Question

How does instantaneous rate of change differ from average rate of change?

Answer 1

Hint: In this question, we have two rates of change instantaneous rate of change and average rate of change.
And we find the difference between instantaneous rate of change and average rate of change. The average rate of change is calculated by the below formula. If the function \[y = f\left( x \right)\] has two points \[x = a\] and \[x = b\], then the average rate of change is
\[average\;rate\;of\;change = \dfrac{{f\left( b \right) - f\left( a \right)}}{{b - a}}\]

Complete step by step answer:
In this question, we used two words: the instantaneous rate of change and average rate of change.
First we know about the average rate of change, the average rate of change is calculated between two points, and defined as the total change of function value divided by the change in the input value.
If the function has two points \[x = a\] and \[x = b\], then the average rate of change is.
\[average\;rate\;of\;change = \dfrac{{f\left( b \right) - f\left( a \right)}}{{b - a}}\]
Now, we come to the instantaneous rate of change, the instantaneous rate of change is defined as the change at a particular point which comes in the function domain, the gradient or a particular moment at that point. If the function is \[y = f\left( x \right)\] has the point \[{x_0}\] in its domain. Then the instantaneous rate of change is calculated as.
\[\mathop {\lim }\limits_{x \to {x_0}} \dfrac{{\Delta y}}{{\Delta x}} = \mathop {\lim }\limits_{x \to {x_0}} \dfrac{{f\left( {{x_0}} \right) - f\left( x \right)}}{{{x_0} - x}}\]
Now come to the question, in the question ask for difference between instantaneous rate of change and average rate of change.
The basic difference in between instantaneous rate of change and average rate of change is below.
The instantaneous rate of change is applicable for a particular point while the average rate of change is applicable for over a range.

Note:
If you want to find the difference between instantaneous rate of change and average rate of change. Then you remember that, when we calculate the average rate of change then it is applicable for over a range while we calculate the instantaneous rate of change then it is applicable for a particular point.