Concept explainers
Explain the difference between average and instantaneous rates of change, and discuss how they are calculated.
The difference between the instantaneous and average rates of change.
Answer to Problem 1RE
The difference between the instantaneous and average rates of change is that the instantaneous rate of change is calculated at a point whereas the average rate of change is calculated over an interval.
Explanation of Solution
The instantaneous rate of change is calculated at a point.
Consider a function,
Let instantaneous rate of change be denoted as
The instantaneous rate of change of
The average rate of change is calculated over an interval.
Consider a function,
Let the average rate of change be denoted as
The average rate of change of
Example: Consider a function,
The aim is to find the instantaneous rate of change in the population of amoeba at
Use the formulae mentioned above for the instantaneous and average rates of changes.
Further,
Now,
Therefore, the instantaneous rate of change in the population of amoeba at
Geometrically, the instantaneous rate of change of y with respect to
The average rate of change of y with respect to
Therefore, the instantaneous rate of change is calculated at a point whereas the average rate of change is calculated over an interval.
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