SELF-CHECK DIAGNOSTIC TEST - The Existence of an I- 123docz.net

THEOREM 1 The Existence of an Inverse Function

0.8 SELF-CHECK DIAGNOSTIC TEST

2. Solve the equation .

3. Find the domain of .

4. Find the inverse of . What is its domain?

5. Express as a single logarithm.

Answers to Self-Check Diagnostic Test 0.8 can be found on page ANS 7.

lnx 1

2 ln(x1)ln cosx f(x)2 ln(x1)

f(x) e1>x 1lnx 200

13e0.3t100 elnxlne2x

Exponential Functions and Their Graphs

Suppose you deposit a sum of $1000 in an account earning interest at the rate of 10%

per year compounded continuously(the way most financial institutions compute interest). Then, the accumulated amount at the end of years is described by the function , whose graph appears in Figure 1.* This function is called an exponen- tial function.Observe that the graph of rises rather slowly at first but very rapidly as time goes by. For purposes of comparison, we have also shown the graph of the function , giving the accumulated amount for the same principal ($1000) but earning simpleinterest at the rate of 10% per year. The moral of the story: It is never too early to save.

Recall that if is a real number and is a positive integer, then

nfactors

In the expression , is called the base and is the exponent or power to which the base is raised. Also, by definition, , and if is a positive integer, then

an 1 an

n a01

n a

⎫⎪⎬⎪⎭

anaⴢaⴢpⴢa n

yt(t)1000(10.10t) f f

(0 t 20)

*We will discuss simple and compound interest later in this section. Continuous compound interest will be FIGURE 1

Under continuous compounding, a sum of money grows exponentially.

0 7000 6000 5000 4000 3000 2000 1000

5 10

Years

Dollars

15 20

y f(t)

y g(t) t

If p>qis a rational number, where and are integers with p q q0, then we define the expression with rational exponent by

To define expressions with irrational exponents such as , we proceed as follows.

Observe that . So can be approximated successively and with increasing accuracy by the rational numbers

1.4, 1.41, 1.414, 1.4142, 1.41421, 1.414213, Thus, we can expect that may be approximated by the numbers

, , , , , ,

In fact, from Table 1 we see that as approaches , the corresponding values of approach the number 2.665143p. It can be shown that this number is unique, and we

2x 12

21.414213 p 21.41421

21.4142 21.414

21.41 21.4

212

p 12

121.414213p

212 ap>q2qap (1qa)p

ap>q

define it to be . Furthermore, Table 1 suggests that correct to five decimal places, 2122.66514

212

x 1.4 1.41 1.414 1.4142 1.41421 1.414213

2x 2.639015 % 2.657371 % 2.664749 % 2.665119 % 2.665137 % 2.665143 % TABLE 1

LAWS OF EXPONENTS

If and are positive numbers and and are real numbers, then

a. b. c.

d. e. aa

bbx ax bx (ab)xaxbx

(ax)yaxy ax

ayaxy axayaxy

y x b

EXAMPLE 1

a. b.

c. d.

Since the number is defined for all real numbers , we can define a function with the rule given by

where is a positive constant and . The domain of is . This function is called an exponential function with base .Examples of exponential functions are

, f(x) a1 , and f(x)px 2bx

f(x)2x

(⬁, ⬁) f

a1 a

f(x)ax f

x (a0)

(4x3)1>2(41>2)(x3>2) 1 2x3>2 (2x)323x

31>2

31>33(1>2)(1>3)31>6 (21>3)(23>5)2(1>3)(2>5)211>15

Similarly, we can define , where is an irrational number. In fact, this procedure can be used to define , where is any positive number and is an irrational number. In this manner, we see that the number can be defined for allreal numbers .

Computations involving exponentials are facilitated by the following laws of exponents.

x ax

x a

x 2x

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An alternative and more rigorous definition of exponential functions is given in Appen- dix C.

Do not confuse an exponential function with a power function such as , encountered in Section 0.6. In the case of the power function the base is a variable, and its exponent is a constant.

f(x)x2

EXAMPLE 2 Sketch the graphs of , , , and on the same set of axes.

