DOCSLIB.ORG

5.4 Power and Radical Functions

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
5.4 Power and Radical Functions 5.4P ower and Radical Functions This is part of 5.2 in the current text In this section, you will learn about the properties and characteristics of power and radical functions. Upon completion you will be able to: • Identify if a given function is a power or a radical function. • Convert between radical and power notation. • Memorize the graphs of the parent even and odd power/radical functions (square root and cube root). • Determine the domain of a function involving power or radical functions, using interval notation. • Identify the conjugate of a given expression involving radicals. • Rationalize the numerator or denominator of a given expression, by multiplying by the conjugate. Defining Power Functions A power function is a function with a single term that is the product of a real number and a variable raised to a fixed real number. Definition A power function is a function that can be represented in the form f (x) = axp where a and p are real numbers , and a is known as the coefficient. Here the left side tells us that x is the variable; a and p are parameters that can be any real numbers. Because a and p can be anything, this is a very general function type meaning that the properties of the function can be very different based on those values of a and p. Before we dive into the properties of power functions, the authors would like to pause and review some basic properties of variables raised to numerical exponents. Review of Exponent Laws Theorem 5.1 Laws of Exponents Let p and q be any real numbers and x and y be any variables with values greater than zero. Then, the following rules hold true. xp xq = xp+q (xp)q = xpq 1 = x−p xp 1 = xp x−p 1 p x p = p x (xy)p = xpyp !p x xp = y yp © TAMU 134 5.4 Power and Radical Functions ! (x + y)p ,(xp + yp) and (x − y)p ,(xp − yp) p 1 Mathematics call p x the radical form and x p the exponent form. Both of these have the same meaning, they just p 1 look a bit different. Anytime you see p x, you can replace it with x p and vice versa. These rules will all be quite handy in calculus. In both integral and differential calculus, we will have rules that work well when we have a power function, but won’t work for other forms of functions. By being able to rewrite 1 functions like f (x) = , as power function ( f (x) = x−2 here), other calculations will be simplified. x2 3 0 4 1 2 B(xy) C Example1 Simplify B C assuming x > 0 and y > 0. @B 1 AC xy 2 p p 2 Example2 Simplify 2x + 3y assuming x > 0 and y > 0. 135 © TAMU Categories of Power Functions Returning to our discussion of power functions, we can categorize a power function, f (x) = axp by the value of its exponent (power). 1. If p = 0 and x , 0, then f (x) is the constant polynomial function, f (x) = ax0 = a. 2. If p is a positive integer, then f (x) is a pth degree polynomial function with only the leading term. 3. If p is a negative integer, then f (x) is a rational function. Each of these types of power functions hold the properties discussed in the Polynomial Functions and Rational Functions Sections, respectively. 4. If p is an irrational number and x > 0, then we have a complicated power function, which is beyond the scope of this text. 1 5. If p = , where n is a natural number, such that n ≥ 2, then f (x) is a radical (root) function. n Describing Properties of Radical Functions Definition A radical (root) function is a function of the form pn 1 f (x) = x = x n where n is a natural number, such that n ≥ 2, called the index or root. • If n is an even number, then we say f (x) has an even root and f (x) is only defined for x ≥ 0. (The even root of a negative number is not a real number.) • If n is an odd number, then we say f (x) has an odd root and f (x) is defined for any real number x. Name Function Graph Table Domain x f (x) p 0 0 Square root f (x) = x [0;1) 1 1 4 2 -9-8-7-6-5-4-3-2-1123456789 Figure 5.4.1 x f (x) -1 -1 p -0.125 -0.5 Cube root f (x) = 3 x (−∞;1) 0 0 0.125 0.5 -9-8-7-6-5-4-3-2-1123456789 1 1 Figure 5.4.2 Table 5.1: A table with 3 rows and 5 columns. The first row includes labels: Name, Function, Graph, Table, and Domain. The second row includes the corresponding properties for a square root parent function, while the third row include the corresponding properties for the cube root parent function. © TAMU 136 5.4 Power and Radical Functions Many times in Calculus we are faced with functions in radical form. It is often necessary to convert from radical form to an equivalent exponent (power) form. Example3 Write the following radicals in their equivalent exponent (power) form. p a. R(p) = 15000 − 1:5p r 3 1 −1 b. h(x) = x2 + 5x 2 1 Sometimes it is necessary to have a function written with non-negative powers using x−p = . xp p p p 2 2 2 N x = ( x) = jxj, not x, as x is non-negative, but if x is negative x is undefined in the real numbers. 