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 willinvestigate 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, and b != 1.

There is an important technical problem with this definition: we know what it means to raisea number to a rational power by using integer roots and powers, but we don’t know whatit means to raise a number to an irrational power. We need to be able to raise to irrationalpowers to talk about exponential functions; for example, if f(x) = 2x then we need to be ableto compute f(") = 2!. One way to think of bx where x is irrational is as a limit:

bx = limr!x

r rational

br.

The “lim” notation will be explored more in Chapter 1. For now you can just imagine that ifx is rational we can approximate bx by looking at quantities br for various rational numbers rthat get closer and close to the irrational number x.For example, 2! can be approximated by2r for rational numbers r that are close to ":

2! " 23.14 = 2314100 =

100#

2314.

As we consider rational numbers r that are closer and closer to ", the expression 2r will getcloser and closer to 2!; see Exercise 4. In Chapter xxx we will give a more rigorous definitionof 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 = 2 or 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 thereforecalled 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 approximationof 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 itsbase is the natural number e:

0.4 Exponential and Trigonometric Functions 48

Theorem 0.22 Natural Exponential Functions

Every exponential function can be written in the form

f(x) = Aekx

for some real number A and some nonzero real number k.

Ev-

ery exponential function has a graph similar to either the exponential growth graph belowleft 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) = bx with b > 1

f(x) = ekx with k < 0,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 naturallogarithmic function

g(x) = lnx.

We require that the base b satisfies b > 0 and b != 1, because these are exactly the conditions wemust have for y = bx to be an exponential function. In Section xxx we will define logarithmsanother way, in terms of integrals and accumulation functions.

You should already be familiar with the algebraic rules of logarithms, but we restate themhere in case you need a refresher; see Exercises 90–94 for proofs.


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:

(a) logb x = y if and only if by = x

(b) logb(bx) = x

(c) blogb x = x

(d) logb(xa) = a logb x

(e) logb(xy) = logb x + logb y

(f) logb(1x) = $ logb x

(g) logb(xy) = logb x$ logb y

(h) logb x = loga xloga b

The first three properties follow from properties of inverse functions, and tell us that logb x isthe exponent to which you have to raise b in order to get x. For example, log2 8 is the powerto which you have to raise 2 to get 8; since 23 = 8 we have log2 8 = 3. All of these rules alsoapply 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 baseconversion 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 thatwe can calculate logarithms on a calculator. For example, log7 2 is equal to ln 7

ln2 , which we canapproximate using the built-in ln key on a calculator.

The graphs of logarithmic functions can be obtained easily from the graphs of exponentialfunctions by reflection over the line y = x, as shown below.

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, ormore generally, as ratios of coordinate lengths on the unit circle. We now provide a quickreview 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 # in standard position, the terminal edge of # intersects the unit circle atsome point (x, y) in the xy-plane. We will define the height y of that point to be the sine of #,while the cosine of # will be defined as the x-coordinate of that point.


Definition 0.25 Trigonometric Functions for Any Angle

Given any angle # measured in radians in standard position, let (x, y) be the point wherethe terminal edge of # intersects the unit circle. The six trigonometric functions of anangle # are the six possible ratios of the coordinates x and y for #:

!

(x,y)(cos !, sin !)

x

y

sin # = y cos# = x tan# =y

x

csc # =1

ysec# =

1

xcot# =

x

y

Notice that the sine and cosine functions determine the remaining four trigonometric func-tions, since, tan# is the ratio sin"

cos" , and the last three trigonometric functions are the reciprocalsof 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 thetrigonometric functions. The sum identities follow from a geometric argument that we willnot get into here. The remaining identities can all be proved from the previous identities. Inthese identities we are using the notation sin2 x as shorthand for (sinx)2.

Theorem 0.26 Basic Trigonometric Identities

Pythagorean Identities Even/Odd Identities Shift Identities

sin2 # + cos2 # = 1 sin($#) = $ sin# cos(# $ !2 ) = sin #

tan2 # + 1 = sec2 # cos($#) = cos# sin(# + !2 ) = cos#

1 + cot2 # = csc2 # tan($#) = $ tan# sin(# + 2") = sin #

cos(# + 2") = cos#

Sum Identities Difference Identities

sin($ + %) = sin$ cos% + sin% cos$ sin($$ %) = sin$ cos% $ sin% cos$

cos($ + %) = cos$ cos% $ sin$ sin% cos($$ %) = cos$ cos% + sin$ sin%

Double Angle Identities Alternate Forms Alternate Forms

sin 2# = 2sin# cos# cos2# = 1$ 2 sin2 # sin2 # = 1"cos2"2

cos2# = cos2 # $ sin2 # cos2# = 2cos2 # $ 1 cos2 # = 1+cos2"2

The graphs of the six trigonometric functions are recorded below. Each of the graphs inthe second row is the reciprocal of the graph immediately above it. Remember that you canuse the graph of a function f to sketch the graph of its reciprocal 1

f. In particular, the zeros of

f will be vertical asymptotes of 1f

, large heights on the graph of f will become small heights

on the graph of 1f , and vice-versa.


