Download 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, trigonometric, and inverse trigonometric functions. Exponential functions are similar to power functions, 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 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:
√
314
100
2314 .
2π ≈ 23.14 = 2 100 =
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 integer, 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 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
Theorem 0.22
48
Ev-
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.
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 upsidedown 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 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
Theorem 0.24
49
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
(e) logb (xy) = logb x + logb y
(b) logb (bx ) = x
(f) logb ( x1 ) = − logb x
(c) blogb x = x
(d) logb (xa ) = a logb x
(g) logb ( xy ) = logb x − logb y
(h) logb x =
loga x
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. For example, log2 8 is the power
to which you have to raise 2 to get 8; since 23 = 8 we have log2 8 = 3. All of these rules also
apply to the natural exponental function, since ln x 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 , which we can
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, as shown below.
g(x) = logb x with b > 1
1
g(x) = logb x with 0 < b < 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
Definition 0.25
50
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
sin θ = y
cos θ = x
tan θ =
y
x
1
x
cot θ =
x
y
θ
x
(x,y)
(cos θ, sin θ)
csc θ =
1
y
sec θ =
Notice that the sine and cosine functions determine the remaining four trigonometric funcsin 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 repeated below for your review; see Exercises 95–100 for proofs. The first Pythagorean identity, 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
2
2
Even/Odd Identities
Shift Identities
tan2 θ + 1 = sec2 θ
sin(−θ) = − sin θ
cos(−θ) = cos θ
cos(θ − π2 ) = sin θ
1 + cot2 θ = csc2 θ
tan(−θ) = − tan θ
sin(θ + 2π) = sin θ
cos(θ + 2π) = cos θ
sin θ + cos θ = 1
Difference Identities
Sum Identities
sin(α + β) = sin α cos β + sin β cos α
cos(α + β) = cos α cos β − sin α sin β
Double Angle Identities
sin 2θ = 2 sin θ cos θ
cos 2θ = cos2 θ − sin2 θ
sin(θ + π2 ) = cos θ
sin(α − β) = sin α cos β − sin β cos α
cos(α − β) = cos α cos β + sin α sin β
Alternate Forms
2
cos 2θ = 1 − 2 sin θ
cos 2θ = 2 cos2 θ − 1
Alternate Forms
sin2 θ =
cos2 θ =
1−cos 2θ
2
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
use the graph of a function f to sketch the graph of its reciprocal f1 . In particular, the zeros of
f will be vertical asymptotes of f1 , large heights on the graph of f will become small heights
on the graph of f1 , and vice-versa.
0.4 Exponential and Trigonometric Functions
51
y = cos x
y = sin x
2
2
1
1
y = tan x
3
2
1
−3π −2π
−π
π
2π
3π
−3π −2π
−π
π
2π
3π
−3π −2π
−π
π
2π
3π
2π
3π
−1
−1
−1
−2
−2
−2
y = csc x
−3π −2π
−3
y = sec x
y = cot x
3
3
3
2
2
2
1
1
1
−π
π
2π
3π
−3π −2π
−π
π
2π
3π
−3π −2π
−π
π
−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 inverses 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 = tan x restricted to
the domain (− π2 , π2 )
y = sin x restricted to
the domain [− π2 , π2 ]
y = sec x restricted to
the domain [0, π2 ) ∪ ( π2 , π]
1
1
!"
"
2
!"
2
"
!"
!"
2
-1
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
(a) The inverse sine function sin−1 x is the inverse of the restriction of the function
sin x to [− π2 , π2 ].
(b) The inverse tangent function tan−1 x is the inverse of the restriction of the
function tan x to (− π2 , π2 ).
(c) The inverse secant function sec−1 x is the inverse of the restriction of the function sec x to [0, π2 ) ∪ ( π2 , π].
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 arcsin x, arctan x, 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 are
the inverses of the restricted functions sin x, tan x, and sec 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, as shown below.
y = sin−1 x
y = tan−1 x
y = sec−1 x
"
"
"
"
2
"
2
"
2
-1
1
!"
2
!"
-1
-
-1
1
1
!"
2
!"
2
!"
!"
Although sin−1 x and (restricted) sin x are transcendental functions, their composition
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.