137 © TAMU As with all functions we need to investigate the domain of radical (root) functions. Our investigation will focus on the differences in domain based on whether the root is even or odd. Example4 State the domain of each function, using interval notation. 2 5 a. f (y) = (y + 1) 4 4 b. h(z) = (6z + 1) 3 4x c. k(x) = p 2x − 1 © TAMU 138 5.4 Power and Radical Functions We find the intercepts of power functions, including radicals, using the same techniques discussed in previous sections. As with rational functions, power functions may or may not have both x-intercepts and y-intercepts. Computing Domains of Algebraic Functions By combining types of functions into a single function, we get, what the authors call, an algebraic function. The domain of an algebraic function is all real numbers, x, which satisfy all parts of the function.So far our only discussed domain restrictions are 1. Division by zero is undefined. 2. Even roots of negative numbers are undefined. Example5 Statep the domain of each algebraic function, using interval notation. a. f (x) = 3 − 18x − 15 4 − x2 b. h(x) = p x − 6 p 7 − x c. j(x) = 3x + 8 3 (5x − 9) 2 d. k(x) = p 7 x + 2 139 © TAMU Rationalizing Numerators and Denominators We can remove radicals from the denominators of fractions using a process call rationalizing the denominator. We know that multiplying by 1 does not change the value of the expression. We use this property of multiplication to change expressions that contain radicals in the denominator. To remove radicals from the denominators of fractions, multiply by the form of 1 that will eliminate the radical. For a denominator containing the sum or difference of two terms where at least one term is a square root radical, multiply the numerator and denominator by the conjugate of the denominator, whichp is found by changing the sign betweenp the two terms of the denominator.p For example, if the denominatorp is a + b c, then the conjugate is a − b c, and if the denominator is a − b c, then the conjugate is a + b c. Similarly, we can rationalize a numerator containing radicals. Example6 Rationalize the denominator and simplify the expression x − 36 p for x ≥ 0 and x , 36 − x + 6 © TAMU 140 5.4 Power and Radical Functions Example7 Find and simplify the difference quotient for the function p r(x) = 2 x + 1 Reflection: • Can you recognize a power/radical function and transition between the two forms of the same function? • Can you select the graphs of the parent power/radical functions from a group of graphs? • Can you explain how to find the domain of a power/radical function based on the index? • Can you explain how a conjugate is used to rationalize an expression? 141 © TAMU.
Load more

Recommended publications
  • Field Theory Pete L. Clark
    Field Theory Pete L. Clark Thanks to Asvin Gothandaraman and David Krumm for pointing out errors in these notes. Contents About These Notes 7 Some Conventions 9 Chapter 1. Introduction to Fields 11 Chapter 2. Some Examples of Fields 13 1. Examples From Undergraduate Mathematics 13 2. Fields of Fractions 14 3. Fields of Functions 17 4. Completion 18 Chapter 3. Field Extensions 23 1. Introduction 23 2. Some Impossible Constructions 26 3. Subfields of Algebraic Numbers 27 4. Distinguished Classes 29 Chapter 4. Normal Extensions 31 1. Algebraically closed fields 31 2. Existence of algebraic closures 32 3. The Magic Mapping Theorem 35 4. Conjugates 36 5. Splitting Fields 37 6. Normal Extensions 37 7. The Extension Theorem 40 8. Isaacs' Theorem 40 Chapter 5. Separable Algebraic Extensions 41 1. Separable Polynomials 41 2. Separable Algebraic Field Extensions 44 3. Purely Inseparable Extensions 46 4. Structural Results on Algebraic Extensions 47 Chapter 6. Norms, Traces and Discriminants 51 1. Dedekind's Lemma on Linear Independence of Characters 51 2. The Characteristic Polynomial, the Trace and the Norm 51 3. The Trace Form and the Discriminant 54 Chapter 7. The Primitive Element Theorem 57 1. The Alon-Tarsi Lemma 57 2. The Primitive Element Theorem and its Corollary 57 3 4 CONTENTS Chapter 8. Galois Extensions 61 1. Introduction 61 2. Finite Galois Extensions 63 3. An Abstract Galois Correspondence 65 4. The Finite Galois Correspondence 68 5. The Normal Basis Theorem 70 6. Hilbert's Theorem 90 72 7. Infinite Algebraic Galois Theory 74 8. A Characterization of Normal Extensions 75 Chapter 9.
