1. Exponential Functions Compound Interest Natural Base (e)
Ch 3 Section 1
Exponential Functions
Compound Interest
Natural Base (e)

2. Exponential Functions
An exponential function with (constant) base b and exponent x is defined by
Notice that the exponent x can be any real number but the output y = bx is always a positive number. That is,

3. Exponential Functions
We will consider the more general exponential function defined by
where A is an arbitrary but constant real number.
Example:

4. Graph of Exponential Functions when b > 1

5. Graph of Exponential Functions when 0 < b < 1

6. Graph of Exponential Functions when b > 1y -4
1/16 -3
1/8 -2
1/4 -1
1/2 1 2 4 3 8 x y -4
1/16 -3
1/8 -2
1/4 -1
1/2 1 2 4 3 8

7. Graphing Exponential Functionsy -3 8 -2 4 -1 2 1
1/2
1/4 3
1/8
1/16 x y -3 8 -2 4 -1 2 1
1/2
1/4 3
1/8
1/16

8. Graphing Exponential Functions

9. Exponential Functions-Examples
A certain bacteria culture grows according to the following exponential growth model. If the bacteria numbered 20 originally, find the number of bacteria present after 6 hours.
When t  6
Thus, after 6 hours there are about 830 bacteria

10. Compound Interest A = the future value P = Present value
r = Annual interest rate (in decimal form)
m = Number of times/year interest is compounded
t = Number of years

11. Compound Interest
Find the accumulated amount of money after 5 years if $4300 is invested at 6% per year and interest is reinvested each month
= $

12. The Number e
The exponential function with base e is called “The Natural Exponential Function”
where e is an irrational constant whose value is

13. The Natural Exponential Function

14. Continuous Compound Interest
A = Future value or Accumulated amount
P = Present value
r = Annual interest rate (in decimal form)
t = Number of years

15. Continuous Compound Interest
Example: Find the accumulated amount of money after 25 years if $7500 is invested at 12% per year compounded continuously.