Solution We first construct a table of values for each of the functions (see Table 2).

With the help of Table 2 we obtain the graphs of , , , and f t h Fshown in Figure 2.

F(x) 1132x

h(x) 1122x

t(x)3x f(x)2x

x 4 3 2 1 0 1 2 3 4

2x 161 18 14 12 1 2 4 8 16

3x 811 271 19 13 0 3 9 27 81

1122x 16 8 4 2 1 12 14 18 161

1132x 81 27 9 3 0 13

1 9

1 27

1 81

TABLE 2

FIGURE 2

The graphs of f(x)2x,t(x)3x, 0

4 3 2 1

1 2

x 1

( )2 1 2

y 3x y 2x

x 1

( )3

y x

, and F(x)1132x

h(x)1122x

The graphs of f(x)2x,t(x)3x,h(x) 1122x, and F(x) 1132xobtained in Exam- ple 2 are special cases of the graphs of , obtained by setting , 3, , and , respectively. In general, the exponential function with has a graph similar to that of or , whereas the graph of for is similar to that of y1122x. If a1, then the function yaxyreduces to the constant functiona 0a1

y3x x

y2x

a1 yax

1 3

a2 2

f(x)ax

. The graphs of for each of these three cases are shown in Figure 3. Observe that all the graphs pass through the point because . Also, as suggested by Figure 2, the larger is , the faster the graph of rises for .

The properties of exponential functions are summarized below.

x0 f(x)ax

(a1) a

a01 (0, 1)

yax y1

y ax (0<a<1)

y ax (a>1)

y 1

x FIGURE 3

The graph of rises from left to right if , is constant if , and falls from left to right if 0a1.

a1 a1

yax

Properties of Exponential Functions

The exponential function has the following properties.

1. Its domain is . 2. Its range is .

3. Its graph passes through the point .

4. Its graph rises from left to right on if and falls from left to right on (⬁, ⬁)if a1.

a1 (⬁, ⬁)

(0, 1) (0, ⬁)

(⬁, ⬁)

(a0, a1) f(x)ax

EXAMPLE 3 Sketch the graph of the function , and find its domain and range.

Solution The required graph is obtained by first reflecting the graph of (see Figure 4a) to obtain the graph of y 2x (see Figure 4b) and then translating this

y2x f(x)12x

graph upward by 1 unit. The resulting graph is shown in Figure 4c. The domain of is (⬁, ⬁)and its range is (⬁, 1).

0 x

4 5 4 3 2 1

2 1 2 3 4 5

4 y 2x

0 x

y 2

4 2 4

y 2x

5 4 3 2 1 2

0 x

1 2

3 1 2 3 4 y 1 2x

y 1

(a) (b) (c)

FIGURE 4

The Natural Exponential Function

Of all the possible choices for the base of an exponential function, there is one that plays an important role in calculus. This base, denoted by the letter , is the irrational number whose value, correct to five decimal places, is given by

As you will see later on, the use of for the base of an exponential function enables us to express some of the formulas of calculus in the simplest form possible. The ration- ale for this choice of the base will be given in Section 2.2.

The function is called the natural exponential function.Since the number lies between 2 and 3, we expect the graph of to lie between the graphs of and , as we will see in Example 4. In the definition of the exponential function , the base can be any positive constant. But as mentioned earlier, the choice of as the base of the exponential function will lead to much simpler calculations in our work ahead. We will give a precise definition of in Section 2.8.e

a f(x)ax

y3x y2x

yex e

f(x)ex

e2.71828

EXAMPLE 4 Sketch the graphs of , , and on the same set of axes.

Solution The values of and for selected values of were found in Example 2. With the aid of a calculator we obtain the following table. The graphs of , , and are shown in Figure 5.

h f t

x h(x)

f(x)

h(x)3x t(x)ex

f(x)2x

0 10

8 6 4 2

2 4

1 3

y 2x y 3x

y ex

2 1 x FIGURE 5

The graph of lies between the graphs of y2xand y3x.

yex

x 3 2 1 0 1 2 3

e 0.05 0.14 0.37 1 2.72 7.39 20.09

Compound Interest

An important application of exponential functions is found in computations involving interest—charges on borrowed money.