y = sinx y = cosx y = tanx

" 2" 3"#"#2"#3"

#2

#1

2

1

" 2" 3"#"#2"#3"

#2

#1

2

1

" 2" 3"#"#2"#3"

#3

3

#2

#1

2

1

y = cscx y = secx y = cotx

" 2" 3"#"#2"#3"

#3

3

#2

#1

2

1

" 2" 3"#"#2"#3"

#3

3

#2

#1

2

1

" 2" 3"#"#2"#3"

#3

3

#2

#1

2

1

Inverse Trigonometric Functions

None of the six trigonometric functions are one-to-one, but after restricting domains we canconstruct the so-called inverse trigonometric functions. In this section we will focus on theinverses of only three of the six inverse trigonometric functions, those for sine, tangent, andsecant. 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 andstick with it. In this text we will use the restricted domains shown below.

y = sinx restricted tothe domain [!!

2 , !2 ]

y = tanx restricted tothe domain (!!

2 , !2 )

y = secx restricted tothe domain [0, !

2 )" (!2 ,!]

-1

1

!" !"2

"2

" !" !"2

"2

"-1

1

-1

1

!" !"2

"2

"

Each of the restricted functions shown above is one-to-one, and thus invertible. The inversesof these restricted functions are the inverse sine, inverse tangent, and inverse secant functions.

Definition 0.27 The Inverse Trigonometric Functions

(a) The inverse sine function sin"1 x is the inverse of the restriction of the functionsinx to [$!

2 , !2 ].

(b) The inverse tangent function tan"1 x is the inverse of the restriction of thefunction tanx to ($!

2 , !2 ).

(c) The inverse secant function sec"1 x is the inverse of the restriction of the func-tion secx to [0, !

2 ) % (!2 ,"].


Notice that since the inputs to the trigonometric functions are angles, it is the outputs of theinverse trigonometric functions that are angles. We will interchangeably use the alternativenotations arcsinx, arctanx, and arcsec x for these inverse trigonometric functions.

All of the properties of sin"1 x, tan"1 x, and sec"1 x come from the fact that they arethe inverses of the restricted functions sinx, tanx, and secx. For example, we can graph theinverse trigonometric functions simply by reflecting the graphs of the restricted trigonometricfunctions over the line y = x, as shown below.

y = sin!1 x y = tan!1 x y = sec!1 x

-1 1

!"

!"2

"2

"

-1 1

-

!"

!"2

"2

"

-1 1

!"

!"2

"2

"

Although sin"1 x and (restricted) sinx are transcendental functions, their compositionsin"1(sinx) = x is algebraic. This is obvious because these functions are inverses of eachother. However, something more general and surprising is true: the composition of anyinverse 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.

(a) log6 3 + log6 12 (b) cos 5!6 (c) sin"1 1

2

Solution.

(a) log6 3 is the exponent to which we would have to raise 6 to get 2; think 6? = 3. It is notimmediately apparent what this exponent is. Similarly, it is not clear how to calculate log6 12without a calculator. However, using the additive property of logs we can write

log6 3 + log6 12 = log6(3 · 12) = log6 36 = 2.

The final equality above holds since 62 = 36.

(b) The diagram below left shows where the angle 5!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 atriangle whose reference angle is 30#. Using the known side lengths of a 30–60–90 trianglewith 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) = ($$

32 , 1

2 ) of

the point at which the terminal edge of # intersects the unit circle. Therefore cos 5!6 = $

$3

2 .


Angle " = 5!6 has

reference angle 30"Side lengths of a 30-60-90triangle with hypotenuse 1

!6 is the angle in [!!

2 , !2 ]

whose sine is equal to 12

x

y

5"6 q =

5"6

30°11

2 x

y

q =5"6

30°

$32

12

$32

# , )(30°

"61

2

(c) If # = sin"1 12 , then we must have sin # = 1

2 . There are infinitely many angles whose sine is12 , but only one of those angles is in the restricted domain [$!

2 , !2 ] of sine. Thus # = sin"1(1

2 )is the unique angle in [$!

2 , !2 ] whose sine is 1

2 , as shown above right. Notice that the trianglemust be a 30–60–90 triangle (since its height is 1

2 ), and therefore the angle # we are lookingfor must be 30#, i.e., !

6 radians. Therefore sin"1 12 = !

6 .

Example 2 Solving equations that involve transcendental functions

Solve each of the following equations:

(a) 3.25(1.72)x = 1000 (b) sin # = cos# (c) sec"1 x = !3

Solution.

(a) To solve for x we will isolate the expression (1.72)x and then apply the natural logarithmso that we can get x out of the exponent:

3.25(1.72)x = 1000 =& ln((1.72)x) = ln!

10003.25

"

=& x ln(1.72) = ln!

10003.25

"

.

It is now a simple matter to solve for x =ln

“

10003.25

”

ln(1.72) " 10.564.

