Math 22 Rates of Change

Average Rate of Change and velocity

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

Instantaneous Rate of Change and Velocity

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

Notes

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

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

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