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

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,

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

Graph of Exponential Functions when b > 1

Graph of Exponential Functions when 0 < b < 1

Graph of Exponential Functions when b > 1 y -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

Graphing Exponential Functions y -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

Graphing Exponential Functions

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

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

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 = $5800.06

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

The Natural Exponential Function

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

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