(b) If sin# = cos#, then # is an angle whose terminal edge intersects the unit circle at a point

(x, y) with x = y. The only such points on the unit circle are ($

22 ,

$2

2 ) and ($$

22 ,$

$2

2 ), asshown below left. The angles that end at these points are all of the form # = !

4 + "k for someinteger k. Thus the solution set for the equation is {. . . ,$ 3!

4 , !4 , 5!

4 , 9!4 , . . .}..

Diagram to solve sin " = cos" Diagram to solve sec!1 x = !3

x

y

45

"4

22$

22$

45

22$

22$

3"4

x

y

30

"3

32$

12

(c) If sec"1 x = !3 , then x = sec !

3 = 1cos !

3= 1

12

= 2. The angle !3 and the reference triangle we

used for this calculation are shown above right.


Example 3 Domains and graphs of transcendental functions

Find the domain of each of the following functions. Then use transformations to sketchcareful graphs of each function by hand, without a graphing utility.

(a) f(x) = 5$ 3e1.7x (b) g(x) = 1ln(x"2) (c) h(x) = 3 sec2x

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 = $3e1.7x can be obtained by reflecting theleftmost graph over the y-axis and stretching vertically by a factor of three, as shown belowmiddle. The graph of f(x) = 5$ 3e1.7x can now be obtained by shifting the middle graph upfive units, as shown below right.

y = e1.7x y = !3e1.7x y = 5! 3e1.7x

1

-3 2

5

(b) For the function g(x) = 1ln(x"2) to be defined at a value x, we must have x $ 2 > 0, and

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

ln(x"2) we start

with the graph of y = lnx shown below right, translate to the right two units as shown belowmiddle, and then sketch the reciprocal as shown below right.

y = lnx y = ln(x! 2) y =1

ln(x! 2)

1 2 3 4 5

-3

-2

-1

1

2

3

1 2 3 4 5

-3

-2

-1

1

2

3

1 2 3 4 5

-3

-2

-1

1

2

3

(c) The function h(x) = 3 sec2x = 3cos2x

is defined when cos2x != 0. This happens when 2x isnot a multiple of !

2 , and thus when x is not a multiple of !4 . Thus the domain of h(x) is x != !

4 kfor positive integers k. To sketch the graph of h(x), we start with the graph of y = secx belowleft, stretch vertically by a factor of 3 as shown below middle, and then compress horizontallyby a factor of 2 as shown below right.


y = secx y = 3secx y = 3sec2x

!" !"

2

"

2"

-9

-6

-3

3

6

9

!" !"

2

"

2"

-9

-6

-3

3

6

9

!" !"

2

"

2"

-9

-6

-3

3

6

9

Example 4 Simplifying compositions of inverse trigonometric and trigonometric functions

Write cos(sin"1 x) as an algebraic function, that is, a function that involves only arithmeticoperations and rational powers.

Solution. If we define # = sin"1 x, then sin# = x and # must be in the interval [$!2 , !

2 ]. Let’sfirst consider the case where # is in the first quadrant [0, !

2 ]; the reference triangle for such a# is shown below left. If we wish # to have a sine of x then the length of the vertical leg ofthe triangle must be x. The hypotenuse of the triangle is length 1, since we are on the unitcircle. We could also have considered that the sine of # is “opposite over hypotenuse”; thusone triangle involving our angle # could have an opposite side of length x and a hypotenuseof length 1. Using the Pythagorean theorem, we find that the length of the remaining leg ofthe triangle is

#1$ x2, as shown below right.

Reference triangle foran angle " in [0, !

2 ]Use Pythagorean Theorem to

determine length of remaining leg

#

#

x1

1!x2$

Now cos# is the horizontal coordinate of the point on the unit circle corresponding to #, or interms of “adjacent over hypotenuse,” we have:

cos# =

#1$ x2

1=

#

1$ x2.

The case where # is in the fourth quadrant, i.e., where # ( [$!2 ,0], is similar and also shows

that cos# =#

1$ x2. Therefore we have shown that cos(sin"1 x) is equal to the algebraicfunction

#1$ x2.

Checking the Answer. To verify the strange fact that cos(sin"1 x) =#

1$ x2, try evaluatingboth sides at some simple x-values. While looking at just a few x-values will not prove thatthe two expressions are equal for all x, it will at least give us some evidence that the equalityis reasonable. For example, at x = 0 we have

cos(sin"1 0) = cos0 = 1 and#

1$ 02 =#

1 = 1,


and at x = 1 we have

cos(sin"1 1) = cos("

2) = 0 and

#

1$ 12 =#

0 = 0.

As a less trivial example, consider x = 12 . At this value we have

cos(sin"1(12 )) = cos(!

6 ) =$

32 and

$

1$ (12 )2 =

$

1$ 14 =

$

34 =

$3

2 .

? Questions. Test your understanding of the reading by answering these questions:

! Why do we require that A != 0 and b > 0, b != 1 in the definition of exponential functions?What would the graphs look like when A = 0, when b < 0, b = 0, or b = 1?