    [Show full text]
  • Section 1.2 – Mathematical Models: a Catalog of Essential Functions
    Section 1-2 © Sandra Nite Math 131 Lecture Notes Section 1.2 – Mathematical Models: A Catalog of Essential Functions A mathematical model is a mathematical description of a real-world situation. Often the model is a function rule or equation of some type. Modeling Process Real-world Formulate Test/Check problem Real-world Mathematical predictions model Interpret Mathematical Solve conclusions Linear Models Characteristics: • The graph is a line. • In the form y = f(x) = mx + b, m is the slope of the line, and b is the y-intercept. • The rate of change (slope) is constant. • When the independent variable ( x) in a table of values is sequential (same differences), the dependent variable has successive differences that are the same. • The linear parent function is f(x) = x, with D = ℜ = (-∞, ∞) and R = ℜ = (-∞, ∞). • The direct variation function is a linear function with b = 0 (goes through the origin). • In a direct variation function, it can be said that f(x) varies directly with x, or f(x) is directly proportional to x. Example: See pp. 26-28 of the text. 1 Section 1-2 © Sandra Nite Polynomial Functions = n + n−1 +⋅⋅⋅+ 2 + + A function P is called a polynomial if P(x) an x an−1 x a2 x a1 x a0 where n is a nonnegative integer and a0 , a1 , a2 ,..., an are constants called the coefficients of the polynomial. If the leading coefficient an ≠ 0, then the degree of the polynomial is n. Characteristics: • The domain is the set of all real numbers D = (-∞, ∞). • If the degree is odd, the range R = (-∞, ∞).
    [Show full text]
  • Trigonometric Functions
    72 Chapter 4 Trigonometric Functions To define the radian measurement system, we consider the unit circle in the xy-plane: ........................ ....... ....... ...... ....................... .............. ............... ......... ......... ....... ....... ....... ...... ...... ...... ..... ..... ..... ..... ..... ..... .... ..... ..... .... .... .... .... ... (cos x, sin x) ... ... 4 ... A ..... .. ... ....... ... ... ....... ... .. ....... .. .. ....... .. .. ....... .. .. ....... .. .. ....... .. .. ....... ...... ....... ....... ...... ....... x . ....... Trigonometric Functions . ...... ....y . ....... (1, 0) . ....... ....... .. ...... .. .. ....... .. .. ....... .. .. ....... .. .. ....... .. ... ...... ... ... ....... ... ... .......... ... ... ... ... .... B... .... .... ..... ..... ..... ..... ..... ..... ..... ..... ...... ...... ...... ...... ....... ....... ........ ........ .......... .......... ................................................................................... An angle, x, at the center of the circle is associated with an arc of the circle which is said to subtend the angle. In the figure, this arc is the portion of the circle from point (1, 0) So far we have used only algebraic functions as examples when finding derivatives, that is, to point A. The length of this arc is the radian measure of the angle x; the fact that the functions that can be built up by the usual algebraic operations of addition, subtraction, radian measure is an actual geometric length is largely responsible for the usefulness of
    [Show full text]
  • Coefficients of Algebraic Functions: Formulae and Asymptotics
    COEFFICIENTS OF ALGEBRAIC FUNCTIONS: FORMULAE AND ASYMPTOTICS CYRIL BANDERIER AND MICHAEL DRMOTA Abstract. This paper studies the coefficients of algebraic functions. First, we recall the too-less-known fact that these coefficients fn always a closed form. Then, we study their asymptotics, known to be of the type n α fn ∼ CA n . When the function is a power series associated to a context-free grammar, we solve a folklore conjecture: the appearing critical exponents α belong to a subset of dyadic numbers, and we initiate the study the set of possible values for A. We extend what Philippe Flajolet called the Drmota{Lalley{Woods theorem (which is assuring α = −3=2 as soon as a "dependency graph" associated to the algebraic system defining the function is strongly connected): We fully characterize the possible singular behaviors in the non-strongly connected case. As a corollary, it shows that certain lattice paths and planar maps can not be generated by a context-free grammar (i.e., their generating function is not N-algebraic). We give examples of Gaussian limit laws (beyond the case of the Drmota{Lalley{Woods theorem), and examples of non Gaussian limit laws. We then extend our work to systems involving non-polynomial entire functions (non-strongly connected systems, fixed points of entire function with positive coefficients). We end by discussing few algorithmic aspects. Resum´ e.´ Cet article a pour h´erosles coefficients des fonctions alg´ebriques.Apr`esavoir rappel´ele fait trop peu n α connu que ces coefficients fn admettent toujours une forme close, nous ´etudionsleur asymptotique fn ∼ CA n .