−1
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 5π
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 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
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 5π
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
√
3 1
shown below middle. This in turn means that we know the coordinates (x, y) = (− 2 , 2 ) of
√
3
the point at which the terminal edge of θ intersects the unit circle. Therefore cos 5π
6 =− 2 .
0.4 Exponential and Trigonometric Functions
53
has
Angle θ = 5π
6
reference angle 30◦
y
5π
6
30°
π
6
is the angle in [− π2 , π2 ]
whose sine is equal to 21
Side lengths of a 30-60-90
triangle with hypotenuse 1
y
(− √32 , 12 )
5π
q=
6
1
2
x
1
30°
π
6
1
2
5π
q=
6
30°
x
√3
2
(c) If θ = sin−1 12 , then we must have sin θ = 12 . There are infinitely many angles whose sine is
−1 1
1
π π
(2)
2 , but only one of those angles is in the restricted domain [− 2 , 2 ] of sine. Thus θ = sin
π π
1
is the unique angle in [− 2 , 2 ] whose sine is 2 , as shown above right. Notice that the triangle
must be a 30–60–90 triangle (since its height is 21 ), and therefore the angle θ we are looking
for must be 30◦ , i.e., π6 radians. Therefore sin−1 12 = π6 .
Example 2
Solving equations that involve transcendental functions
Solve each of the following equations:
(a) 3.25(1.72)x = 1000
(c) sec−1 x =
(b) sin θ = cos θ
π
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:
"
!
"
!
=⇒ x ln(1.72) = ln 1000
3.25(1.72)x = 1000 =⇒ ln((1.72)x ) = ln 1000
3.25
3.25 .
It is now a simple matter to solve for x =
”
“
1000
ln 3.25
ln(1.72)
≈ 10.564.
(b) If sin θ = cos θ, then θ is an angle whose terminal edge intersects
the unit circle
at a√ point
√
√
√
2
2
2
(x, y) with x = y. The only such points on the unit circle are ( 2 , 2 ) and (− 2 , − 22 ), as
shown below left. The angles that end at these points are all of the form θ = π4 + πk for some
π 5π 9π
integer k. Thus the solution set for the equation is {. . . , − 3π
4 , 4 , 4 , 4 , . . .}..
Diagram to solve sin θ = cos θ
Diagram to solve sec−1 x =
π
3
y
y
π
π
4
3
√2
2
√2
2
45
√2
2
45
√2
2
1
2
30
x
√3
2
x
3π
4
(c) If sec−1 x = π3 , then x = sec π3 = cos1 π = 11 = 2. The angle
3
2
used for this calculation are shown above right.
π
3
and the reference triangle we
0.4 Exponential and Trigonometric Functions
Example 3
54
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.
(a) f (x) = 5 − 3e1.7x
(b) g(x) =
1
ln(x−2)
(c) h(x) = 3 sec 2x
Solution.
(a) The domain of f (x) = 5 − 3e1.7x is all of R, and its graph is a transformation of the exponential 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 shown below
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
1
to be defined at a value x, we must have x − 2 > 0, and
(b) For the function g(x) = ln(x−2)
thus x > 2. We must also have ln(x − 2) != 0, which means that x − 2 != 1, and thus x != 3.
1
Therefore the domain of g(x) is (2, 3) ∪ (3, ∞). To sketch the graph of g(x) = ln(x−2)
we start
with the graph of y = ln x shown below right, translate to the right two units as shown below
middle, and then sketch the reciprocal as shown below right.
y = ln x
y=
y = ln(x − 2)
3
3
3
2
2
2
1
1
1
1
2
3
4
5
1
2
3
4
5
1
-1
-1
-1
-2
-2
-2
-3
-3
-3
1
ln(x − 2)
2
3
4
5
(c) The function h(x) = 3 sec 2x = cos32x is defined when cos 2x != 0. This happens when 2x is
not a multiple of π2 , and thus when x is not a multiple of π4 . Thus the domain of h(x) is x != π4 k
for positive integers k. To sketch the graph of h(x), we start with the graph of y = sec x 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
!Π
Example 4
!
Π
2
55
y = sec x
y = 3 sec x
y = 3 sec 2x
9
9
9
6
6
6
3
3
3
-3
Π
2
Π
!Π
!
