Continuous Exponential Growth or Decay

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					2.8 Continuous Exponential Growth or Decay
Some growth occurs exponentially at specific intervals: interest is accrued at the end of
every fourth month. Other exponential growth or decay occurs continuously: the number
of bacteria in a sample grows continuously. (At any time, the population is increasing,
not just every 4th minute or something like that…)

If we have an investment that offers interest at 100%/a compounded as often as you’d
like we can see what happens as we increase the number of compounding periods to
infinity.

Principal # of Compounding       Rate of Interest    Amount in Dollars
             Periods             per Period
$1           1                   1.00
$1
$1
$1
$1



As the number of compounding periods approaches infinity, the value of the expression
(1 + 1/n)n approaches 2.7182818… This is an irrational number (like pi) that is named
euler’s number or e. Calculators have this number displayed as ex . (To solve
exponential expressions with a base of e, we use loge or ln.)

Usage of e

Whenever an amount is continuously growing exponentially, where the initial amount is
Ao , the time is t (in years) and the growth rate is r% per year, we use the following
expression: A(t) = Ao e(r/100)t

Example 1

If a population of 5000 bacteria is growing continuously at a rate of 45% per hour, then
what is the population after 3 hours? 3 days?

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When there is exponential decay that occurs continuously, we can also use the euler’s
number. When there is continuous exponential decay there is a negative exponent.

Example 2

A snowball melts such that its volume is a function of the time it is left in the warm air,
where V(t) is the volume at time t (in hours):
V(t) = Vo e-2.7t

   a) A snowball has a diameter of 4cm. What is its volume?
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   b) What is its volume after 2 hours?
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   c) By using your graphing calculator, calculate how many hours will pass by for
       85% to be gone.
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   d) Using logs, determine how many hours it will take to reduce the snowball to 50%
       of its original volume.
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