The rate of change at one known instant is the Instantaneous rate of change, and it is equivalent to the value of the derivative at that specific point. So it can be said that, in a function, the slope, m of the tangent is equivalent to the instantaneous rate of change at a specific point. One more method to comprehend this concept clearly is with the difference quotient and limits. The average rate of change of y with respect to x is the difference quotient. Now if one looks at the difference quotient and lets Delta x->0, this will be the instantaneous rate of change. In guileless words, the time interval gets lesser and lesser.
The Instantaneous Rate of Change Formula provided with limit exists in,
When y = f(x), with regards to x, when x = a.
Instantaneous Rate of Change – Solved Examples
Underneath are given the problems on Instantaneous Rate of Change:
Problem 1: Compute the Instantaneous rate of change of the function f(x) = 3x2 + 12 at x = 4 ?
f(x) = 3x2 + 12
f'(x) = 3(2x) + 0 = 6x
Thus, the instantaneous rate of change at x = 4
f'(4) = 6(4) = 24
Problem 2: Compute the Instantaneous rate of change of the function f(x) = 5x3 – 4x2 + 2x + 1 at x = 2?
Known Function, f(x) = 5x3 – 4x2 + 2x + 1
f'(x) = 5(3x2) – 4(2x) + 2 + 0 = 15x2 – 8x + 2
Thus, the instantaneous rate of change at x = 2
f'(2) = 15(2)2 – 8(2) + 2 = 60 – 16 + 2 = 46