4.1 Exponential functions  (Page 8/16)

The continuous growth/decay formula

For all real numbers $t,$ and all positive numbers $a$ and $r,$ continuous growth or decay is represented by the formula

where

• $a$ is the initial value,
• $r$ is the continuous growth rate per unit time,
• and $t$ is the elapsed time.

If $r>0$ , then the formula represents continuous growth. If $r<0$ , then the formula represents continuous decay.

For business applications, the continuous growth formula is called the continuous compounding formula and takes the form

where

• $P$ is the principal or the initial invested,
• $r$ is the growth or interest rate per unit time,
• and $t$ is the period or term of the investment.