HSC Mathematics Advanced, Extension 1, and Extension 2/3-Unit/HSC/Applications of calculus to the physical world
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Exponential Growth and Decay
edit2 unit course
The exponential function can be used to show the growth or decay of a given variable, including the growth or decay of population in a city, the heating or cooling of a body, radioactive decay of radioisotopes in nuclear chemistry, and amount of bacteria in a culture.
The exponential growth and decay formula is 0ekt
where:
0 is the first value of N (where )
represents time in given units (seconds, hours, days, years, etc.)
is the exponential constant ( ), and
is the growth ( ) or decay( ) constant.
Differentiation can be used to show that the rate of change (with respect to time, ) of is proportional (∞) to .
if:
0ekt,
then the derivative of can be shown as:
dN 0ekt
dt
, substituting 0ekt.
(note the derivative of e is the variable of the power of e times and are constant.)
3 Unit applications
editnot yet complete
The variable of a given application can be proportionate to the difference between the variable and a constant. An example of this is the internal cooling of a body as it adjusts to the external room temperature.
dN =
dt
0ekt
where = the external constant (e.g., the external room temperature)
using natural logarithms, e , we can find any variable when given certain information.
Example:
A cup of boiling water is initially oC. The external room temperature is oC. after 10 minutes, the temperature of the water is oC. find
(i) k
(ii)how many minutes it takes for the temperature to equal 30 degrees.
(i) e10k
e10k
e e
= .34567359... (store in memory)
(ii) 30=24-100e^(.34657359t)
incomplete 10th august '08