! In the reading we calculated log7 2 by finding ln 7ln2 with a calculator. Would we get the

same answer if we computed log10 7log10 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 graphs of y = 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-sions in the form Abx for some real numbers A and b.

# 32x+1 # 5x23!x # (23x!5)4

#1

2(3x!4) #4(3x)2

2x#

( 18 )x

3(23x+1)

Inverse functions: Suppose f and g are inverses of eachother.

# What can you say about f(g(x)) and g(f(x))?

# If f has a horizontal asymptote at y = 0, what canyou say about g?

# If f has a y-intercept at y = 1, what can you say aboutg?

Famous triangles, degrees, and radians: The following exer-cises will help you review and recall basic trigonometry.

# Suppose a right triangle has angles 30", 60", and 90"

and a hypotenuse of length 1. What are the lengthsof the remaining legs of the triangle?

# Suppose a right triangle has angles 45", 45", and 90"

and a hypotenuse of length 1. What are the lengthsof the remaining legs of the triangle?

# What is a radian? Is it larger or smaller than a de-gree? Compare an angle of one degree with an angleof one radian in standard position.

# Show each of the following angles in standard posi-tion on the unit circle, in radians:# 3!

4 # ! 4!3 # 17!

6 # 21!

Concepts

0. Problem Zero: Read the section and make your ownsummary of the material.

1. True/False: Determine whether each of the followingstatements is true or false. If a statement is true, ex-plain why. If a statement is false, provide a coun-terexample.

(a) True or False: The function f(x) = 3e0.5x ! 2 isan exponential function.

(b) True or False: Every exponential functionf(x) = Aekx has a horizontal asymptote at y=0.

(c) True or False: For all x > 0, ln(x3) = 3 lnx.

(d) True or False: For all x > 0,log2 xlog2 3

=log6 xlog6 3

.

(e) True or False: If (x, y) is the point on the unitcircle corresponding to the angle ! 7!

3 , then x ispositive and y is negative.

(f) True or False: The sine of an angle " is alwaysequal to the sine of the reference angle for ".

(g) True or False: For any x, 1 ! cos2(5x3) =sin2(5x3).

(h) True or False: sec!1 x =1

cos!1 x.

2. Examples: Give examples of each of the following. Tryto find examples that are different than any in thereading.

(a) Two exponential functions and their inverses.

(b) Two x-values at which tanx is not defined.

(c) Two x-values at which sec!1 x is not defined.

3. What is the definition of an exponential function, andhow is it different from a power function? Is the func-tion f(x) = xx a power function, an exponential func-tion, or neither, and why?

4. In this exercise we will examine two ways to think ofab when b is an irrational number, and in particularwhat the quantity 2! represents.

(a) One way to define 2! is to think of it as a limit.If we take a sequence a1, a2, a3, . . . of rationalnumbers that approaches !, then the sequence2a1 ,2a2 ,2a3 , . . . should approach 2! . Said interms of limits, this means that:

2! = lima#!

2a,

where each a is assumed to be a rational num-ber. Can you think of a sequence of rationalnumbers that get closer and closer to !? (Hint:Think about the decimal expansion of !.)

(b) Another way to consider 2! is to write it as aninfinite product:

2! = 23 2110 2

4100 2

11000 2

510000 2

9100000 · · · .

What will the next term in the product be? Howcould 2! equal the product of infinitely manynumbers? Wouldn’t that make 2! infinitelylarge? Calculate some of the later terms in the

product (for example, 25

10000 or 29

100000 ) and usethese calculations to argue that even though 2!

can be written as a product of infinitely manynumbers, it is not necessarily infinitely large.

5. Approximate 2$

3 by calculating 2r for rational val-ues r that get closer and closer to

#3. (Hint: You can

use the decimal expansion of#

3 to get a sequence of ratio-nal numbers that approaches

#3.)

6. Why can’t we define the number e just by writing itdown in decimal notation to lots of decimal places?


7. Write the exponential function f(x) = 3e!2x in theform Abx for some real numbers A and b. Then writethe exponential function g(x) = !2(3x) in the formAekx for some real numbers A and k.

8. Fill in each blank with an interval of real numbers.

(a) An exponential function f(x) = Abx representsexponential growth if b $ , and exponentialdecay if b $ .

(b) An exponential function f(x) = Aekx repre-sents exponential growth if k $ , and ex-ponential decay if k $ .

(c) Suppose that ekx = bx for some real numbers kand b. Then k $ (0,%) if and only if b $ .

(d) Suppose that ekx = bx for some real numbers kand b. Then k $ (!%,0) if and only if b $ .

9. In the definition of the logarithmic function logb x,what are the allowable values for the base b, andwhy?

10. Fill in the blanks in each statement below.

(a) For all x $ , log2 x = y if and only if x =.

(b) For all x $ , 3log3 x = .

(c) For all x $ , log4(4x) = .

(d) log2 3 is the exponent to which you have to raiseto get .

