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8-3: The Number ‘e’ (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. Investigating the Natural Base e - PowerPoint PPT Presentation
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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