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

Exponential Growth and DecayEdit

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 applicationsEdit

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

 

 e e 

 

= .34567359... (store in memory)

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

incomplete 10th august '08