11. The graphs of y = log2 x and y = log4 x are shown be-low. Determine which graph is which, without usinga calculator. (Hint: Think about the graphs y = 2x

and y = 4x, and then reflect those graphs over theline y = x.)

y = log2 x and y = log4 x

1 2 3 4

-2

-1

1

2

12. State the algebraic properties of the natural logarithmfunction that correspond to the eight properties oflogarithmic functions in Theorem 0.24.

13. Use algebraic properties of logarithms, the graph ofy = lnx, and your knowledge of transformations tosketch graphs of f(x) = ln(x2) and g(x) = ln( 1

x ).

14. Solve the inequality ln( x+1x!1 ) & 0.

15. Give a mathematical definition of sin " for any angle". Your definition should include the words “unitcircle,” “standard position,” “terminal,” and “coor-dinate.”

16. Give a mathematical definition of tan " for any an-gle ". Your definition should include the words “unitcircle,” “standard position,” “terminal,” and “coordi-nate.”

17. Use the definition of the sine function to explain whysin(!

4 ) is equal to sin( 9!4 ) and sin(! 7!

4 ).

18. Fill in each blank with an interval of real numbers.

(a) The function f(x) = cosx has domain andrange .

(b) The function f(x) = cscx has domain andrange .

(c) The restricted tangent function has domainand range .

(d) The function f(x) = sec!1 x has domainand range .

19. Suppose " is an angle in standard position whose ter-minal edge intersects the unit circle at the point (x, y).If y = ! 1

3 , what are the possible values of cos"? Ifyou know that the terminal edge of " is in the thirdquadrant, what can you say about cos"? What if theterminal edge of " is in the fourth quadrant? Couldthe terminal edge of " be in the first or second quad-rant?

20. Show that !#

3 is in the range of tangent by findingan angle " for which tan " = !

#3.

21. Describe restricted domains for sinx, tanx, and secxon which each function is invertible. Then describethe corresponding domains and ranges for arcsinx,arctanx, and arcsec x.

22. Fill in the blanks:

(a) sin!1 x is the angle in the interval whoseis x.

(b) y = arcsinx if and only if siny = , for allx $ and y $ .

(c) If tan!1 x = " and tan" is positive, then " is inthe quadrant.

(d) If arctanx = " and sin " = 13 , then cos " = .

23. Which of the following expressions are defined?Why or why not?

(a) sin!1(! 125 ) (b) sin!1 3

2

(c) tan!1 100 (d) sec!1 !4

24. Sketch a graph of the restricted cosine function on thedomain [0,!] and argue that this restricted function isone-to-one. Then sketch a graph of cos!1 x, and listthe domains and ranges of the inverse cos!1 x of thisrestricted function.

25. Without calculating the exact or approximate valuesof the following expressions, use the unit circle to de-termine whether each of the following quantities ispositive or negative.

(a) sin!1(! 15 ) (b) sin!1(! 2

3 )

(c) tan!1 2 (d) sec!1(!5)

26. Find all angles whose secant is 2, and then findsec!1(2).


Skills

Find the domains of the functions in Exercises 27–32.

27. f(x) =ln(x + 1)

ln(x! 2)28. f(x) =

1ex ! e2x

29. f(x) =1

p

ln(x! 1)30. f(x) =

11! tan "

31. f(x) =#

sec" 32. f(x) = 2sin!1(x! 3)

Find the exact values of each of the quantities in Exer-cises 33–44. Do not use a calculator.

33. ln( 1e2 ) 34. log 1

24

35. 4 log2 6! 2 log2 9 36.log7 9

log713

+ log3 1

37. tan(!!4 ) 38. cos( 48!

3 )

39. csc(! 5!4 ) 40. sin(201!)

41. cos!1(!1) 42. sin!1(!1)

43. arcsec (! 2$2) 44. arctan(! 1$

3)

Solve the equations in Exercises 45–50 by hand. When youare finished, check your answers either by testing your so-lutions or by graphing an appropriate function.

45. 2x = 3x!1 46. 2 = 10(1 + 0.1912 )12x

47. log2(x!1x+1 ) = 4 48. sinx = 1

2

49. cos 2x = 1 50. sec!1 x = !

Suppose that cos(") = 16 , sin(") > 0, sin($) = 3

5 , andcos($) < 0. Use trigonometric identities to identify thequantities in Exercises 51–56.

51. sin(") 52. sin(!$)

53. cos(2") 54. sin(" + !2 )

55. the sign of cos(" + $) 56. the sign of tan(" + !)

Write each of the expressions Exercises 57–60 as an alge-braic expression that does not involve trigonometric or in-verse trigonometric functions.

57. sin(cos!1 x) 58. tan(tan!1 2x)

59. sec2(tan!1 x) 60. sin2(tan!1 x)

61. sin(sec!1 x3 ) 62. csc(2 tan!1 x)

63. cos(2 sin!1 5x) 64. tan2(2 sec!1 x3 )

Sketch graphs of the functions in Exercises 65–72 by hand,without using a calculator or graphing utility. Indicate anyroots, intercepts, and asymptotes on your graphs.

