8-3: The Number ‘e’8-3: The Number ‘e’(Day 1)(Day 1)

Objective (Ca. Standard 12): Students know the laws of fractional exponents,

understand exponential function, and use these functions in problems in problems involving exponential growth and decay.

Like and , the number is denoted by a letter.

The number is called the natural base , or the

Euler number, after its discoverer, Leonard Euler.

i e

e

Investigating the Natural Base e

Turn to page 480 complete the table and answer the question in part 2.

1 2 3 4 10 10 10 10

11 2.594 2.705 2.718 2.718

n

n

n

Do the values in the table appear to be approaching a fixed decimal number?

Yes, the number 2.718.

The Natural Base e

The natural base e is irrational. It is defined as follows:

1As n approaches , 1 approaches

2.718281828459

n

n

e

Example 1: Simplifying Natural Base Expressions

3 4a) e e 3 4 7e e 3

2

10b)

5

e

e 3 22 2e e

24c) 3 xe 4 22

8

93 x

xe

e

Simplify

Example 2: Evaluating Natural Base Expressions

2a) e 7.3890560.06b) e 0.941765

( ) rxf x ae

Natural base exponential function has the form

The function is an exponential growth function if a > 0 and r > 0.

8

6

4

2

Exponential Growth

f x = e x

(2, 7.29)

(1, 2.7)

(0, 1)

The function is an exponential decay function if a > 0 and r < 0.

8

6

4

2

-2

5

Exponential Decayf x = e -x

(-1, 2.7)

(0, 1)

(-2, 7.29)

Example 3: Graphing Natural Base Functions

Graph the function. State the domain and range.

0.75a) ( ) 2 xf x eSolution:

Because a = 2 is positive and r = 0.75, the function is an exponential growth function.

0.75

0.75 0 0

0.75 1 0.75

( ) 2

0 (0) 2 2 2 0,2

1 (1) 2 2 4.73 1,4.73

xx f x e

f e e

f e e

Plot the points and sketch the graph.

8

6

4

2

f x = 2e 0.75x

Domain: all real #’sRange: y > 0

0.5 2b) 1xy e

Solution:Because a= 1 is positive and r = - 0.5 is negative, the function is an exponential decay function.

0.5

0.5 0 0

0.5 3 1.5

0 0 1 0,1

3 3 4.48 3,4.48

xx f x e

f e e

f e e

8

6

4

2

5

0, 1

-3, 4.48

q x = e -0.5x

Translate the graph to the right by 2 units and up 1.

8

6

4

22, 2

?

-1, 5.48

r x = e -0.5 x-2 +1

0, 1

-3, 4.48

q x = e -0.5x

Domain = all real #’s Range = y > 1

Homework: P.483 #17-31 Odd, #49-59 Odd