Π
2
-3
Π
2
Π
!Π
!
Π
2
-3
-6
-6
-6
-9
-9
-9
Π
2
Π
Simplifying compositions of inverse trigonometric and trigonometric functions
Write cos(sin−1 x) as an algebraic function, that is, a function that involves only arithmetic
operations and rational powers.
Solution. If we define θ = sin−1 x, then sin θ = x and θ must be in the interval [− π2 , π2 ]. Let’s
first consider the case where θ is in the first quadrant [0, π2 ]; 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
an angle θ in [0, π2 ]
Use Pythagorean Theorem to
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, or in
terms of “adjacent over hypotenuse,” we have:
√
1 − x2 #
= 1 − x2 .
cos θ =
1
The case where θ is in the fourth quadrant, i.e., where θ ∈ [− π2 , 0], is similar and also shows
√
that cos θ√= 1 − x2 . Therefore we have shown that cos(sin−1 x) is equal to the algebraic
function 1 − x2 .
√
Checking the Answer. To verify the strange fact that cos(sin−1 x) = 1 − x2 , 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 = 0 we have
#
√
1 − 02 = 1 = 1,
cos(sin−1 0) = cos 0 = 1 and
0.4 Exponential and Trigonometric Functions
56
and at x = 1 we have
π
cos(sin−1 1) = cos( ) = 0
2
and
#
√
1 − 12 = 0 = 0.
As a less trivial example, consider x = 12 . At this value we have
cos(sin−1 ( 21 )) = cos( π6 ) =
?
√
3
2
and
$
$
1 − ( 12 )2 = 1 −
1
4
=
$
3
4
=
√
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
log10 7
same answer if we computed log
2?
ln 7
ln 2
with a calculator. Would we get the
10
! 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 = csc x, y = sec x,
and y = cot x?
! How are the unit circle definitions of the trigonometric functions related to the righttriangle definitions of trigonometric functions?
Exercises 0.4
Thinking Back
Algebra with exponents: Write each of the following expressions in the form Abx for some real numbers A and b.
# 32x+1
#
1
2(3x−4 )
# 5x 23−x
#
4(3x )2
2x
# (23x−5 )4
( 81 )x
3(23x+1 )
#
Inverse functions: Suppose f and g are inverses of each
other.
# What can you say about f (g(x)) and g(f (x))?
# If f has a horizontal asymptote at y = 0, what can
you say about g?
# If f has a y-intercept at y = 1, what can you say about
g?
Famous triangles, degrees, and radians: The following exercises 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 lengths
of 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 lengths
of the remaining legs of the triangle?
# What is a radian? Is it larger or smaller than a degree? Compare an angle of one degree with an angle
of one radian in standard position.
# Show each of the following angles in standard position on the unit circle, in radians:
# 3π
# − 4π
# 17π
# 21π
4
3
6
Concepts
0. Problem Zero: Read the section and make your own
summary of the material.
1. True/False: Determine whether each of the following
statements is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.
True or False: The function f (x) = 3e0.5x − 2 is
an exponential function.
(b) True or False: Every exponential function
f (x) = Aekx has a horizontal asymptote at y =0.
(a)
3
4. In this exercise we will examine two ways to think of
ab when b is an irrational number, and in particular
what the quantity 2π represents.
(a)
One way to define 2π is to think of it as a limit.
If we take a sequence a1 , a2 , a3 , . . . of rational
numbers that approaches π, then the sequence
2a1 , 2a2 , 2a3 , . . . should approach 2π . Said in
terms of limits, this means that:
2π = lim 2a ,
a→π
True or False: For all x > 0, ln(x ) = 3 ln x.
log 2 x
log6 x
(d) True or False: For all x > 0,
=
.
log2 3
log6 3
(e) True or False: If (x, y) is the point on the unit
circle corresponding to the angle − 7π
, then x is
3
positive and y is negative.
(f) True or False: The sine of an angle θ is always
equal to the sine of the reference angle for θ.
where each a is assumed to be a rational number. Can you think of a sequence of rational
numbers that get closer and closer to π? (Hint:
Think about the decimal expansion of π.)
