Question

What is instantaneous rate of change as the limit of average rate of change?

I understand what the instantaneous rate of change and average rate of change are, I just don't know how to represent the instantaneous as the limit of the average.

Answer 1

It is the limit as the denominator goes to `0`.

Explanation:

There are several choices for notation:

Using `y=f(x)` we can write the average rate of change as

First Notation
`x` goes from `a` to `x`

Average Rate: `(f(x)-f(a))/(x-a)`

Instantaneous Rate at `a`: `lim_(xrarra)(f(x)-f(a))/(x-a)`

Second Notation
`x` goes from `x` to `x+h`

Average Rate: `(f(x+h)-f(x))/h`

Instantaneous Rate at `a`: `lim_(hrarr0)(f(x+h)-f(x))/h`

Third Notation

Average rate of change of `y` with respect to `x`: `(Deltay)/(Deltax)`

Instantaneous rate of change of `y` with respect to `x`: `lim_(Deltaxrarr0)(Deltay)/(Deltax)`