Simple interestis interest that is computed on the original principal only. Thus, if denotes the interest on a principal (in dollars) at an interest rate of per year for years, then

IPrt

t r

P I

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The accumulated amount , the sum of the principal and interest after years, is given by

(1) and is a linear function of .

In contrast to simple interest, earned interest that is periodically added to the principal and thereafter itself earns interest at the same rate is called compound interest.

To find a formula for the accumulated amount, suppose that dollars (the principal) is deposited in a bank for a term of years, earning interest at the rate of per year (called the nominalor stated rate) compounded annually. Then, using Equation (1), we see that the accumulated amount at the end of the first year is

To find the accumulated amount at the end of the second year, we use Equation (1) again, this time with , since the principal andinterest now earn interest over the second year. We obtain

Continuing, we see that the accumulated amount after years is

(2) Equation (2) was derived under the assumption that interest was compounded annu- ally.In practice, however, interest is usually compounded more than once a year. The interval of time between successive interest calculations is called the conversion period.

If interest at a nominal rate of per year is compounded times a year on a principal of dollars, then the simple interest rate per conversion period is

For example, if the nominal rate is 8% per year and interest is compounded quarterly , then

or 2% per period.

To find a general formula for the accumulated amount when a principal of dollars is deposited in a bank for a term of years and earns interest at the (nominal) rate of per year compounded times per year, we proceed as before, using Equation (2) repeatedly with the interest rate . We see that the accumulated amount at the end of each period is as follows:

First period:

Second period:

th period:

But there are periods in years (number of conversion periods times the term).

Therefore, the accumulated amount at the end of years is given by

(3) APa1 r

mbmt t t

nmt

AnAn1(1i)[P(1i)n1](1i)P(1i)n n

o o

A2A1(1i)[P(1i)](1i)P(1i)2 A1P(1i)

ir>m m

P i r

m 0.08 4 0.02 (m4)

(r0.08)

annual interest rate number of periods per year

i r m P

m r

AP(1r)t t A

A2A1(1rt)P(1rt)(1rt)P(1rt)2 PA1

A1P(1rt)

r t

P t

P(1rt) APIPPrt

t A

EXAMPLE 5 Find the accumulated amount after 3 years if $1000 is invested at 8%

per year compounded annually, semiannually, quarterly, monthly, and daily. (Assume that there are 365 days in a year.)

Solution We use Equation (3) with , , and , 2, 4, 12, and 365 in succession to obtain the results summarized in Table 3.

Logarithmic Functions

If you examine the graph of the exponential function where and (see Figure 3), you will see that it passes the horizontal line test, and so the function

is one-to-one and therefore possesses an inverse function . This function is called the logarithmic function with base .The graph of is obtained by reflecting the graph of about the line . The graph of for the case is given in Figure 6.

The function is called the logarithmic function with base and is denoted by . Using the definition of an inverse function given in Section 0.7,

if and only if we are led to the following:

f(y)x f1(x)y

loga

a f1

ylogax yx

f(x)ax

f1(x)logax a

f1 f

a1 a0

f(x)ax

m1 r0.08

P1000 TABLE 3 The accumulated

amount after 3 years when interest is converted mtimes/year

m A(dollars)

1 2 4 12 365

1259.71 1265.32 1268.24 1270.24 1271.22

0 1

1 y y ax

y x

y loga x x

FIGURE 6

The graphs of and

are mirror reflections about the line yx.

f(x)ax

f1(x)logax

if and only if ayx logaxy

Thus, if , then is the exponent to which must be raised to obtain . Also because f(x)axand t(x)logaxare inverses of each other, we have

x a

logax x0

for all in and

for all in (x ⬁, ⬁) loga(ax)x

(0, ⬁) x

alogaxx

A summary of the properties of logarithmic functions follows.