(b) Another way to consider 2π is to write it as an
infinite product:
For any x, 1 − cos2 (5x3 ) =
What will the next term in the product be? How
could 2π equal the product of infinitely many
numbers? Wouldn’t that make 2π infinitely
large? Calculate some of the later terms in the
5
9
product (for example, 2 10000 or 2 100000 ) and use
these calculations to argue that even though 2π
can be written as a product of infinitely many
numbers, it is not necessarily infinitely large.
(c)
(g) True or False:
sin2 (5x3 ).
1
.
cos−1 x
2. Examples: Give examples of each of the following. Try
to find examples that are different than any in the
reading.
(h) True or False: sec−1 x =
(a)
Two exponential functions and their inverses.
(b) Two x-values at which tan x is not defined.
(c)
Two x-values at which sec−1 x is not defined.
3. What is the definition of an exponential function, and
how is it different from a power function? Is the function f (x) = xx a power function, an exponential function, or neither, and why?
1
4
1
5
9
2π = 23 2 10 2 100 2 1000 2 10000 2 100000 · · · .
√
r
for rational val5. Approximate 2 3 by calculating 2√
ues r that get closer and closer
to 3. (Hint: You can
√
use the decimal expansion of √3 to get a sequence of rational numbers that approaches 3.)
6. Why can’t we define the number e just by writing it
down in decimal notation to lots of decimal places?
0.4 Exponential and Trigonometric Functions
7. Write the exponential function f (x) = 3e−2x in the
form Abx for some real numbers A and b. Then write
the exponential function g(x) = −2(3x ) in the form
Aekx 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 represents
exponential growth if b ∈
, and exponential
decay if b ∈
.
(b) An exponential function f (x) = Aekx repre, and exsents exponential growth if k ∈
ponential decay if k ∈
.
(c)
Suppose that ekx = bx for some real numbers k
and b. Then k ∈ (0, ∞) if and only if b ∈
.
kx
x
(d) Suppose that e = b for some real numbers k
.
and 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, and
why?
10. Fill in the blanks in each statement below.
(a)
For all x ∈
.
, log2 x = y if and only if x =
, 3log3 x =
(b) For all x ∈
(c)
.
x
, log4 (4 ) =
For all x ∈
.
(d) log2 3 is the exponent to which you have to raise
to get
.
11. The graphs of y = log2 x and y = log4 x are shown below. Determine which graph is which, without using
a calculator. (Hint: Think about the graphs y = 2x
and y = 4x , and then reflect those graphs over the
line y = x.)
y = log2 x and y = log4 x
58
17. Use the definition of the sine function to explain why
sin( π4 ) is equal to sin( 9π
) and sin(− 7π
).
4
4
18. Fill in each blank with an interval of real numbers.
and
The function f (x) = cos x has domain
range
.
(b) The function f (x) = csc x has domain
and
.
range
(c) The restricted tangent function has domain
and range
.
(a)
(d) The function f (x) = sec−1 x has domain
and range
.
19. Suppose θ is an angle in standard position whose terminal edge intersects the unit circle at the point (x, y).
If y = − 13 , what are the possible values of cos θ? If
you know that the terminal edge of θ is in the third
quadrant, what can you say about cos θ? What if the
terminal edge of θ is in the fourth quadrant? Could
the terminal edge of θ be in the first or second quadrant?
√
20. Show that − 3 is in the range√of tangent by finding
an angle θ for which tan θ = − 3.
21. Describe restricted domains for sin x, tan x, and sec x
on which each function is invertible. Then describe
the corresponding domains and ranges for arcsin x,
arctan x, and arcsec x.
22. Fill in the blanks:
sin−1 x is the angle in the interval
is x.
(b) y = arcsin x if and only if sin y =
x∈
and y ∈
.
(a)
(c)
2
whose
, for all
If tan−1 x = θ and tan θ is positive, then θ is in
the
quadrant.
(d) If arctan x = θ and sin θ = 31 , then cos θ =
1
1
2
3
4
-1
-2
12. State the algebraic properties of the natural logarithm
function that correspond to the eight properties of
logarithmic functions in Theorem 0.24.
