# 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

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
_{0}*e*^{kt}

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:

_{0}*e*^{kt},

then the derivative of can be shown as:

__dN__ _{0}*e*^{kt}

dt

, substituting _{0}*e*^{kt}.

(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

_{0}*e*^{kt}

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 ^{o}C. The external room temperature is ^{o}C. after 10 minutes, the temperature of the water is ^{o}C. find

(i) k

(ii)how many minutes it takes for the temperature to equal 30 degrees.

(i) *e*^{10k}

*e*^{10k}

_{e} _{e}

= .34567359... (store in memory)

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

*incomplete 10th august '08*