65. f(x) = !( 12 )x + 10 66. f(x) = !0.25(3x!2)

67. f(x) = 20! 5e!2x 68. f(x) = log 12

x

69. f(x) = ! log2(x! 3) 70. f(x) = sin(2x) + 4

71. f(x) = 2cos(x! !4 ) 72. f(x) = tan!1(x!2) + !

For each graph in Exercises 73–78, find a function whosegraph looks like the one shown. When you are finished,use a graphing utility to check that your function f has theproperties and features of the given graph.

73. -4 -3 -2 -1 1 2 3 4

-4

-3

-2

-1

1

2

3

4

74. -4 -3 -2 -1 1 2 3 4

-4

-3

-2

-1

1

2

3

4

75.

-1 1 2

-5

5

10

15

76.-5 -4 -3 -2 -1 1 2 3

-4

-3

-2

-1

1

2

77.

-1

1

!" !"2

"2

" 78.

-2

-1

1

2

3

4

5

6

!2" !" " 2"

Applications


79. Ten years ago, Jenny deposited $10,000 into an in-vestment account. Her investment account nowholds $22,609.80. Her accountant tells her that herinvestment account balance I(t) is an exponentialfunction.

(a) Find an exponential function of the form I(t) =Aekt to model Jenny’s investment account bal-ance.

(b) Find an exponential function of the form I(t) =Abt to model Jenny’s investment account bal-ance.

80. Suppose there were 500 rats on a certain island in1973, and 1697 rats on the same island ten years later.Assume that the number R(t) of rats on the island tyears after 1973 is an exponential function.

(a) Find an equation for the exponential functionR(t) that describes the number of rats on the is-land. Let t = 0 represent the year 1973.

(b) According to your function R(t), how manyrats will be on the island in 2020?

(c) How long did it take for the population of ratsto double from its 1973 amount? How long didit take for it to double again? And again?

81. Suppose a rock sample initially contains 250 gramsof the radioactive substance unobtainium, and thatthe amount of unobtainium after t years is given byan exponential function of the form S(t) = Aekt. Thehalf-life of unobtainium is 29 years, which means thatit takes 29 years for the amount of the substance todecrease by half.

(a) Find an equation for the exponential functionS(t).

(b) What percentage of unobtainium decays eachyear?

(c) How long will it be before the rock sample con-tains only 6 grams of unobtainium?

82. Again considering the rock sample described in Ex-ercise 81, answer the following questions:

(a) At one point the rock sample contained 900grams of unobtainium; how long ago?

(b) What percentage of the unobtainium will be leftin 300 years?

(c) How long will it be before 95% of the unob-tainium has decayed?

83. Alina is flying a kite, and has managed to get her kiteso high in the air that she has let out 400 feet of kitestring. If the angle made by the ground and the lineof kite string is 32 degrees, how high is the kite?

84. Suppose two stars are each 60 light years away fromEarth. The angle between the line of sight to the firststar and the line of sight to the second star is two de-grees. In other words, if you look at the first star, thenturn your head to look at the second star, your headwill move through an angle of two degrees. How farapart are the stars?

Proofs

85. Prove by contradiction that every exponential func-tion f(x) = Abx has the property that f(x) is neverzero.

86. Use the definition of a one-to-one function to provethat every exponential function f(x) = Abx is one-to-one.

87. Use the base conversion formula for logarithms toprove that the function f(x) = log2 x is equal to thefunction g(x) = log3 x only when x = 1.

88. Use logarithms to prove that every exponential func-tion of the form f(x) = Abx can be written in the formf(x) = Aekx, and vice-versa.

89. Use the definition of a logarithmic function y = logb xto prove that for any b > 0 with b '= 1, the quantitylogb 1 is equal to zero.

In Exercises 90–94, assume that x, y, a, and b are valueswhich make sense in the expressions involved.

90. Use the fact that logarithmic functions are the in-verses of exponential functions to prove that:

(a) logb x = y if and only if by = x

(b) logb(bx) = x

(c) blogb x = x

91. Prove that logb(xa) = a logb x. (Hint: Start with

logb(xa) and replace x with blogb x.)

92. Prove that logb(xy) = logb x + logb y. (Hint: Showthis is equivalent to the statement xy = blogb x+logb y , andprove this new statement instead.)

93. Use the results of the two problems above to provethat:

(a) logb(1x ) = ! logb x

(b) logb(xy ) = logb x! logb y

94. Prove the base conversion formula logb x = loga xloga b .

(Hint: Set y = logb x and then show that by = x.)


95. Use the unit circle definitions of sine and cosine toprove the identity sin2 " + cos2 " = 1.

96. Use the first Pythagorean identity sin2 " + cos2 " = 1to prove the second and third Pythagorean identitieslisted in Theorem 0.26. (Hint: To prove the second iden-tity, divide both sides of the first identity by cos2 x. Asimilar trick will prove the third identity.)

