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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. ≈ As we consider rational numbers r that are closer and closer to π, the expression 2r will get closer and closer to 2π; see Exercise 4. In Chapter xxx we will give a more rigorous definition of exponential functions as the inverses of certain accumulation integrals. Interestingly, the most natural base b to use for an exponential function isn’t a simple inte- ger, like b =2or b =3. Instead, for reasons that will become clear when we study derivatives, the most natural base is the irrational number known as e, and the function ex is therefore called the natural exponential function. The first 75 decimal places of the number e are: 2.71828182845904523536028747135266249775724709369995957496696762772407663035.... Of course, since e is an irrational number, we cannot define e just by writing an approximation of e in decimal notation; we will define e properly once we cover limits in Chapter xxx. In Exercise 88 you will prove that every exponential function can be written so that its base is the natural number e: 0.4 Exponential and Trigonometric Functions 48 Theorem 0.22 Natural Exponential Functions Ev- Every exponential function can be written in the form f(x)=Aekx for some real number A and some nonzero real number k. ery exponential function has a graph similar to either the exponential growth graph below left or the exponential decay graph below right, depending on the value of k or b. Of course, if the coefficient A is negative, then the graph of f(x)=Aekx or f(x)=Abx will be an upside- down version of one of these two graphs. f(x)=ekx with k>0, f(x)=ekx with k<0, f(x)=bx with b>1 f(x)=bx with 0 <b<1 1 1 Logarithmic Functions Since every exponential function bx is one-to-one, every exponential function has an inverse. These inverses are what we call the logarithmic functions: Definition 0.23 Logarithmic Functions as Inverses of Exponential Functions The inverse of the exponential function f(x)=bx is the logarithmic function g(x) = logb x. As a special case, the inverse of the natural exponential function f(x)=ex is the natural logarithmic function g(x) = ln x. We require that the base b satisfies b>0 and b =1, because these are exactly the conditions we ! must have for y = bx to be an exponential function. In Section xxx we will define logarithms another way, in terms of integrals and accumulation functions. You should already be familiar with the algebraic rules of logarithms, but we restate them here in case you need a refresher; see Exercises 90–94 for proofs. 0.4 Exponential and Trigonometric Functions 49 Theorem 0.24 Algebraic Rules for Logarithmic Functions For all values of x, y, b and a for which these expressions are defined, we have: y (a) logb x = y if and only if b = x (e) logb(xy) = logb x + logb y (b) log (bx)=x (f) log ( 1 )= log x b b x − b x (c) blogb x = x (g) log ( ) = log x log y b y b − b a log x (d) logb(x )=a logb x (h) log x = a b loga b The first three properties follow from properties of inverse functions, and tell us that logb x is the exponent to which you have to raise b in order to get x.Forexample,log2 8 is the power 3 to which you have to raise 2 to get 8;since2 =8we have log2 8=3. All of these rules also apply to the natural exponental function, since lnx is just logb x with base b = e. Properties (d) and (e) follow from the algebraic rules of exponents, and properties (f) and (g) are their immediate consequences. The final property in Theorem 0.24 is called the base conversion formula, because it allows us to translate from one logarithmic base to another. The base conversion formula is especially helpful for converting to base e or base 10 so that ln 7 we can calculate logarithms on a calculator. For example, log7 2 is equal to ln 2 ,whichwecan approximate using the built-in ln key on a calculator. The graphs of logarithmic functions can be obtained easily from the graphs of exponential functions by reflection over the line y = x,asshownbelow. g(x)=logb x with b>1 g(x)=logb x with 0 <b<1 1 1 Trigonometric Functions There are six trigonometric functions defined as ratios of side lengths of right triangles, or more generally, as ratios of coordinate lengths on the unit circle. We now provide a quick review of the definitions of these functions and their graphical and algebraic properties. Throughout most of this book we will be using radian measure for angles (not degrees). Given any angle θ in standard position, the terminal edge of θ intersects the unit circle at some point (x, y) in the xy-plane. We will define the height y of that point to be the sine of θ, while the cosine of θ will be defined as the x-coordinate of that point. 0.4 Exponential and Trigonometric Functions 50 Definition 0.25 Trigonometric Functions for Any Angle Given any angle θ measured in radians in standard position, let (x, y) be the point where the terminal edge of θ intersects the unit circle. The six trigonometric functions of an angle θ are the six possible ratios of the coordinates x and y for θ: y y sin θ = y cos θ = x tan θ = θ x x 1 1 x (x,y) csc θ = sec θ = cotθ = (cos θ, sin θ) y x y Notice that the sine and cosine functions determine the remaining four trigonometric func- sin θ tions, since, tan θ is the ratio cos θ , and the last three trigonometric functions are the reciprocals of the first three. You should already be familiar with the basic trigonometric identities, but they are re- peated below for your review; see Exercises 95–100 for proofs. The first Pythagorean iden- tity, the even/odd identities, and the shift identities follow easily from the definitions of the trigonometric functions. The sum identities follow from a geometric argument that we will not get into here. The remaining identities can all be proved from the previous identities. In these identities we are using the notation sin2 x as shorthand for (sin x)2. Theorem 0.26 Basic Trigonometric Identities Pythagorean Identities Even/Odd Identities Shift Identities sin2 θ +cos2 θ = 1 sin( θ)= sin θ cos(θ π ) = sin θ − − − 2 tan2 θ +1=sec2 θ cos( θ)=cosθ sin(θ + π )=cosθ − 2 1+cot2 θ =csc2 θ tan( θ)= tan θ sin(θ +2π) = sin θ − − cos(θ +2π)=cosθ Sum Identities Difference Identities sin(α + β) = sinα cos β + sinβ cos α sin(α β) = sin α cosβ sin β cos α − − cos(α + β)=cosα cos β sin α sin β cos(α β)=cosα cos β + sinα sin β − − Double Angle Identities Alternate Forms Alternate Forms 2 2 1 cos 2θ sin 2θ = 2 sinθ cos θ cos 2θ =1 2 sin θ sin θ = − − 2 cos 2θ =cos2 θ sin2 θ cos 2θ =2cos2 θ 1cos2 θ = 1+cos 2θ − − 2 The graphs of the six trigonometric functions are recorded below. Each of the graphs in the second row is the reciprocal of the graph immediately above it. Remember that you can 1 use the graph of a function f to sketch the graph of its reciprocal f . In particular, the zeros of 1 f will be vertical asymptotes of f , large heights on the graph of f will become small heights 1 on the graph of f ,andvice-versa. 0.4 Exponential and Trigonometric Functions 51 y =sinx y =cosx y =tanx 2 2 3 2 1 1 1 −3π −2π −π π 2π 3π −3π −2π −π π 2π 3π −3π −2π −π π 2π 3π −1 −1 −1 −2 −2 −2 −3 y =cscxy=secx y =cotx 3 3 3 2 2 2 1 1 1 −3π −2π −π π 2π 3π −3π −2π −π π 2π 3π −3π −2π −π π 2π 3π −1 −1 −1 −2 −2 −2 −3 −3 −3 Inverse Trigonometric Functions None of the six trigonometric functions are one-to-one, but after restricting domains we can construct the so-called inverse trigonometric functions.
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