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