97. Use the unit circle definitions of the trigonometricfunctions to prove the even/odd identities and theshift identities listed in Theorem 0.26.

98. Use the sum identities and the even/odd identities toprove the difference identities listed in Theorem 0.26.

99. Use the sum identities to prove the double angleidentities listed in Theorem 0.26. (Hint: Note that 2"is equal to " + ".)

100. The four identities listed as alternate forms in Theo-rem 0.26 are alternate ways of writing the double an-gle identity cos 2" = cos2 " ! sin2 ". Use this doubleangle identity, algebra, and the Pythagorean identi-ties to prove these four alternate forms.

Thinking Forward

# A special exponential limit: Use a calculator to approx-

imate eh!1h for the following values of h: (a) h = 0.1;

(b) h = 0.01; (c) h = 0.001. As h gets closer to zero,what number do your approximations seem to ap-proach?

# Logarithms with absolute values: Sketch a graph of thefunction f(x) = ln |x|. What is the domain of thisfunction? Is this function even, odd, or neither, andwhy?

# Rewriting trigonometric expressions: Use the double an-gle identity sin2 x = 1!cos 2x

2 to rewrite the expres-sion sin4 x cos2 x in terms of a sum of expressions ofthe form A coskx. (Note: You’ll have to multiply outsome expressions, and use the double angle identitymore than once.)

Appendix A

Answers To Odd Problems

83. If f(x) = Ax3 + lower-degree terms andg(x) = Bx3 + lower-degree terms, thenf(x)g(x) = ABx6 + lower-degree terms.Since f and g are cubic we know that Aand B are nonzero. Thus AB must also benonzero, and therefore fg is of degree 6.

85. (a) If f(x) = k for all k, then f(x) = k = 0x+k is also a linear function. (b) If f(x) = mx+b is a linear function with m != 0, then f isa polynomial of degree 1 with coefficientsa1 = m and a0 = b. If f(x) = mx + b withm = 0 then f is a polynomial of degree zerowith sole coefficient a0 = b.

86. The domain of a quotient f(x) = p(x)q(x)

of functions is {x | x " Domain(p(x)) #Domain(q(x)) and q(x) != 0}. Since p(x)and q(x) are polynomials, they are definedon ($%,%); thus the domain of f is {x |q(x) != 0}.

Section 0.4

1. F, T, T, T, T, F, T, F.

3. A function is exponential if it can be writtenin the form f(x) = Abx; the variable is in theexponent and a constant is in the base. For apower function, the situation is reversed. xx

is neither a power nor an exponential func-tion because a variable appears in both thebase and the exponent.

5.&

3 ' 1.73205. We have 21.7 ' 3.2490,21.73 ' 3.3173, 21.732 ' 3.3219, 21.7320 '3.3219, 21.73205 ' 3.3220, and so on. Eachof these approximations gets closer to the

value of 2!

3.7. f(x) = 3(e"2)x ' 3(0.135)x , g(x) =

$2e(ln3)x ' $2e1.0986x .9. We must have b > 0 and b != 1, since those

conditions are necessary for bx to be an ex-ponential function.

11. The graph that passes through (2,1) islog2 x; the graph that passes through (2, 1

2 )is log4 x.

13. f(x) = 2 lnx is the graph of lnx stretchedvertically by a factor of 2; g(x) = $ lnx isthe graph of lnx reflected over the x-axis.

15. If ! is an angle in standard position, thensin ! the vertical coordinate y of the point(x, y) where the terminal edge of ! intersectsthe unit circle.

17. The terminal edges of the angles !4 , 9!

4 , and$ 7!

4 all meet the unit circle at the samepoint (and in particular, at the same y-coordinate).

19. cos! = x is $!

83 if ! is in the third quadrant;

cos! =!

83 if ! is in the fourth quadrant; !

cannot be in the first or second quadrant.

21. See Definition 0.27 for the restricted do-mains of the trigonometric functions, andthus the ranges of the inverse trigonometricfunctions. The domain of arcsinx is [$1,1],the domain of arctanx is all of R, and thedomain of arcsec x is ($%,$1] ( [1,%).Their ranges are the restricted domains ofsinx, tanx, and secx, respectively.

23. Only (a) and (c) are defined.

25. (a) negative; (b) negative; (c) positive; (d)positive

27. (2,3) ( (3,%)

29. (2,%)

31. . . .( ($ 5!2 ,$ 3!

2 )( ($!2 , !

2 )( ( 5!2 , 7!

2 )( . . .

33. $2

35. 4

37. $1

39.&

2

41. "

43. 3!4

45. x = ln3ln( 3

2 )' 2.70951

47. x = $ 1715

49. x = "k, where k is any integer

51.!

356

53. $ 1718

55. Negative

57.&

1$ x2

59. x2 + 1

61.p

1$ (x3 )2

63. 1$ 2(5x)2

65. Start with the graph of y = ( 12 )x, then reflect

over the x-axis and shift up by 10 units.

67.

-1 1 2

15

20

69.1 2 3 4 5 6 7 8

-3

-2

-1

1

2

3

4

5

6

71.

