ACTIVITY 36

Exponential Functions (Section 5.1, pp. 384-392)

Exponential Functions:

Let a > 0 be a positive number with a ≠ 1. The exponential function with base a is defined by

f(x) = ax

for all real numbers x.

In Activity 4 (Section P.4), we defined ax for a > 0 and x a rational number. So, what does, for instance, 5√2 mean? When x is irrational, we successively approximate x by rational numbers. For instance, as

√2 ≈ 1.41421 . . .we successively approximate 5√2 with

51.4, 51.41, 51.414, 51.4142, 51.41421, . . .In practice, we simply use our calculator and find out

5√2 ≈ 9.73851 . . .

Example 1

Let f(x) = 2x. Evaluate the following:

)2(f

)(f

3

1f

3f

22 42 824978.8

3

1

2

31

2

1

3 2

1 793700526.0

3232

1 321997.3

Example 2:

Draw the graph of each function:

xxf 2x y

0 1

1 2

-1 1/22 4

-2 1/4

3 8

x

xg

2

1

Notice that

x

xg

2

1 x12

x2 )( xf

Consequently, the graph of g is the graph of f flipped over the y – axis.

Example 3:

Use the graph of f(x) = 3x to sketch the graph of each function:xxg 31)(

xxh 3)( xxk 32)(

xy 3 xy 3 xy 32xy 3 xy 31 xy 3

The Natural Exponential Function:

The natural exponential function is the exponential functionf(x) = ex

with base e ≈ 2.71828. It is often referred to as the exponential function.

Example 4:

Sketch the graph of

1 xey

xey xey 1 xey

Example 5:

When a certain drug is administered to a patient, the number of milligrams remaining in the patient’s bloodstream after t hours is modeled by

D(t) = 50 e−0.2t.How many milligrams of the drug remain in the patient’s bloodstream after 3 hours?

)3(D 3*2.050 e 6.050 e 548811636.*50milligrams 4405818.27

Compound Interest:

Compound interest is calculated by the formula:nt

n

rPtA

1)(

whereA(t) = amount after t yearsP = principalr = interest rate per yearn = number of times interest is

compounded per year

t = number of years

Example 6:

Suppose you invest $2, 000 at an annual rate of 12% (r = 0.12) compounded quarterly (n = 4). How much money would you have one year later?

What if the investment was compounded monthly (n = 12)?

1*4

4

12.012000)1(

A

nt

n

rPtA

1)(

403.12000 1255.1*2000 02.2251

1*12

12

12.012000)1(

A 1201.12000 1268.1*2000 65.2253

Compounding Continuously

Continuously Compounded interest is calculated by the formula:

whereA(t) = amount after t yearsP = principalr = interest rate per yeart = number of years

rtPetA )(

Example 7:

Suppose you invest $2,000 at an annual rate of 12% compounded continuously. How much money would you have after one years?

rtPetA )(2000P

12.0r1t

1*12.02000)1( eA 12749.1*2000

99.2254