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 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 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. In this section we will focus on the inverses of only three of the six inverse trigonometric functions, those for sine, tangent, and secant. There are many different restricted domains that we could use to obtain partial in- verses to these three functions. We need to pick one restricted domain for each function and stick with it. In this text we will use the restricted domains shown below. y =sinx restricted to y =tanx restricted to y =secx restricted to the domain [ π , π ] the domain ( π , π ) the domain [0, π ) ( π , π] − 2 2 − 2 2 2 ∪ 2 1 1 1 !" !" " " !" !" " " !" !" " " 2 2 2 -1 2 2 -1 2 -1 Each of the restricted functions shown above is one-to-one, and thus invertible. The inverses of these restricted functions are the inverse sine, inverse tangent, and inverse secant functions. Definition 0.27 The Inverse Trigonometric Functions 1 (a) The inverse sine function sin− x is the inverse of the restriction of the function sin x to [ π , π ]. − 2 2 1 (b) The inverse tangent function tan− x is the inverse of the restriction of the function tan x to ( π , π ). − 2 2 1 (c) The inverse secant function sec− x is the inverse of the restriction of the func- tion sec x to [0, π ) ( π , π]. 2 ∪ 2 0.4 Exponential and Trigonometric Functions 52 Notice that since the inputs to the trigonometric functions are angles, it is the outputs of the inverse trigonometric functions that are angles. We will interchangeably use the alternative notations arcsinx, arctanx,andarcsecx for these inverse trigonometric functions. 1 1 1 All of the properties of sin− x, tan− x,andsec− x come from the fact that they are the inverses of the restricted functions sin x, tan x,andsec x. For example, we can graph the inverse trigonometric functions simply by reflecting the graphs of the restricted trigonometric functions over the line y = x,asshownbelow. 1 1 1 y =sin− xy=tan− xy=sec− x " " " " " " 2 2 2 -1 1 -1 1 -1 1 !" - !" !" 2 2 2 !" !" !" 1 Although sin− x and (restricted) sin x are transcendental functions, their composition 1 sin− (sin x)=x is algebraic. This is obvious because these functions are inverses of each other. However, something more general and surprising is true: the composition of any inverse trigonometric function with any trigonometric function is algebraic; see Example 4. Examples and Explorations Example 1 Finding values of transcendental functions by hand Calculate each of the following by hand, without a calculator. 5π 1 1 (a) log6 3 + log6 12 (b) cos 6 (c) sin− 2 Solution. ? (a) log6 3 is the exponent to which we would have to raise 6 to get 2; think 6 =3. It is not immediately apparent what this exponent is. Similarly, it is not clear how to calculate log6 12 without a calculator. However, using the additive property of logs we can write log 3 + log 12 = log (3 12) = log 36 = 2. 6 6 6 · 6 The final equality above holds since 62 =36. 5π (b) The diagram below left shows where the angle 6 lies on the unit circle. If we draw a line from the point (x, y) where the angle meets the unit circle to the x-axis, we obtain a triangle whose reference angle is 30◦. Using the known side lengths of a 30–60–90 triangle with hypotenuse of length one, we can label the side lengths of our reference triangle, as shown below middle. This in turn means that we know the coordinates (x, y)=( √3 , 1 ) of − 2 2 the point at which the terminal edge of θ intersects the unit circle. Therefore cos 5π = √3 . 