7.3 Use Functions Involving e

Natural base e

An irrational number, where approaches e as n increases

THE NATURAL BASE e

The natural base e is irrational. It is defined as follows:

As n approaches + ¥ approaches

e » _2.718281828_.

Example 1

Simplify the expression.

a. e⁶ · e³ = e _6 + 3_ = _e⁹_

b. = _9e^{6 - 4}_ = _9e²_

c. (4e^3x)² = _4²e^(3x)(2)_

= _16e^6x_

Example 2

Use a calculator to evaluate the expression.

Expression	Keystrokes	Display
a. e^-2	[e^x] _2_	_0.1353352832_
b. e^0..3	[e^x] _0.3_	_0.349858808_

NATURAL BASE FUNCTIONS

A function of the form y = ae^rx is called a natural base exponential function.

· If a > 0 and r > 0, the function is an exponential _growth_ function.

· If a > 0 and r < 0, the function is an exponential _decay_ function.

Example 3

Graph the function. State the domain and range.

a. y = 2e^0.6x

b. y = e^{-0..35(x + 1)} - 2

Solution

a. Because a = _2_ is _positive_ and r = _0.6_ is _positive_, the function is an exponential _growth_ function. Plot the points (0, _2_) and (1, _3.64_) and draw the curve. The domain is _all real numbers_, and the range is _y > 0.

b. Because a = _1_ is __positive__ and r =__-0.35__ is __negative__, the function is an exponential _decay_function. Translate the graph of y = e^-0.35x __left 1 unit__ and __down 2 units.__ The domain is __all real numbers__, and the range is __y > -2__ .

CONTINUOUSLY COMPOUNDED INTEREST

When interest is compounded continuously, the amount A in an account after t years is given by the formula

A = _Pe^rt_ where P is the _principal_ and t is the _annual interest rate_ expressed as a decimal.

Example 4

Model continuously compounded interest

Compound Interest You deposit $3500 in an account that pays 4% annual interest compounded continuously. What is the balance after 1 year?

Solution

Use the formula for continuously compounded interest.

A = Pe^rt Write formula.

= _3500e^0.04(1)_ Substitute for P, r, and t.

» _3642.84_

The balance at the end of 1 year is _$3642.84_.