    [Show full text]
  • Complex Algebraic Geometry
    Complex Algebraic Geometry Jean Gallier∗ and Stephen S. Shatz∗∗ ∗Department of Computer and Information Science University of Pennsylvania Philadelphia, PA 19104, USA e-mail: [email protected] ∗∗Department of Mathematics University of Pennsylvania Philadelphia, PA 19104, USA e-mail: [email protected] February 25, 2011 2 Contents 1 Complex Algebraic Varieties; Elementary Theory 7 1.1 What is Geometry & What is Complex Algebraic Geometry? . .......... 7 1.2 LocalStructureofComplexVarieties. ............ 14 1.3 LocalStructureofComplexVarieties,II . ............. 28 1.4 Elementary Global Theory of Varieties . ........... 42 2 Cohomologyof(Mostly)ConstantSheavesandHodgeTheory 73 2.1 RealandComplex .................................... ...... 73 2.2 Cohomology,deRham,Dolbeault. ......... 78 2.3 Hodge I, Analytic Preliminaries . ........ 89 2.4 Hodge II, Globalization & Proof of Hodge’s Theorem . ............ 107 2.5 HodgeIII,TheK¨ahlerCase . .......... 131 2.6 Hodge IV: Lefschetz Decomposition & the Hard Lefschetz Theorem............... 147 2.7 ExtensionsofResultstoVectorBundles . ............ 162 3 The Hirzebruch-Riemann-Roch Theorem 165 3.1 Line Bundles, Vector Bundles, Divisors . ........... 165 3.2 ChernClassesandSegreClasses . .......... 179 3.3 The L-GenusandtheToddGenus .............................. 215 3.4 CobordismandtheSignatureTheorem. ........... 227 3.5 The Hirzebruch–Riemann–Roch Theorem (HRR) . ............ 232 3 4 CONTENTS Preface This manuscript is based on lectures given by Steve Shatz for the course Math 622/623–Complex Algebraic Geometry, during Fall 2003 and Spring 2004. The process for producing this manuscript was the following: I (Jean Gallier) took notes and transcribed them in LATEX at the end of every week. A week later or so, Steve reviewed these notes and made changes and corrections. After the course was over, Steve wrote up additional material that I transcribed into LATEX. The following manuscript is thus unfinished and should be considered as work in progress.
    [Show full text]
  • Finite Fields and Function Fields
    Copyrighted Material 1 Finite Fields and Function Fields In the first part of this chapter, we describe the basic results on finite fields, which are our ground fields in the later chapters on applications. The second part is devoted to the study of function fields. Section 1.1 presents some fundamental results on finite fields, such as the existence and uniqueness of finite fields and the fact that the multiplicative group of a finite field is cyclic. The algebraic closure of a finite field and its Galois group are discussed in Section 1.2. In Section 1.3, we study conjugates of an element and roots of irreducible polynomials and determine the number of monic irreducible polynomials of given degree over a finite field. In Section 1.4, we consider traces and norms relative to finite extensions of finite fields. A function field governs the abstract algebraic aspects of an algebraic curve. Before proceeding to the geometric aspects of algebraic curves in the next chapters, we present the basic facts on function fields. In partic- ular, we concentrate on algebraic function fields of one variable and their extensions including constant field extensions. This material is covered in Sections 1.5, 1.6, and 1.7. One of the features in this chapter is that we treat finite fields using the Galois action. This is essential because the Galois action plays a key role in the study of algebraic curves over finite fields. For comprehensive treatments of finite fields, we refer to the books by Lidl and Niederreiter [71, 72]. 1.1 Structure of Finite Fields For a prime number p, the residue class ring Z/pZ of the ring Z of integers forms a field.