6 − 2 0.4 Exponential and Trigonometric Functions 53 5π π π π Angle θ = 6 has Side lengths of a 30-60-90 6 is the angle in [ 2 , 2 ] − 1 reference angle 30◦ triangle with hypotenuse 1 whose sine is equal to 2 y y √3 1 π − , 6 5π ( 2 2 ) 1 5π 5π 2 6 q = 1 1 q = 6 6 30° 30° x 2 30° x √3 2 1 1 1 (c) If θ = sin− 2 , then we must have sin θ = 2 . There are infinitely many angles whose sine is 1 π π 1 1 2 , but only one of those angles is in the restricted domain [ 2 , 2 ] of sine. Thus θ = sin− ( 2 ) π π 1 − is the unique angle in [ 2 , 2 ] whose sine is 2 , as shown above right. Notice that the triangle − 1 must be a 30–60–90 triangle (since its height is 2 ), and therefore the angle θ we are looking π 1 1 π for must be 30◦,i.e., 6 radians. Therefore sin− 2 = 6 . Example 2 Solving equations that involve transcendental functions Solve each of the following equations: x 1 π (a) 3.25(1.72) =1000 (b) sin θ =cosθ (c) sec− x = 3 Solution. (a) To solve for x we will isolate the expression (1.72)x and then apply the natural logarithm so that we can get x out of the exponent: 3.25(1.72)x =1000 = ln((1.72)x) = ln 1000 = x ln(1.72) = ln 1000 . ⇒ 3.25 ⇒ 3.25 1000 ln ! " ! " It is now a simple matter to solve for x = “ 3.25 ” 10.564. ln(1.72) ≈ (b) If sin θ =cosθ, then θ is an angle whose terminal edge intersects the unit circle at apoint √2 √2 √2 √2 (x, y) with x = y. The only such points on the unit circle are ( 2 , 2 ) and ( 2 , 2 ),as π − − shown below left. The angles that end at these points are all of the form θ = 4 + πk for some integer k. Thus the solution set for the equation is ..., 3π , π , 5π , 9π ,... .. { − 4 4 4 4 } 1 π Diagram to solve sin θ =cosθ Diagram to solve sec− x = 3 y y π 4 π 3 √ √2 2 1 2 45 2 2 x 30 x √2 √ 45 √3 2 2 2 2 3π 4 1 π π 1 1 π (c) If sec− x = 3 , then x =sec3 = cos π = 1 =2. The angle 3 and the reference triangle we 3 2 used for this calculation are shown above right. 0.4 Exponential and Trigonometric Functions 54 Example 3 Domains and graphs of transcendental functions Find the domain of each of the following functions. Then use transformations to sketch careful graphs of each function by hand, without a graphing utility. 1.7x 1 (a) f(x)=5 3e (b) g(x)= ln(x 2) (c) h(x)=3sec2x − − Solution. (a) The domain of f(x)=5 3e1.7x is all of R, and its graph is a transformation of the expo- − nential growth function e1.7x shown below left. y = 3e1.7x can be obtained by reflecting the − leftmost graph over the y-axis and stretching vertically by a factor of three, as shownbelow middle. The graph of f(x)=5 3e1.7x can now be obtained by shifting the middle graph up − five units, as shown below right. y = e1.7x y = 3e1.7x y =5 3e1.7x − − 5 -3 2 1 (b) For the function g(x)= 1 to be defined at a value x, we must have x 2 > 0,and ln(x 2) − thus x>2. We must also have−ln(x 2) =0, which means that x 2 =1, and thus x =3. − ! − ! ! Therefore the domain of g(x) is (2, 3) (3, ). To sketch the graph of g(x)= 1 we start ∪ ∞ ln(x 2) with the graph of y = lnx shown below right, translate to the right two units as shown− below middle, and then sketch the reciprocal as shown below right. 1 y =lnxy=ln(x 2) y = − ln(x 2) − 3 3 3 2 2 2 1 1 1 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 -1 -1 -1 -2 -2 -2 -3 -3 -3 3 (c) The function h(x)=3sec2x = cos 2x is defined when cos2x =0.Thishappenswhen2x is π π ! π not a multiple of , and thus when x is not a multiple of . Thus the domain of h(x) is x = k 2 4 ! 4 for positive integers k. To sketch the graph of h(x), we start with the graph of y =secx below left, stretch vertically by a factor of 3 as shown below middle, and then compress horizontally by a factor of 2 as shown below right. 0.4 Exponential and Trigonometric Functions 55 y =secx y =3secxy=3sec2x 9 9 9 6 6 6 3 3 3 Π Π Π Π Π Π !Π ! Π !Π ! Π !Π ! Π 2 -3 2 2 -3 2 2 -3 2 -6 -6 -6 -9 -9 -9 Example 4 Simplifying compositions of inverse trigonometric and trigonometric functions 1 Write cos(sin− x) as an algebraic function, that is, a function that involves only arithmetic operations and rational powers. 