Question

How you used the limit definition of a derivative to calculate the instantaneous acceleration?

Answer 1

Average acceleration is defined as

`a_(\rm ave) = (\Delta v)/(\Delta t)`

Instantaneous acceleration is obtained by letting
`\Delta t\to0`.

`a = \displaystyle \lim_(\Delta t \to 0) (\Delta v)/(\Delta t)`

As velocity is a function
`v(t)` of time
`t`, we have for some time interval
`\Delta t`,

`\Delta v = v(t+\Delta t) - v(t)`

so that

`a = \displaystyle \lim_(\Delta t \to 0) (v(t+\Delta t) - v(t))/(\Delta t)`

and the right side is exactly the derivative of
`v(t)`.