# Average and Instantaneous Rate of Change

A variable which can assign any value independently is called the independent variable, and the variable which depends on some independent variable is called the dependent variable.

For Example:
$y = f(x) = {x^2} – 1$
If $x = 0,1,2,3, \ldots$ etc, then

$\begin{array}{*{20}{c}} {f(0) = – 1} \\ {f(1) = 0} \\ {f(2) = 3} \\ {f(3) = 6} \\ \vdots\end{array}$

We see that as $x$ behaves independently, we call it the independent variable. But the behavior of $y$ or $f(x)$ depends on the variable $x$, so we call it the dependent variable.

Increment:

Literally the word increment means increase, but in mathematics this word covers both increase as well as decrease because the increment may be positive or negative. Simply put, the word increment in mathematics means “the difference between two values of variables.”

The final value minus the initial value is called an increment in the variable. The increment in $x$ is denoted by the symbols $\delta x$ or $\Delta x$ (read as “delta $x$”).
If $y = f(x)$, and $x$ changes from an initial value ${x_0}$ to the final value ${x_1}$, then $y$ changes from an initial value ${y_0} = f({x_0})$ to the final value ${y_1} = f({x_1})$.
Thus, the increment in ‘$x$’
$\Delta x = {x_1} – {x_0}$
produces a corresponding increment in ‘$y$’
$\Delta y = {y_1} – {y_0} = f({x_1}) – f({x_0})$

Average Rate of Change:
If $y = f(x)$ is a real valued continuous function in the interval $({x_0},{x_1})$, then the average rate of change of ‘$y$’ with respect to ‘$x$’ over this interval is
$\frac{{f({x_1}) – f({x_0})}}{{{x_1} – {x_0}}}$
But $\Delta x = {x_1} – {x_0}$
$\Rightarrow {x_1} = {x_0} + \Delta x$
$\therefore \frac{{f({x_0} + \Delta x) – f({x_0})}}{{\Delta x}}$

Instantaneous Rate of Change:

If $y = f(x)$ is a real valued continuous function in the interval $({x_0},{x_1})$, then the average rate of change of ‘$y$’ with respect to ‘$x$’ over this interval is
$\mathop {\lim }\limits_{{x_1} \to {x_0}} \frac{{f({x_1}) – f({x_0})}}{{{x_1} – {x_0}}}$
But $\Delta x = {x_1} – {x_0}$
$\Rightarrow {x_1} = {x_0} + \Delta x$
This shows that $\Delta x \to 0$, as ${x_1} \to {x_0}$
$\therefore \mathop {\lim }\limits_{\Delta x \to 0} \frac{{f({x_0} + \Delta x) – f({x_0})}}{{\Delta x}}$

Average or Instantaneous Rate of Change of Distance
OR
Average or Instantaneous Velocity:

Suppose a particle (or an object) is moving in a straight line and its positions (from some fixed point) after times ${t_0}$ and ${t_1}$ are given by $S({t_0})$ and $S({t_1})$, then the average rate of change or the average velocity is
${V_{ave}} = \frac{{S({t_1}) – S({t_0})}}{{{t_1} – {t_0}}}$
Also, the instantaneous rate of change of distance or instantaneous velocity is
${V_{ins}} = \mathop {\lim }\limits_{{t_1} \to {t_0}} \frac{{S({t_1}) – S({t_0})}}{{{t_1} – {t_0}}}$