1 π π Solution. If we define θ = sin− x, then sin θ = x and θ must be in the interval [ 2 , 2 ]. Let’s π − first consider the case where θ is in the first quadrant [0, 2 ]; the reference triangle for such a θ is shown below left. If we wish θ to have a sine of x then the length of the vertical leg of the triangle must be x. The hypotenuse of the triangle is length 1, since we are on the unit circle. We could also have considered that the sine of θ is “opposite over hypotenuse”; thus one triangle involving our angle θ could have an opposite side of length x and a hypotenuse of length 1. Using the Pythagorean theorem, we find that the length of the remaining leg of the triangle is √1 x2, as shown below right. − Reference triangle for Use Pythagorean Theorem to π an angle θ in [0, 2 ] determine length of remaining leg 1 # x # $1!x2 Now cos θ is the horizontal coordinate of the point on the unit circle corresponding to θ,orin terms of “adjacent over hypotenuse,” we have: √1 x2 cos θ = − = 1 x2. 1 − # π The case where θ is in the fourth quadrant, i.e., where θ [ 2 , 0],issimilarandalsoshows 2 ∈ − 1 that cos θ = √1 x . Therefore we have shown that cos(sin− x) is equal to the algebraic − function √1 x2. − 1 2 Checking the Answer. To verify the strange fact that cos(sin− x)=√1 x , try evaluating − both sides at some simple x-values. While looking at just a few x-values will not prove that the two expressions are equal for all x, it will at least give us some evidence that the equality is reasonable. For example, at x =0we have 1 2 cos(sin− 0) = cos0 = 1 and 1 0 = √1=1, − # 0.4 Exponential and Trigonometric Functions 56 and at x =1we have 1 π 2 cos(sin− 1) = cos( )=0 and 1 1 = √0=0. 2 − # 1 As a less trivial example, consider x = 2 . At this value we have 1 1 π √3 1 2 1 3 √3 cos(sin− ( )) = cos( )= and 1 ( ) = 1 = = . 2 6 2 − 2 − 4 4 2 $ $ $ ? Questions. Test your understanding of the reading by answering these questions: ! Why do we require that A =0and b>0, b =1in the definition of exponential functions? ! ! What would the graphs look like when A =0,whenb<0, b =0,orb =1? ln 7 ! In the reading we calculated log7 2 by finding ln 2 with a calculator. Would we get the same answer if we computed log10 7 ? log10 2 ! How do you convert from radians to degrees, or vice-versa? ! How is the graph of the reciprocal of a function related to the graph of that function? How can that information be useful for remembering the graphsofy =cscx, y =secx, and y =cotx? ! How are the unit circle definitions of the trigonometric functions related to the right- triangle definitions of trigonometric functions? Exercises 0.4 Thinking Back Algebra with exponents: Write each of the following expres- Famous triangles, degrees, and radians: The following exer- sions in the form Abx for some real numbers A and b. cises will help you review and recall basic trigonometry. 2x+1 x 3 x 3x 5 4 # 3 # 5 2 − # (2 − ) # Suppose a right triangle has angles 30◦, 60◦,and90◦ 1 x 2 1 x and a hypotenuse of length .Whatarethelengths 1 4(3 ) ( 8 ) # x 4 # # of the remaining legs of the triangle? 2(3 − ) 2x 3(23x+1) # Suppose a right triangle has angles 45◦, 45◦,and90◦ and a hypotenuse of length 1.Whatarethelengths Inverse functions: Suppose f and g are inverses of each of the remaining legs of the triangle? other. # What is a radian? Is it larger or smaller than a de- # What can you say about f(g(x)) and g(f(x))? gree? Compare an angle of one degree with an angle # If f has a horizontal asymptote at y =0,whatcan of one radian in standard position. you say about g? # Show each of the following angles in standard posi- # If f has a y-intercept at y =1, what can you say about tion on the unit circle, in radians: g? # 3π # 4π # 17π # 21π 4 − 3 6 Concepts 0. Problem Zero: Read the section and make your own 4. In this exercise we will examine two