HSC Extension 1 and 2 Mathematics/3-Unit/HSC/Applications of calculus to the physical world

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Exponential Growth and Decay[edit]

2 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 show 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[edit]

not 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

ee

= .34567359... (store in memory)

(ii) 30=24-100e^(.34657359t)

incomplete 10th august '08