# What is the relationship between average rate of change and instantaneous change; explain using the case of velocity?

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The average rate of change of a variable X over a given duration of time t is the difference between the initial value of the variable `X_0` and the final value of the variable `X_t` divided by t.

In the case of average velocity and instantaneous velocity the variable X is the displacement. Average velocity is the displacement in time t divide by t. To determine the instantaneous velocity, the duration of time t has to be reduced and made infinitesimally small; this is denoted as `dt` . The displacement in time `dt` is denoted as `dX` and the instantaneous velocity is V = `(dX)/(dt)`

Alternately, instantaneous velocity V = `lim_(t->0) (X_t - X_0)/t`

Let the velocity be given by y=f(x) where x is time

the velocity at any time x1 and x3 i.e. the start and the end of time interval are given by y1 & y3 coresponding to time x1 and x3

**The instantaneous velocity at the mid point** of the time interval x2=(x3-x1)/2** is y2=f((x3-x1)/2)**

**whereas the average velocity of the object between point x1 and x3 is** constant and is equal to the distance covered between time interval x3-x1 divided by x3-x1 and isequal to:

** Sf(x).dx/(x3-x1) with limits x1 to x3**

Please note that 'S' is used inplace of integral sign above.

I hope it helps.