ways to think of summary of the material. ab when b is an irrational number, and in particular π 1. True/False: Determine whether each of the following what the quantity 2 represents. statements is true or false. If a statement is true, ex- (a) One way to define 2π is to think of it as a limit. plain why. If a statement is false, provide a coun- If we take a sequence a1,a2,a3,... of rational terexample. numbers that approaches π, then the sequence a1 a2 a3 π (a) True or False: The function f(x)=3e0.5x 2 is 2 , 2 , 2 ,... should approach 2 .Saidin − an exponential function. terms of limits, this means that: (b) True or False: Every exponential function 2π =lim2a, kx a π f(x)=Ae has a horizontal asymptote at y=0. → (c) TrueorFalse: Forall x>0, ln(x3)=3lnx. where each a is assumed to be a rational num- log2 x log6 x ber. Can you think of a sequence of rational (d) Trueor False: For all x>0, = . numbers that get closer and closer to π?(Hint: log2 3 log6 3 Think about the decimal expansion of π.) (e) True or False: If (x, y) is the point on the unit π circle corresponding to the angle 7π ,thenx is (b) Another way to consider 2 is to write it as an − 3 positive and y is negative. infinite product: (f) True or False: The sine of an angle θ is always π 3 1 4 1 5 9 2 =2 2 10 2 100 2 1000 2 10000 2 100000 . equal to the sine of the reference angle for θ. ··· (g) True or False: For any x, 1 cos2(5x3)= What will the next term in the product be? How − sin2(5x3). could 2π equal the product of infinitely many π 1 1 numbers? Wouldn’t that make 2 infinitely − (h) True or False: sec x = 1 . cos− x large? Calculate some of the later terms in the 5 9 2. Examples: Give examplesof each of the following. Try product (for example, 2 10000 or 2 100000 ) and use π to find examples that are different than any in the these calculations to argue that even though 2 reading. can be written as a product of infinitely many (a) Two exponential functions and their inverses. numbers, it is not necessarily infinitely large. (b) Two x-values at which tan x is not defined. 5. Approximate 2√3 by calculating 2r for rational val- 1 ues r that get closer and closer to √3. (Hint: You can (c) Two x-values at which sec− x is not defined. use the decimal expansion of √3 to get a sequence of ratio- 3. What is the definition of an exponential function, and nal numbers that approaches √3.) how is it different from a power function? Is the func- tion f(x)=xx a power function, an exponential func- 6. Why can’t we define the number e just by writing it tion, or neither, and why? down in decimal notation to lots of decimal places? 0.4 Exponential and Trigonometric Functions 58 2x 7. Write the exponential function f(x)=3e− in the 17. Use the definition of the sine function to explain why form Abx for some real numbers A and b.Thenwrite sin( π ) is equal to sin( 9π ) and sin( 7π ). 4 4 − 4 the exponential function g(x)= 2(3x) in the form − 18. Fill in each blank with an interval of real numbers. Aekx for some real numbers A and k. (a) The function f(x)=cosx has domain and 8. Fill in each blank with an interval of real numbers. range . (a) An exponential function f(x)=Abx represents (b) The function f(x)=cscx has domain and exponential growth if b ,andexponential ∈ range . decay if b . ∈ kx (c) The restricted tangent function has domain (b) An exponential function f(x)=Ae repre- and range . sents exponential growth if k ,andex- ∈ 1 ponential decay if k . (d) The function f(x)=sec− x has domain ∈ and range . (c) Suppose that ekx = bx for some real numbers k and b.Thenk (0, ) if and only if b . 