Properties of Logarithmic Functions

The logarithmic function has the following prop-

erties.

1. Its domain is . 2. Its range is .

3. Its graph passes through the point .

4. Its graph rises from left to right on if and falls from left to right if a1.

a1 (0, ⬁)

(1, 0) (⬁, ⬁)

(0, ⬁)

f(x)logax(a0, a1)

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As in the case of exponentials, computations involving logarithms are facilitated by the following laws of logarithms. (These laws are proved in Appendix C.)

0 4 3 2 1 1 2 3 4

8 7 6 5 4 3 2 1 y

y log4 x y log2 x y log6 x y log10 x x

FIGURE 7

The graphs of ylogaxfor a2, 4, 6, and 10

LAWS OF LOGARITHMS

If and are positive numbers, then a.

c. where is any real number

e. logaa1 loga10

r logaxrrlogax

loga

ylogaxlogay logaxylogaxlogay

y x

EXAMPLE 6 Use the laws of logarithms to evaluate . Solution We have

Use Law b.

Use Law c.

Use Law e.

Before turning to another example, we mention that the two widely used systems of logarithms are the system of common logarithms,which uses the number 10 as the base, and the system of natural logarithms,which uses the number as the base. It is standard practice to write logfor and lnfor loge. As in the case of exponentials, the use of natural logarithms rather than logarithms with other bases leads to simpler expressions.

log10

e 3(1)3

3 log22

log28log223 log240log25log2

40 5

log240log25 The graphs of ylogaxfor different bases are shown in Figure 7.a

EXAMPLE 7 Expand and simplify the following expressions:

a. b.

Solution

a. Use Law b.

Use Law c.

Use Law e.

b. Rewrite.

Use Laws a, b, and c.

Use Law c.

Use Law e.

Properties Relating the Natural Exponential and the Natural Logarithmic Functions

The following properties follow as an immediate consequence of the definition of the natural logarithm of a number.

2 lnx1

2ln(x21)x 2 lnx1

2ln(x21)xlne lnx2 1

2ln(x21)xlne ln x22x21

ex ln x2(x21)1>2 ex log2(x21)x log2(x21)xlog22 log2

x21

2x log2(x21)log22x ln x22x21

ex log2

x21 2x

Properties Relating exand ln x

(4)

for any real number (5)

lnexx

x0 elnxx

The relationships expressed in Equations (4) and (5) are useful in solving equations that involve exponentials and logarithms.

EXAMPLE 8 Solve the equation .

Solution We first divide both sides of the equation by 2 to obtain

Next, taking the natural logarithm of each side of the equation and using Equation (5), we have

x 2ln 2.51.08 x2ln 2.5

lnex2ln 2.5 ex2 5

22.5 2ex25

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PROOF Let . Then . Taking the natural logarithm of both sides of this equation gives , and so, solving for , we obtain

and this proves the result.

y lnx lna

y lnxlnayylna

xay ylogax

EXAMPLE 9 Solve the equation . Solution We have

Change of Base Formula

As we mentioned earlier, it is sometimes preferable to use one base rather than another when solving a problem. More specifically, we mentioned that we often use natural logarithms to simplify formulas in calculus. The following formula enables us to write the logarithms with any base in terms of natural logarithms.

x 1

3 (e7.55)604.347 3x5e7.5

ln(3x5)7.5 2 ln(3x5)15

2 ln(3x5)15

Change of Base Formula

If is a positive number and , then logax lnx

lna a1

EXAMPLE 10 Evaluate correct to five decimal places.

Solution We have

log97ln 7

ln 90.88562 log97

1. Define the number . What is its approximate value?

2. Define the natural exponential function . What are its domain and range?

3. State the laws of exponents.

4. What is the relationship between the graph of and that of t(x)lnx? Sketch the graphs on the same set of axes.

f(x)ex f(x)ex

e 5. Define the natural logarithmic function . What are

its domain and range?

6. State the laws of logarithms.

f(x)lnx