13. Use algebraic properties of logarithms, the graph of
y = ln x, and your knowledge of transformations to
sketch graphs of f (x) = ln(x2 ) and g(x) = ln( x1 ).
x+1
14. Solve the inequality ln( x−1
) ≥ 0.
15. Give a mathematical definition of sin θ for any angle
θ. Your definition should include the words “unit
circle,” “standard position,” “terminal,” and “coordinate.”
16. Give a mathematical definition of tan θ for any angle θ. Your definition should include the words “unit
circle,” “standard position,” “terminal,” and “coordinate.”
.
23. Which of the following expressions are defined?
Why or why not?
(a)
1
sin−1 (− 25
)
(b) sin−1
3
2
(c)
tan−1 100
(d) sec−1
π
4
24. Sketch a graph of the restricted cosine function on the
domain [0, π] and argue that this restricted function is
one-to-one. Then sketch a graph of cos−1 x, and list
the domains and ranges of the inverse cos−1 x of this
restricted function.
25. Without calculating the exact or approximate values
of the following expressions, use the unit circle to determine whether each of the following quantities is
positive or negative.
(a)
sin−1 (− 51 )
(b) sin−1 (− 32 )
(c)
tan−1 2
(d) sec−1 (−5)
26. Find all angles whose secant is 2, and then find
sec−1 (2).
0.4 Exponential and Trigonometric Functions
59
Skills
Find the domains of the functions in Exercises 27–32.
27. f (x) =
ln(x + 1)
ln(x − 2)
29. f (x) = p
31. f (x) =
√
1
ln(x − 1)
sec θ
28. f (x) =
1
ex − e2x
1
30. f (x) =
1 − tan θ
32. f (x) = 2 sin−1 (x − 3)
Find the exact values of each of the quantities in Exercises 33–44. Do not use a calculator.
33. ln( e12 )
34. log 1 4
35. 4 log2 6 − 2 log2 9
36.
37. tan(− π4 )
)
38. cos( 48π
3
39.
csc(− 5π
)
4
2
log 7 9
+ log3 1
log 7 13
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 any
roots, intercepts, and asymptotes on your graphs.
65. f (x) = −( 21 )x + 10
66. f (x) = −0.25(3x−2 )
67. f (x) = 20 − 5e−2x
68. f (x) = log 1 x
69. f (x) = − log2 (x − 3)
70. f (x) = sin(2x) + 4
71. f (x) = 2 cos(x −
40. sin(201π)
41. cos−1 (−1)
42. sin−1 (−1)
43. arcsec (− √22 )
44. arctan(− √13 )
2
π
)
4
72. f (x) = tan−1 (x−2) + π
For each graph in Exercises 73–78, find a function whose
graph looks like the one shown. When you are finished,
use a graphing utility to check that your function f has the
properties and features of the given graph.
4
Solve the equations in Exercises 45–50 by hand. When you
are finished, check your answers either by testing your solutions or by graphing an appropriate function.
45. 2x = 3x−1
46. 2 = 10(1 +
47. log2 ( x−1
)=4
x+1
48. sin x =
4
3
73.
-4
-3
0.19 12x
)
12
1
2
2
1
1
-1
-2
3
2
1
2
3
4
74.
-4
-3
-2
-1
1
-1
-1
-2
-2
-3
-3
-4
-4
15
2
3
4
1
2
3
2
1
49. cos 2x = 1
50. sec−1 x = π
10
75.
76.
5
-5
-4
-3
-1
-1
1
2
-3
-5
-4
1
52. sin(−φ)
53. cos(2θ)
54. sin(θ + π2 )
55. the sign of cos(θ + φ)
56. the sign of tan(θ + π)
Applications
6
5
4
3
77.
!"
"
2
!"
2
"
78.
2
1
!2"
-1
Write each of the expressions Exercises 57–60 as an algebraic expression that does not involve trigonometric or inverse trigonometric functions.
-1
-2
Suppose that cos(θ) = 61 , sin(θ) > 0, sin(φ) = 35 , and
cos(φ) < 0. Use trigonometric identities to identify the
quantities in Exercises 51–56.
51. sin(θ)
-2
!"