19. Suppose θ is an angle in standard position whose ter- ∈ ∞ ∈ kx x minal edge intersects the unit circle at the point (x, y). e = b k 1 (d) Suppose that for some real numbers If y = , what are the possible values of cos θ?If and b.Thenk ( , 0) if and only if b . − 3 ∈ −∞ ∈ you know that the terminal edge of θ is in the third 9. In the definition of the logarithmic function logb x, quadrant, what can you say about cos θ?Whatifthe what are the allowable values for the base b,and terminal edge of θ is in the fourth quadrant? Could why? the terminal edge of θ be in the first or second quad- 10. Fill in the blanks in each statement below. rant? 20.Showthat √3 is in the range of tangent by finding (a) For all x , log2 x = y if and only if x = − ∈ an angle θ for which tan θ = √3. . − (b) For all x , 3log3 x = . 21. Describe restricted domains for sin x, tan x,andsec x ∈ on which each function is invertible. Then describe (c) For all x , log (4x)= . ∈ 4 the corresponding domains and ranges for arcsin x, (d) log2 3 is the exponent to which you have to raise arctan x,andarcsecx. to get . 22.Fillintheblanks: 1 11.Thegraphsofy =log2 x and y =log4 x are shown be- (a) sin− x is the angle in the interval whose low. Determine which graph is which, without using is x. y =2x a calculator. (Hint: Think about the graphs (b) y = arcsin x if and only if sin y = ,forall y =4x and ,andthenreflectthosegraphsoverthe x and y . line y = x.) ∈ 1 ∈ (c) If tan− x = θ and tan θ is positive, then θ is in y =log2 x and y =log4 x 2 the quadrant. 1 1 (d) If arctan x = θ and sin θ = 3 ,thencos θ = . 1 2 3 4 23.Whichofthefollowingexpressionsaredefined? -1 Why or why not? 1 1 1 3 -2 (a) sin− ( ) (b) sin− − 25 2 12. State the algebraic properties of the natural logarithm 1 1 π (c) tan− 100 (d) sec− 4 function that correspond to the eight properties of logarithmic functions in Theorem 0.24. 24. Sketch a graph of the restricted cosine function on the domain [0, π] and argue that this restricted function is 13. Use algebraic properties of logarithms, the graph of 1 one-to-one. Then sketch a graph of cos− x,andlist y =lnx, and your knowledge of transformations to 1 the domains and ranges of the inverse cos− x of this sketch graphs of f(x)=ln(x2) and g(x)=ln(1 ). x restricted function. x+1 14. Solve the inequality ln( x 1 ) 0. 25. Without calculating the exact or approximate values − ≥ 15. Give a mathematical definition of sin θ for any angle of the following expressions, use the unit circle to de- θ. Your definition should include the words “unit termine whether each of the following quantities is circle,” “standard position,” “terminal,” and “coor- positive or negative. 1 1 1 2 dinate.” (a) sin− ( ) (b) sin− ( ) − 5 − 3 16. Give a mathematical definition of tan θ for any an- 1 1 (c) tan− 2 (d) sec− ( 5) gle θ. Your definition should include the words “unit − circle,” “standard position,” “terminal,” and “coordi- 26.Findallangleswhosesecantis2,andthenfind 1 nate.” sec− (2). 0.4 Exponential and Trigonometric Functions 59 Skills Find the domains of the functions in Exercises 27–32. 1 1 57. sin(cos− x)58. tan(tan− 2x) ln(x +1) 1 2 1 2 1 27. f(x)= 28. f(x)= 59. sec (tan− x)60. sin (tan− x) ln(x 2) ex e2x − − 1 x 1 61. sin(sec− ) − 1 1 3 62. csc(2 tan x) 29. f(x)= 30. f(x)= 1 2 1 x ln(x 1) 1 tan θ 64 tan (2 sec− ) − − 63. cos(2 sin− 5x) . 3 1 31. f(x)=√psec θ 32. f(x)=2sin− (x 3) − Sketch graphs of the functions in Exercises 65–72 by hand, without using a calculator or graphing utility. Indicate any Find the exact values of each of the quantities in Exer- roots, intercepts, and asymptotes on your graphs. cises 33–44. Do not use a calculator. 1 1 x x 2 33. ln( ) 34. log 1 4 65. f(x)= ( 2 ) +10 66. f(x)= 0.25(3 − ) e2 2 − − 2x − 68. f(x)=log1 x log7 9 67. f(x)=20 5e 2 35. 4log2 6 2log2 9 36. 1 +log3 1 − − log7 3 69. f(x)= log2(x 3) 70. f(x)=sin(2x)+4 π 48π − − π 37. tan( 4 ) 38. cos( 3 ) 1 − 71. f(x)=2cos(x 4 ) 72. f(x)=tan− (x 2) + π − − 39. csc( 5π )40. sin(201π) − 4 For each graph in Exercises 73–78, find a function whose 1 1 41. cos− ( 1) 42. sin− ( 1) graph looks like the one shown. When you are finished, − − use a graphing utility to check that your function f has the 43.arcsec( 2 )44. arctan( 1 ) − √2 − √3 properties and features of the given graph. 