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

Exponential Growth and Decay
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 $$ N = N$$0ekt

where:  $$N$$0 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: $$ N = N$$0ekt, then the derivative of $$N$$ can be shown as: dN      $$=kN$$0ekt dt $$=kN$$, substituting $$ N = N$$0ekt.

(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
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 $$= kN$$0ekt$$- kP$$ where $$P$$ = the external constant (e.g., the external room temperature)

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

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

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

$$log$$e$$1 - log$$e$$(-2) = 10k$$

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

= .34567359... (store in memory) (ii) 30=24-100e^(.34657359t)

incomplete 10th august '08