-1
-2
"
2"
0.4 Exponential and Trigonometric Functions
79. Ten years ago, Jenny deposited $10, 000 into an investment account. Her investment account now
holds $22, 609.80. Her accountant tells her that her
investment account balance I(t) is an exponential
function.
Find an exponential function of the form I(t) =
Aekt to model Jenny’s investment account balance.
(b) Find an exponential function of the form I(t) =
Abt to model Jenny’s investment account balance.
80. Suppose there were 500 rats on a certain island in
1973, and 1697 rats on the same island ten years later.
Assume that the number R(t) of rats on the island t
years after 1973 is an exponential function.
(a)
(a)
Find an equation for the exponential function
R(t) that describes the number of rats on the island. Let t = 0 represent the year 1973.
(b) According to your function R(t), how many
rats will be on the island in 2020?
(c) How long did it take for the population of rats
to double from its 1973 amount? How long did
it take for it to double again? And again?
60
81. Suppose a rock sample initially contains 250 grams
of the radioactive substance unobtainium, and that
the amount of unobtainium after t years is given by
an exponential function of the form S(t) = Aekt . The
half-life of unobtainium is 29 years, which means that
it takes 29 years for the amount of the substance to
decrease by half.
(a)
Find an equation for the exponential function
S(t).
(b) What percentage of unobtainium decays each
year?
(c) How long will it be before the rock sample contains only 6 grams of unobtainium?
82. Again considering the rock sample described in Exercise 81, answer the following questions:
(a)
At one point the rock sample contained 900
grams of unobtainium; how long ago?
(b) What percentage of the unobtainium will be left
in 300 years?
(c) How long will it be before 95% of the unobtainium has decayed?
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 degrees. 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 function f (x) = Abx has the property that f (x) is never
zero.
86. Use the definition of a one-to-one function to prove
that every exponential function f (x) = Abx is one-toone.
87. Use the base conversion formula for logarithms to
prove that the function f (x) = log 2 x is equal to the
function g(x) = log3 x only when x = 1.
88. Use logarithms to prove that every exponential function of the form f (x) = Abx can be written in the form
f (x) = Aekx , and vice-versa.
89. Use the definition of a logarithmic function y = logb x
to prove that for any b > 0 with b '= 1, the quantity
logb 1 is equal to zero.
In Exercises 90–94, assume that x, y, a, and b are values
which make sense in the expressions involved.
90. Use the fact that logarithmic functions are the inverses of exponential functions to prove that:
(a)
log b x = y if and only if by = x
(b) log b (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 log b (xy) = logb x + log b y. (Hint: Show
this is equivalent to the statement xy = blogb x+logb y , and
prove this new statement instead.)
93. Use the results of the two problems above to prove
that:
(a)
log b ( x1 ) = − logb x
(b) log b ( xy ) = logb x − logb y
94. Prove the base conversion formula logb x =
(Hint: Set y = logb x and then show that by = x.)
log a x
.
loga b
0.4 Exponential and Trigonometric Functions
61
95. Use the unit circle definitions of sine and cosine to
prove the identity sin2 θ + cos2 θ = 1.
98. Use the sum identities and the even/odd identities to
prove the difference identities listed in Theorem 0.26.
96. Use the first Pythagorean identity sin2 θ + cos2 θ = 1
to prove the second and third Pythagorean identities
listed in Theorem 0.26. (Hint: To prove the second identity, divide both sides of the first identity by cos2 x. A
similar trick will prove the third identity.)
99. Use the sum identities to prove the double angle
identities listed in Theorem 0.26. (Hint: Note that 2θ
is equal to θ + θ.)
97. Use the unit circle definitions of the trigonometric
functions to prove the even/odd identities and the
shift identities listed in Theorem 0.26.
100. The four identities listed as alternate forms in Theorem 0.26 are alternate ways of writing the double angle identity cos 2θ = cos2 θ − sin2 θ. Use this double
angle identity, algebra, and the Pythagorean identities to prove these four alternate forms.
Thinking Forward
# A special exponential limit: Use a calculator to approxh
imate e h−1 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 approach?
# Logarithms with absolute values: Sketch a graph of the
function f (x) = ln |x|. What is the domain of this
function? Is this function even, odd, or neither, and
why?