    [Show full text]
  • D-Finite Numbers 3
    D-finite Numbers∗ Hui Huang† David R. Cheriton School of Computer Science, University of Waterloo 200 University Avenue West, Waterloo, Ontario, N2L 3G1,Canada [email protected] Manuel Kauers‡ Institute for Algebra, Johannes Kepler University Altenberger Strasse 69, 4040 Linz, Austria [email protected] Abstract D-finite functions and P-recursive sequences are defined in terms of linear differential and recurrence equations with polynomial coefficients. In this paper, we introduce a class of numbers closely related to D-finite functions and P-recursive sequences. It consists of the limits of convergent P-recursive sequences. Typically, this class contains many well-known mathematical constants in addition to the algebraic numbers. Our definition of the class of D-finite numbers depends on two subrings of the field of complex numbers. We investigate how different choices of these two subrings affect the class. Moreover, we show that D- finite numbers are essentially limits of D-finite functions at the point one, and evaluating D-finite functions at non-singular algebraic points typically yields D-finite numbers. This result makes it easier to recognize certain numbers to be D-finite. Keywords: Complex numbers; D-finite functions; P-recursive sequences; algebraic numbers; evaluation of special functions. Mathematics Subject Classification 2010: 11Y35, 33E30, 33F05, 33F10 1 Introduction D-finite functions have been recognized long ago [23, 15, 30, 19, 16, 24] as an especially attractive class of functions. They are interesting on the one hand because each of them can arXiv:1611.05901v3 [math.NT] 26 May 2018 be easily described by a finite amount of data, and efficient algorithms are available to do exact as well as approximate computations with them.
    [Show full text]
  • Algebraic Function Fields
    Math 612 { Algebraic Function Fields Julia Hartmann May 12, 2015 An algebraic function field is a finitely generated extension of transcendence degree 1 over some base field. This type of field extension occurs naturally in various branches of mathematics such as algebraic geometry, number theory, and the theory of compact Riemann surfaces. In this course, we study algebraic function fields from an algebraic point of view. Topics include: • Valuations on a function field: valuation rings, places, rational function fields, weak approximation • Divisors of a function field: divisor group, dimension of a divisor • Genus of a function field: definition, adeles, index of speciality • Riemann-Roch theorem: Weil differentials, divisor of a differential, duality theorem and Riemann-Roch theorem, canonical class, strong approxima- tion • Fields of genus 0: divisor classes, rational function fields • Extensions of valuations: ramification index and relative degree, funda- mental equation, extensions and automorphisms • Completions: definition and existence of completions, complete discretely valued case, Hensel-Lemma, completions of function fields • The different: different exponent, Dedekind's different theorem • Hurwitz genus formula: divisors in extensions, lifting of differentials, genus formula, L¨uroth'stheorem, constant extensions, inseparable extensions • Differentials of algebraic function fields: derviations, differentials, residues • The residue theorem: connection between differentials and Weil differen- tials, residue theorem • Congruence zeta function: finiteness theorems, zeta function, smallest positive divisor degree, functional equation, Riemann hypothesis for con- gruence function fields (Hasse-Weil) 1 • Applications to coding theory: gemetric Goppa codes, rational codes, de- coding, automorphisms, asymptotic bounds • Elliptic function fields and elliptic curves Additional topics will be added as time permits and based on students' interests.
    [Show full text]
  • The Uncountability of Transcendental Numbers Zhuyu Ye
    The Uncountability of Transcendental Numbers Zhuyu Ye The history of transcendental number The name "transcendental" comes from the root trans meaning across and length of numbers and Leibniz in his 1682 paper where he proved that sin(x) is not an algebraic function of x.Euler was probably the first person to define transcendental numbers in the modern sense. After the Lauville number proved, many mathematicians devoted themselves to the study of transcen- dental numbers. In 1873, the French mathematician Hermite (CharlesHermite, 1822-1901) proved the transcendence of natural logarithm, so that people know more about the number of transcendence. In 1882, the German mathematician Lindemann proved that pi is also a transcendental number (com- pletely deny the "round for the party" mapping possibilities).In the process of studying transcendental numbers, David Hilbert has conjectured that a is an algebraic number that is not equal to 0 and 1, and b is an irrational algebra, then abis the number of transcendences (Hilbert's problem Of the seventh question).This conjecture has been proved, so we can conclude that e, pi is the transcendental number. π The first to get pi = 3.14 ... is the Greek Archimedes (about 240 BC), the first to give pi decimal after the four accurate value is the Greek Ptolemy (about 150 BC), the earliest Calculate the pi decimal after seven accurate value is Chinese ancestor Chong (about 480 years), in 1610 the Dutch mathemati- cian Rudolph application of internal and external tangent polygon calculation of pi value, through the calculation of ? to 35 decimal digits.
