Let us find the instantaneous rate of change for different functions
Instantaneous rate of change calculus gives the change in rate at a given instant. Before proceeding further, let us define rate of change in math. It is given by the change of a quantity between two instances.
The instantaneous rate of change of a function gives the derivative of the same function at particular point according to the instantaneous rate of change definition. The formula for instantaneous rate of change will help to calculate the same for any given function.
Consider a function f(x) for which we will be finding instantaneous rate of change over the interval (a,b). We have Δf = f(b) – f(a) and Δx = b – a. Hence the average rate of change over an interval (a,b) will be while the instantaneous rate of change will be This instantaneous rate of change formula is similar to that of differentiation. Thus we have seen the calculus instantaneous rate of change for the function. So its time to move on with related examples.
Evaluation of instantaneous rate of change examples
The instantaneous rate of change examples given below will help to have a profound knowledge of the topic. So it is advised to study them seriously.
(1) Find the instantaneous rate of change for y = x at x = 4:
Here y = x, Δy = f(x) – f(4) = x – 4 and Δx = x – 4. So Δy/Δx = 1 => the limit value is also 1. Thus the rate of change for above equation is 1 irrespective of the points.
(2) Find instantaneous rate of change for y = x2 – x + 1 at x = 3:
We have Δy = f(x) – f(3) = (x2 – x + 1) – (32 – 3 + 1) = x2 – x + 1 – 7 = x2 – x – 6. Factorizing the above equation, Δy = (x + 2)(x – 3). Now Δx = (x – 3) since we need to find it for x = 3. So the ratios will be Δy/Δx = (x + 2)(x – 3) / (x – 3) = (x + 2). Therefore R.O.Cinst = [Δy / Δx]x->3 = [x + 2]x->3 = 3 + 2 = 5.
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