4 4 Solve the equations in Exercises 45–50 by hand. When you 3 3 2 2 are finished, check your answers either by testing your so- 1 1 73. -4 -3 -2 -1 1 2 3 4 74. -4 -3 -2 -1 1 2 3 4 lutions or by graphing an appropriate function. -1 -1 -2 -2 x x 1 46 2=10(1+0.19 )12x -3 -3 45. 2 =3 − . 12 -4 -4 x 1 1 47. log2( x−+1 )=4 48. sin x = 2 15 2 1 1 10 49. cos 2x =1 50. sec− x = π -5 -4 -3 -2 -1 1 2 3 75. 5 76. -1 1 3 -2 -1 1 2 Suppose that cos(θ)= 6 , sin(θ) > 0, sin(φ)= 5 ,and -3 cos(φ) < 0. Use trigonometric identities to identify the -5 -4 quantities in Exercises 51–56. 1 6 5 51. sin(θ)52. sin( φ) 4 − 3 π 2 53. cos(2θ) 54. sin(θ + 2 ) 77. !" !" " " 78. 2 2 1 !2" !" " 2" 55.thesignofcos(θ + φ)56.thesignoftan(θ + π) -1 -1 -2 Write each of the expressions Exercises 57–60 as an alge- braic expression that does not involve trigonometric or in- verse trigonometric functions. Applications 0.4 Exponential and Trigonometric Functions 60 79.Tenyearsago,Jennydeposited$10, 000 into an in- 81. Suppose a rock sample initially contains 250 grams vestment account. Her investment account now of the radioactive substance unobtainium, and that holds $22, 609.80. Her accountant tells her that her the amount of unobtainium after t years is given by investment account balance I(t) is an exponential an exponential function of the form S(t)=Aekt.The function. half-life of unobtainium is 29 years, which means that (a) Find an exponential function of the form I(t)= it takes 29 years for the amount of the substance to Aekt to model Jenny’s investment account bal- decrease by half. ance. (a) Find an equation for the exponential function (b) Find an exponential function of the form I(t)= S(t). t Ab to model Jenny’s investment account bal- (b) What percentage of unobtainium decays each ance. year? 80. Suppose there were 500 rats on a certain island in (c) How long will it be before the rock sample con- 1973, and 1697 rats on the same island ten years later. tains only 6 grams of unobtainium? Assume that the number R(t) of rats on the island t 82 years after 1973 is an exponential function. . Again considering the rock sample described in Ex- ercise 81, answer the following questions: (a) Find an equation for the exponential function R(t) that describes the number of rats on the is- (a) At one point the rock sample contained 900 land. Let t =0represent the year 1973. grams of unobtainium; how long ago? (b) What percentage of the unobtainium will be left (b) According to your function R(t), how many rats will be on the island in 2020? in 300 years? (c) How long did it take for the population of rats (c) How long will it be before 95% of the unob- to double from its 1973 amount? How long did tainium has decayed? it take for it to double again? And again? 83. Alina is flying a kite, and has managed to get her kite so high in the air that she has let out 400 feet of kite string. If the angle made by the ground and the line of kite string is 32 degrees, how high is the kite? 84. Suppose two stars are each 60 light years away from Earth. The angle between the line of sight to the first star and the line of sight to the second star is two de- grees. In other words, if you look at the first star, then turn your head to look at the second star, your head will move through an angle of two degrees. How far apart are the stars? Proofs 85. Prove by contradiction that every exponential func- 90. Use the fact that logarithmic functions are the in- tion f(x)=Abx has the property that f(x) is never verses of exponential functions to prove that: zero. y (a) logb x = y if and only if b = x 86. Use the definition of a one-to-one function to prove (b) log (bx)=x that every exponential function f(x)=Abx is one-to- b log x one. (c) b b = x 87. Use the base conversion formula for logarithms to a 91.Provethatlogb(x )=a logb x. (Hint: Start with prove that the function f(x)=log2 x is equal to the a logb x logb(x ) and replace x with b .) function g(x)=log3 x only when x =1. 