# Rewriting trigonometric expressions: Use the double an2x
gle identity sin2 x = 1−cos
to rewrite the expres2
4
2
sion sin x cos x in terms of a sum of expressions of
the form A cos kx. (Note: You’ll have to multiply out
some expressions, and use the double angle identity
more than once.)
Appendix A
Answers To Odd Problems
83. If f (x) = Ax3 + lower-degree terms and
g(x) = Bx3 + lower-degree terms, then
f (x)g(x) = ABx6 + lower-degree terms.
Since f and g are cubic we know that A
and B are nonzero. Thus AB must also be
nonzero, and therefore f g 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 is
a polynomial of degree 1 with coefficients
a1 = m and a0 = b. If f (x) = mx + b with
m = 0 then f is a polynomial of degree zero
with 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 defined
on (−∞, ∞); 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 written
in the form f (x) = Abx ; the variable is in the
exponent and a constant is in the base. For a
power function, the situation is reversed. xx
is neither a power nor an exponential function because a variable appears in both the
base and the exponent.
√
5.
3 ≈ 1.73205. We have 21.7 ≈ 3.2490,
1.73
2
≈ 3.3173, 21.732 ≈ 3.3219, 21.7320 ≈
3.3219, 21.73205 ≈ 3.3220, and so on. Each
of these approximations
gets closer to the
√
value of 2 3 .
7. f (x) = 3(e−2 )x ≈ 3(0.135)x , g(x) =
−2e(ln 3)x ≈ −2e1.0986x .
9. We must have b > 0 and b != 1, since those
conditions are necessary for bx to be an exponential function.
11. The graph that passes through (2, 1) is
log2 x; the graph that passes through (2, 21 )
is log4 x.
13. f (x) = 2 ln x is the graph of ln x stretched
vertically by a factor of 2; g(x) = − ln x is
the graph of ln x reflected over the x-axis.
15. If θ is an angle in standard position, then
sin θ the vertical coordinate y of the point
(x, y) where the terminal edge of θ intersects
the unit circle.
17. The terminal edges of the angles π4 , 9π
, and
4
− 7π
all
meet
the
unit
circle
at
the
same
4
point (and in particular, at the same ycoordinate).
√
19. cos θ = x is − 38 if θ is in the third quadrant;
√
cos θ = 38 if θ is in the fourth quadrant; θ
cannot be in the first or second quadrant.
21. See Definition 0.27 for the restricted domains of the trigonometric functions, and
thus the ranges of the inverse trigonometric
functions. The domain of arcsin x is [−1, 1],
the domain of arctan x is all of R, and the
domain of arcsec x is (−∞, −1] ∪ [1, ∞).
Their ranges are the restricted domains of
sin x, tan x, and sec x, respectively.
23. Only (a) and (c) are defined.
25. (a) negative; (b) negative; (c) positive; (d)
positive
27. (2, 3) ∪ (3, ∞)
29. (2, ∞)
31. . . . ∪ (− 5π
, − 3π
) ∪ (− π2 , π2 ) ∪ ( 5π
, 7π
) ∪ ...
2
2
2
2
33. −2
35. 4
37. −1
√
2
39.
41. π
43. 3π
4
ln 3
45. x = ln(
3 ) ≈ 2.70951
2
47. x = − 17
15
49. x = πk, where k is any integer
51.
53.
55.
57.
59.
61.
63.
65.
√
35
6
− 17
18
Negative
√
1 − x2
2
x +1
p
1 − ( x3 )2
1 − 2(5x)2
Start with the graph of y = ( 12 )x , then reflect
over the x-axis and shift up by 10 units.
20
15
67.
-1
69.
1
2
6
5
4
3
2
1
-1
-2
-3
1
2
4
3
5
7
6
8
2
1
71.
!5"
4
!3"
4
!"
4
"
4
3"
4
5"
4
-1
-2
73.
75.
77.
79.
f (x) = 2e−x − 3
f (x) = −5e−x + 10
f (x) = − cos 2x
(a) I(t) ≈ 10, 000e0.08t ;
10, 000(1.085)t .
