6.6 The Natural Base, e

Natural Base, e

The natural base, e, is used to estimate the ages of artifacts and to calculate interest that is compounded continuously.

2.718e

Natural Exponential Function The exponential function with base e, f(x)=ex,

is called the natural exponential function.

And e is called the natural base.You calculator has a FANCY ex button you can use!!!

Evaluate f(x)=ex to the nearest thousandth for each value of x below: x = 2

x = -1/3

x = 0.25

2(2) 7.389f e

0.25(0.25) 1.284f e

1

310.717

3f e

Simplify the expression.

a. e2 e5 = e2 + 5

= e7

b. 12e4

3e3 = e4 – 34

= 4e

(5 )c. e –3x 2 = 52 (e –3x )2

= 25e –6x

= 25e6x

Exponential Properties

Simplify the expression.

1. e7 e4 e11

3. 24e8

4e5

12e22. 2e–3 6e5

6 e3

Try This!

4. (10e –4x )3 1000e12x

2.117

5. Use a calculator to evaluate 4

3e

The Graph of the Exponential Function

Graph the function. State the domain and range.

a. y = 3e 0.25x

SOLUTION

Because a = 3 is positive and r = 0.25 is positive, the function is an exponential growth function. Plug in 0 and 1.

The domain is all real numbers, and the range is y > 0.

Graph natural base functions

x y

01

Graph the function. State the domain and range.

SOLUTION

The domain is all real numbers, and the range is y > 1.

b. y = e –0.75(x – 2) + 1

a = 1 is positive and r = –0.75 is negative, so the function is an exponential decay function. Translate the graph of y = right 2 units and up 1 unit.

e –0.75x

Graph natural base functions

Graph the function. State the domain and range.

y = 2e 0.5x

domain: all real numbers,range: y>0

Try This!

Graph the function. State the domain and range.12

f(x)= e -x +1

domain: all real numbers,range: y>1

Try This!

Continuous Compounding Formula If P dollars are invested at an interest rate, r,

that is compounding continuously, then the amount, A, of the investment at time t is given by:

A = Pert

An investment of $1000 earns an annual interest rate of 7.6% Compare the final amounts after 8 years for interest

compounded quarterly and for interest compounded continuously.

Compounded Quarterly Continuously

1nt

rtrA P A Pe

n

Find the value:

Of $500 after 4 years invested at an annual interest rate of 9% compounded continuously.

$716.66

Homework

Green Book!!Green Book!!Page 142 # 1-25odd

#27 and 28