Download - 8.3 the number e
What is e?Much like π and i, e is a special number
used in math.
Discovered by mathematician Leonhard
Euler. (Sounds like “oiler”)
Called the natural base e or the Euler
number.
Investigating e
n 10 100 1000 10,000 100,000 1,000,000
•As n approaches +∞, approaches
e≈2.718281828459
•The natural base e is irrational. (It cannot be
expressed as a fraction.)
Simplifying Natural Base
ExpressionsFollow the same exponent rules as with
other bases.
Examples:
Your Turn! Simplify
Evaluating Natural Base
ExpressionsUse a calculator to evaluate each
expression.
Press “2nd” then “LN” key to get to ex .
Examples:
e2
e-0.06
Natural Base Exponential
FunctionsFunctions of the form f(x) = aerx are called
natural base exponential functions.
If r > 0 it is exponential growth.
If r < 0 it is exponential decay.
Graphing Natural Base Functions
Plot points (0, a) and (1, ___)
If points are too close together, you may
choose a different x for the 2nd point.
Shift parent graph using h and k if needed.
Examples:Graph. Then state the domain and range.
Your Turn!Graph. Then state the domain and range.
Continuous InterestRemember, compound interest uses the
equation:
As n approaches +∞ it is called continuously
compounded interest.
It is then modeled by:
Example:You deposit $1000 in an account that pays
8% annual interest compounded
continuously.
What is the balance after 1 year?
Your Turn!You deposit $1500 in an account that pays
7.5% annual interest compounded
continuously.
What is the balance after 3 years?