92.Provethatlogb(xy)=logb x +logb y. (Hint: Show 88. Use logarithms to prove that every exponential func- logb x+logb y x this is equivalent to the statement xy = b ,and tion of the form f(x)=Ab can be written in the form prove this new statement instead.) f(x)=Aekx,andvice-versa. 93. Use the results of the two problems above to prove 89 y =log x . Use the definition of a logarithmic function b that: to prove that for any b>0 with b =1, the quantity 1 ' (a) logb( x )= logb x logb 1 is equal to zero. − x (b) logb( y )=logb x logb y In Exercises 90–94, assume that x, y, a,andb are values − 94. Prove the base conversion formula log x = loga x . which make sense in the expressions involved. b loga b y (Hint: Set y =logb x and then show that b = x.) 0.4 Exponential and Trigonometric Functions 61 95. Use the unit circle definitions of sine and cosine to 98. Use the sum identities and the even/odd identities to prove the identity sin2 θ +cos2 θ =1. prove the difference identities listed in Theorem 0.26. 96.UsethefirstPythagoreanidentitysin2 θ +cos2 θ =1 99. Use the sum identities to prove the double angle to prove the second and third Pythagorean identities identities listed in Theorem 0.26. (Hint: Note that 2θ listed in Theorem 0.26. (Hint: To prove the second iden- is equal to θ + θ.) 2 tity, divide both sides of the first identity by cos x.A 100. The four identities listed as alternate forms in Theo- similar trick will prove the third identity.) rem 0.26 are alternate ways of writing the double an- 97. Use the unit circle definitions of the trigonometric gle identity cos 2θ =cos2 θ sin2 θ. Use this double − functions to prove the even/odd identities and the angle identity, algebra, and the Pythagorean identi- shift identities listed in Theorem 0.26. ties to prove these four alternate forms. Thinking Forward # A special exponential limit: Use a calculator to approx- # Rewriting trigonometric expressions: Use the double an- eh 1 2 1 cos 2x imate − for the following values of h:(a)h =0.1; gle identity sin x = − 2 to rewrite the expres- h 4 2 (b) h =0.01;(c)h =0.001.Ash gets closer to zero, sion sin x cos x in terms of a sum of expressions of what number do your approximations seem to ap- the form A cos kx. (Note: You’ll have to multiply out proach? some expressions, and use the double angle identity more than once.) # Logarithms with absolute values: Sketch a graph of the function f(x)=lnx . What is the domain of this | | function? Is this function even, odd, or neither, and why? Appendix A Answers To Odd Problems 83. If f(x)=Ax3 + lower-degree terms and 21. See Definition 0.27 for the restricted do- g(x)=Bx3 + lower-degree terms, then mains of the trigonometric functions, and f(x)g(x)=ABx6 + lower-degree terms. thus the ranges of the inverse trigonometric Since f and g are cubic we know that A functions. The domain of arcsin x is [ 1, 1], − and B are nonzero. Thus AB must also be the domain of arctan x is all of R,andthe nonzero, and therefore fg is of degree 6. domain of arcsec x is ( , 1] [1, ). −∞ − ∪ ∞ 85. (a) If f(x)=k for all k,thenf(x)=k =0x + Their ranges are the restricted domains of k is also a linear function. (b) If f(x)=mx + sin x, tan x,andsec x,respectively. b is a linear function with m =0,thenf is 23. Only (a) and (c) are defined. ! a polynomial of degree 1 with coefficients 25. (a) negative; (b) negative; (c) positive; (d) a1 = m and a0 = b.Iff(x)=mx + b with positive m =0then f is a polynomial of degree zero 27. (2, 3) (3, ) with sole coefficient a0 = b. ∪ ∞ 29. (2, ) 86. The domain of a quotient f(x)= p(x) ∞ q(x) 31. ... ( 5π , 3π ) ( π , π ) ( 5π , 7π ) ... of functions is x x Domain(p(x)) ∪ − 2 − 2 ∪ − 2 2 ∪ 2 2 ∪ { | ∈ ∩ 2 Domain(q(x)) and q(x) =0.Sincep(x) 33. ! } − and q(x) are polynomials, they are defined 35. 