGOREAN TRIPLES KEITH CONRAD
PYTHAGOREAN TRIPLES KEITH CONRAD 1. Introduction A Pythagorean triple is a triple of positive integers (a, b, c) where a + b = c. Examples include (3, 4, 5), (5, 1, 13), and (8, 15, 17). Below is an ancient
More informationPrice Elasticity of Demand MATH 104 and MATH 184 Mark Mac Lean (with assistance from Patrick Chan) 2011W
Price Elasticity of Demand MATH 104 and MATH 184 Mark Mac Lean (with assistance from Patrick Chan) 2011W The rice elasticity of demand (which is often shortened to demand elasticity) is defined to be the
More informationJoint Distributions. Lecture 5. Probability & Statistics in Engineering. 0909.400.01 / 0909.400.02 Dr. P. s Clinic Consultant Module in.
3σ σ σ +σ +σ +3σ Joint Distributions Lecture 5 0909.400.01 / 0909.400.0 Dr. P. s Clinic Consultant Module in Probabilit & Statistics in Engineering Toda in P&S 3σ σ σ +σ +σ +3σ Dealing with multile
More information3 1. Note that all cubes solve it; therefore, there are no more
Math 13 Problem set 5 Artin 11.4.7 Factor the following polynomials into irreducible factors in Q[x]: (a) x 3 3x (b) x 3 3x + (c) x 9 6x 6 + 9x 3 3 Solution: The first two polynomials are cubics, so if
More informationEquations and Inequalities
Rational Equations Overview of Objectives, students should be able to: 1. Solve rational equations with variables in the denominators.. Recognize identities, conditional equations, and inconsistent equations.
More informationChapter 2 Limits Functions and Sequences sequence sequence Example
Chapter Limits In the net few chapters we shall investigate several concepts from calculus, all of which are based on the notion of a limit. In the normal sequence of mathematics courses that students
More informationCore Maths C1. Revision Notes
Core Maths C Revision Notes November 0 Core Maths C Algebra... Indices... Rules of indices... Surds... 4 Simplifying surds... 4 Rationalising the denominator... 4 Quadratic functions... 4 Completing the
More information5.1 The Remainder and Factor Theorems; Synthetic Division
5.1 The Remainder and Factor Theorems; Synthetic Division In this section you will learn to: understand the definition of a zero of a polynomial function use long and synthetic division to divide polynomials
More informationSlant asymptote. This means a diagonal line y = mx + b which is approached by a graph y = f(x). For example, consider the function: 2x 2 8x.
Math 32 Curve Sketching Stewart 3.5 Man vs machine. In this section, we learn methods of drawing graphs by hand. The computer can do this much better simply by plotting many points, so why bother with
More informationChapter 3. Special Techniques for Calculating Potentials. r ( r ' )dt ' ( ) 2
Chater 3. Secial Techniues for Calculating Potentials Given a stationary charge distribution r( r ) we can, in rincile, calculate the electric field: E ( r ) Ú Dˆ r Dr r ( r ' )dt ' 2 where Dr r 'r. This
More information