# Math 22 Rates of Change

## Average Rate of Change and velocity

 If ${\displaystyle y=f(x)}$, then the average rate of change of ${\displaystyle y}$ with respect to
${\displaystyle x}$ on the interval ${\displaystyle [a,b]}$ is ${\displaystyle {\frac {f(b)-f(a)}{b-a}}={\frac {\Delta y}{\Delta x}}}$


## Instantaneous Rate of Change and Velocity

 The instantaneous rate of change of ${\displaystyle y=f(x)}$ at ${\displaystyle x}$ on the interval ${\displaystyle [a,b]}$ is
${\displaystyle \lim _{\Delta x\to 0}{\frac {f(x+\Delta x)-f(x)}{\Delta x}}}$ for ${\displaystyle \Delta x=b-a}$


## Notes

1) If ${\displaystyle p=f(x)}$ is the demand function, then Revenue function will be ${\displaystyle R(x)=x.p=x.f(x)}$

2) The profit function ${\displaystyle P(x)}$ can be calculate as ${\displaystyle P(x)=R(x)-C(x)}$ where ${\displaystyle R(x)}$ is revenue function and ${\displaystyle C(x)}$ is the cost function.