(b)
I(t) =
ln 2
81. (a) S(t) = 250e− 29 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 and
b are nonzero real numbers with Abx = 0 for
some real number x. Since A '= 0 we know
bx = 0, and therefore that x(ln b) = ln(bx ) =
ln 0. But this is a contradiction because ln 0
is undefined, so the product x(ln b) of real
numbers cannot equal ln 0.
ln x
x
87. log2 x = log3 x
⇔
= ln
⇔
ln 2
ln 3
(ln 3)(ln x) = (ln 2)(ln x) ⇔ (ln x)(ln 3 −
ln 2) = 0 ⇔ ln x = 0 ⇔ x = 1.
89. y = logb x if and only if by = x. Since the
only solution to by = 1 is y = 0 (if b '= 0), we
know 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 ( x1 ) = logb (x−1 ) = − logb x; (b)
logb ( xy ) = logb (xy −1 ) = logb x − log b y.
95. (a) For any angle θ, (cos θ, sin θ) are the coordinates of the point where the terminal edge
of θ meets the unit circle. Since the equation of the unit circle is x2 + y 2 = 1 ad we
have x = cos θ and y = sin θ, we must have
sin2 θ + cos2 θ = 1.
97. For any angle θ, sin θ is the y-coordinate
where the terminal edge of θ meets the unit
circle. The angle −θ is the angle of the same
magnitude as θ but opening in the clockwise direction from the x-axis, and therefore
its terminal edge will be the same as the terminal edge of θ except flipped over the xaxis. Therefore the y-coordinates of these
two terminal edges have the same magnitude but opposite signs; in other words,
sin(−θ) = − sin θ. The remaining even/odd
identities can be proved in a similar fashion.
99. sin 2θ = sin(θ + θ) = sin θ cos θ + sin θ cos θ =
2 sin θ cos θ.
The identity for cos 2θ is
proved similarly, and the alternate forms
follow from the first two forms and the
Pythagorean 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 is
false, then D may or may not be true.
5. “For all x > 0, we have x > −2.” and “If
x > 0, then x > −2.”
7. The original statement is true. The converse
is “Every rectangle is a square,” which is
false. The contrapositive is “Everything that
is not a rectangle is not a square,” which is
true.
9.
11.
13.
15.
17.
19.
21.
23.
25.
27.
29.
31.
33.
35.
37.
39.
41.
43.
45.
47.
49.
51.
53.
55.
57.
59.
The contrapositive is “Not(Q) ⇒ Not(P ),”
which is logically equivalent to P ⇒ Q.
It is always better to switch! Can you explain
why?
All integers greater than or equal to 4.
True. The negation is “For all real numbers
x, x ≤ 2 and x ≥ 3.”
True. The negation is “There exists a real
number that is both rational and irrational.”
True. The negation is “There exists x such
that, for all y, y '= x2 .” (In other words,
“There exists x for which there is no y with
y = x2 .)”
True. The negation is “There is some integer
x greater than 1 for which x < 2.”
False. One counterexample is x = 1.35.
False.
False. One counterexample is x = −1.
True. One example is x = 3.
True. The negation is “There exist real numbers a and b such that a < b but 3a + 1 ≥
3b + 1.”
True.
False.
True.
True. One example is x = −1, since for all y
we have |y| > −1.
True.
False. The only counterexample where the
two sides of the double implication are not
equivalent is x = 0, y = 0.
(a) B ⇒ (Not A); (b) (Not B) ⇒ A
(a) (Not A) ⇒ (Not B); (b) A ⇒ B
(a) C ⇒ (A and B); (b) Not(C) ⇒ (Not A) or
(Not B)
(a) (B and C) ⇒ A; (b) ((Not B) or (Not C))
⇒ (Not A)
(a) The converse is “If x is rational, then x is
a real number.” (b) The contrapositive is “If
x is irrational, then x is not a real number.”
(c) x = π is a counterexample to the original
and the contrapositive.
(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 counterexample to both the original and the contrapositive.
√
(a) The converse is “If x is not a real number, then x is negative.” (b) The contraposi√
tive is “If x is a real number, then x is nonnegative. (c) No possible counterexamples
for any of the statements.
(a) The converse is “If |x| = −x, then x ≤ 0.”
(b) The contrapositive is “If |x| '= −x, then
x > 0.” (c) No possible counterexamples for
any of the statements.