-2

-1

1

2

!3"4

!"4

"4

3"4

!5"4

5"4

73. f(x) = 2e"x $ 3

75. f(x) = $5e"x + 10

77. f(x) = $ cos2x

79. (a) I(t) ' 10,000e0.08t ; (b) I(t) =10,000(1.085)t .

81. (a) S(t) = 250e!ln 229 t ( 250e!0.0239t , or

equivalently, S(t) = 250(0.97638)t ; (b)2.39%; (c) 156 years

83. 211.97 feet

85. Seeking a contradiction, suppose that A andb are nonzero real numbers with Abx = 0 forsome real number x. Since A '= 0 we knowbx = 0, and therefore that x(ln b) = ln(bx) =ln0. But this is a contradiction because ln0is undefined, so the product x(ln b) of realnumbers cannot equal ln 0.

87. log2 x = log3 x ) lnxln2 = lnx

ln3 )(ln3)(lnx) = (ln2)(lnx) ) (lnx)(ln3 !ln2) = 0 ) lnx = 0 ) x = 1.

89. y = logb x if and only if by = x. Since theonly solution to by = 1 is y = 0 (if b '= 0), weknow that logb 1 = 0.

91. Since x = blogb x, we have logb(xa) =

logb((blogb x)a) = logb(b

(logb x)a =(logb x)a = a logb x.

93. (a) logb(1x ) = logb(x

!1) = ! logb x; (b)logb(

xy ) = logb(xy!1) = logb x! logb y.

95. (a) For any angle ", (cos", sin ") are the coor-dinates of the point where the terminal edgeof " meets the unit circle. Since the equa-tion of the unit circle is x2 + y2 = 1 ad wehave x = cos " and y = sin ", we must havesin2 " + cos2 " = 1.

97. For any angle ", sin " is the y-coordinatewhere the terminal edge of " meets the unitcircle. The angle !" is the angle of the samemagnitude as " but opening in the clock-wise direction from the x-axis, and thereforeits terminal edge will be the same as the ter-minal edge of " except flipped over the x-axis. Therefore the y-coordinates of thesetwo terminal edges have the same magni-tude but opposite signs; in other words,sin(!") = ! sin". The remaining even/oddidentities can be proved in a similar fashion.

99. sin 2" = sin(" + ") = sin " cos " +sin" cos " =2sin " cos ". The identity for cos 2" isproved similarly, and the alternate formsfollow from the first two forms and thePythagorean identity.

Section 0.5

1. F, T, T, F, T, T, F, F.

3. If C is true, then D must be true. If C isfalse, then D may or may not be true.

5. “For all x > 0, we have x > !2.” and “Ifx > 0, then x > !2.”

7. The original statement is true. The converseis “Every rectangle is a square,” which isfalse. The contrapositive is “Everything thatis not a rectangle is not a square,” which istrue.

9. The contrapositive is “Not(Q)*Not(P ),”which is logically equivalent to P * Q.

11. It is always better to switch! Can you explainwhy?

13. All integers greater than or equal to 4.

15. True. The negation is “For all real numbersx, x + 2 and x & 3.”

17. True. The negation is “There exists a realnumber that is both rational and irrational.”

19. True. The negation is “There exists x suchthat, for all y, y '= x2.” (In other words,“There exists x for which there is no y withy = x2.)”

21. True. The negation is “There is some integerx greater than 1 for which x < 2.”

23. False. One counterexample is x = 1.35.

25. False.

27. False. One counterexample is x = !1.

29. True. One example is x = 3.

31. True. The negation is “There exist real num-bers a and b such that a < b but 3a + 1 &3b + 1.”

33. True.

35. False.

37. True.

39. True. One example is x = !1, since for all ywe have |y| > !1.

41. True.

43. False. The only counterexample where thetwo sides of the double implication are notequivalent is x = 0, y = 0.

45. (a) B * (Not A); (b) (Not B) * A

47. (a) (Not A) * (Not B); (b) A * B

49. (a) C * (A and B); (b) Not(C) * (Not A) or(Not B)

51. (a) (B and C) * A; (b) ((Not B) or (Not C))* (Not A)

53. (a) The converse is “If x is rational, then x isa real number.” (b) The contrapositive is “Ifx is irrational, then x is not a real number.”(c) x = ! is a counterexample to the originaland the contrapositive.

55. (a) The converse is “If x & 3, then x > 2.” (b)The contrapositive is “If x < 3, then x + 2.”which is false. (c) x = 2.5 is a counterexam-ple to both the original and the contraposi-tive.

57. (a) The converse is “If#

x is not a real num-ber, then x is negative.” (b) The contraposi-tive is “If

#x is a real number, then x is non-

negative. (c) No possible counterexamplesfor any of the statements.

59. (a) The converse is “If |x| = !x, then x + 0.”(b) The contrapositive is “If |x| '= !x, thenx > 0.” (c) No possible counterexamples forany of the statements.

0.4 Exponential and Trigonometric Functions -...

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