4 on ( , ); thus the domain of f is x 37. 1 −∞ ∞ { | − q(x) =0 . √ ! } 39. 2 Section 0.4 41. π 3π 1. F, T, T, T, T, F, T, F. 43. 4 x = ln 3 2.70951 3. A function is exponential if it can be written 45. ln( 3 ) 2 ≈ in the form f(x)=Abx;thevariableisinthe 47. x = 17 exponent and a constant is in the base. For a − 15 power function, the situation is reversed. xx 49. x = πk,wherek is any integer √35 is neither a power nor an exponential func- 51. 6 tion because a variable appears in both the 17 53. 18 base and the exponent. − 55. Negative √ 1.7 5. 3 1.73205.Wehave2 3.2490, √ 2 1.73≈ 1.732 ≈1.7320 57. 1 x 2 3.3173, 2 3.3219, 2 2 − ≈ 1.73205 ≈ ≈ 59. x +1 3.3219, 2 3.3220,andsoon.Each ≈ 61. 1 ( x )2 of these approximations gets closer to the − 3 √3 2 value of 2 . 63. 1p 2(5x) 2 x x − 1 x 7. f(x)=3(e− ) 3(0.135) , g(x)= 65. Start with the graph of y =( ) ,thenreflect ≈ 2 2e(ln 3)x 2e1.0986x. over the x-axis and shift up by 10 units. − ≈− 9. We must have b>0 and b =1,sincethose ! x 20 conditions are necessary for b to be an ex- 15 ponential function. 67. 11. The graph that passes through (2, 1) is -1 1 2 1 log2 x; the graph that passes through (2, 2 ) is log x. 6 4 5 f(x)=2lnx ln x 4 13. is the graph of stretched 3 vertically by a factor of 2; g(x)= ln x is 69. 2 − 1 ln x x 1 2 3 4 5 6 7 8 the graph of reflected over the -axis. -1 -2 15. If θ is an angle in standard position, then -3 sin θ the vertical coordinate y of the point 2 (x, y) where the terminal edge of θ intersects 1 the unit circle. 71. !5" !3" !" " 3" 5" π 9π 4 4 4 4 4 4 17. The terminal edges of the angles 4 , 4 ,and -1 7π 4 all meet the unit circle at the same − -2 point (and in particular, at the same y- x 73. f(x)=2e− 3 coordinate). −x 75. f(x)= 5e− +10 19. cos θ = x is √8 if θ is in the third quadrant; − − 3 77. f(x)= cos 2x cos θ = √8 if θ is in the fourth quadrant; θ − 3 79. (a) I(t) 10, 000e0.08t ;(b)I(t)= cannot be in the first or second quadrant. ≈ 10, 000(1.085)t . ln 2 t 0.0239t 81. (a) S(t)=250e− 29 250e− ,or 9. The contrapositive is “Not(Q) Not(P ),” ≈ t ⇒ equivalently, S(t)=250(0.97638) ;(b) which is logically equivalent to P Q. ⇒ 2.39%;(c)156 years 11. It is always better to switch! Can you explain 83. 211.97 feet why? 85. Seeking a contradiction, suppose that A and 13. All integers greater than or equal to 4. x b are nonzero real numbers with Ab =0for 15. True. The negation is “For all real numbers some real number x.SinceA =0we know x, x 2 and x 3.” x ' x ≤ ≥ b =0,andthereforethatx(ln b)=ln(b )= 17. True. The negation is “There exists a real ln 0. But this is a contradiction because ln 0 number that is both rational and irrational.” is undefined, so the product x(ln b) of real 19. True. The negation is “There exists x such numbers cannot equal ln 0. that, for all y, y = x2.” (In other words, ' 87. log x =logx ln x = ln x “There exists x for which there is no y with 2 3 ⇔ ln 2 ln 3 ⇔ 2 (ln 3)(ln x)=(ln2)(lnx) (ln x)(ln3 y = x .)” ⇔ − ln 2) = 0 ln x =0 x =1. 21. True. The negation is “There is some integer ⇔ ⇔ y x greater than 1 for which x<2.” 89. y =logb x if and only if b = x.Sincethe only solution to by =1is y =0(if b =0), we 23. False. One counterexample is x =1.35. ' know that logb 1=0. 25. False. logb x a 91. Since x = b ,wehavelogb(x )= 27. False. One counterexample is x = 1. logb x a (logb x)a − logb((b ) )=logb(b = 29. True. One example is x =3. (log x)a = a log x. b b 31. True. The negation is “There exist real num- 1 1 93. (a) logb( x )=logb(x− )= logb x;(b) bers a and b such that a