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  N = N0ekt

where:
N0
is the first value of N (where t = 0)
t
represents time in given units (seconds, hours, days, years, etc.)
e is the exponential constant (e = 2.718281828...), and
k is the growth (k +ve) or decay(k -ve) constant.

Differentiation can be used to show that the rate of change (with respect to time, t) of N is proportional (∞) to N. if:
 N = N0ekt,
then the Derivative of N can be show as:
dN =kN0ekt
dt
=kN, substituting  N = N0ekt.

(note the derivative of e is the variable k of the power of e times e^{kx} and  N, t 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 =  k(N - P)
dt
= kN0ekt- kP

where P = the external constant (e.g., the external room temperature)


using natural logarithms, logex, we can find any variable when given certain information.
Example:
A cup of boiling water is initially  100oC. The external room temperature is  24oC. after 10 minutes, the temperature of the water is  74oC. find
(i) k
(ii)how many minutes it takes for the temperature to equal 30 degrees.

(i)74=24-100e10k
50=-100e10k

\frac {50} {-100} = e^{10k}

loge1 - loge(-2) = 10k

k = \frac{log_e 1 - log_e(-2)}{2}

= .34567359... (store in memory)

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

incomplete 10th august '08