8.3 the number e

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8.3 The Number e

8.3 The Number eWhat is e?Much like 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 en10100100010,000100,0001,000,000

As n approaches +, approaches e2.718281828459The 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-0.06Natural 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 FunctionsPlot 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?