    [Show full text]
  • ON EFROYMSON's EXTENSION THEOREM for NASH Ddpartement De Math6matiques, Universit
    Journal of Pure and Applied Algebra 37 (1985) 193-203 193 North-Holland ON EFROYMSON'S EXTENSION THEOREM FOR NASH FUNCTIONS Daniel PECKER Ddpartement de Math6matiques, Universit~de Dakar, Dakar-Fann, S~n~gal Communicated by M.F. Coste-Roy Received 26 May 1984 1. Introduction Recall that a subset of ~n is called semi-algebraic if it can be written as a finite union of sets of the form {XE [Rn[Pi(x1,...,Xn)=O, Pj(x 1, ..., Xn)>0, i= 1, ...,m, j=m+ 1, ...,S}. where the Pi are arbitrary polynomials in ~[xl, ..., xn]. A semi-algebraic function is a function whose graph is semi-algebraic. If U is an open subset of ~n, a Nash function f defined on U is an analytic real function which satisfies locally a real polynomial P(x,f(x))- 0 (in n + 1 variables). It is known that if U is an open semi-algebraic subset of IRn, a Nash function on U is semi-algebraic. One frequently uses the principle of Tarski-Seidenberg: "A first-order formula in the language of ordered fields is equivalent to a first-order formula without quan- tifiers in the theory of real closed fields." (A first-order formula in the language of ordered fields is a formula obtained from a finite number of polynomial inequalities, negations, conjunctions, disjunc- tions and quantifiers ~r or V, the variables being elements of the ordered field K.) The classical Tietze-Urysohn theorem is a model for 'extension theorems'. Let us state it in a semi-algebraic form.
    [Show full text]
  • Chapter 4: Transcendental Functions
    4 Transcendental Functions So far we have used only algebraic functions as examples when finding derivatives, that is, functions that can be built up by the usual algebraic operations of addition, subtraction, multiplication, division, and raising to constant powers. Both in theory and practice there are other functions, called transcendental, that are very useful. Most important among these are the trigonometric functions, the inverse trigonometric functions, exponential functions, and logarithms. º½ When you first encountered the trigonometric functions it was probably in the context of “triangle trigonometry,” defining, for example, the sine of an angle as the “side opposite over the hypotenuse.” While this will still be useful in an informal way, we need to use a more expansive definition of the trigonometric functions. First an important note: while degree measure of angles is sometimes convenient because it is so familiar, it turns out to be ill-suited to mathematical calculation, so (almost) everything we do will be in terms of radian measure of angles. 71 72 Chapter 4 Transcendental Functions To define the radian measurement system, we consider the unit circle in the xy-plane: ........................ ....... ....... ...... ....................... .............. ............... ......... ......... ....... ....... ....... ...... ...... ...... ..... ..... ..... ..... ..... ..... .... ..... ..... .... .... .... .... ... (cos x, sin x) ... ... ... A ..... ... ....... ... ... ....... ... ... ....... .. .. ....... .. .. ....... .. .
    [Show full text]
  • 0.4 Exponential and Trigonometric Functions
    0.4 Exponential and Trigonometric Functions ! Definitions and properties of exponential and logarithmic functions ! Definitions and properties of trigonometric and inverse trigonometric functions ! Graphs and equations involving transcendental functions Exponential Functions Functions that are not algebraic are called transcendental functions. In this book we will investigate four basic types of transcendental functions: exponential, logarithmic, trigono- metric, and inverse trigonometric functions. Exponential functions are similar to power func- tions, but with the roles of constant and variable reversed in the base and exponent: Definition 0.21 Exponential Functions An exponential function is a function that can be written in the form f(x)=Abx for some real numbers A and b such that A =0, b>0,andb =1. ! ! There is an important technical problem with this definition: we know what it means to raise a number to a rational power by using integer roots and powers, but we don’t know what it means to raise a number to an irrational power. We need to be able to raise to irrational powers to talk about exponential functions; for example, if f(x)=2x then we need to be able to compute f(π)=2π. One way to think of bx where x is irrational is as a limit: bx = lim br. r x r rational→ The “lim” notation will be explored more in Chapter 1. For now you can just imagine that if x is rational we can approximate bx by looking at quantities br for various rational numbers r that get closer and close to the irrational number x.For example, 2π can be approximated by 2r for rational numbers r that are close to π: π 3.14 314 100 2 2 =2100 = √2314.
    [Show full text]