Rate of change
Introduction:
Definition of Rate of Change
A function varies in accordance with the variation of the variable. The nature of variation of the function depends on the relation of the function. The rate of change of a function, shortly described as rate of change or change rate is the ratio of the change in values of the function and the change in the values of the variable in that interval. To be more explicit, if f(x) is a function of x, and f(a), f(b) are the values of the function when x changes from a to b, the change rate is, [f(b) – f(a)]/[b – a]. This is called as the rate of change formula.
We must consider one fact when we talk of change rates of functions of different types. Excepting for linear functions, the change rate will not be constant. That is, for any function other than linear, the change rate from points a to b will not necessarily be same as the change rate for another set of points from c to d.
Also the change rate in a given interval may not be same even in a sub interval within the main interval. Therefore, what we really calculate from the definition is the average rate of change and the formula [f(b) – f(a)]/[b – a] will only be an average rate of change formula.
One can now easily predict that the average rate change of a function will be more meaningful only if the considered interval is as short as possible. For example, if f(x) = (x – 3)2 – 4, then as per the formula, the average change rate of the function in the interval [1, 5] is [{(5 – 3)2 – 4} – {(1 – 3)2 – 4}]/[5 – 1] = 0.
It is ridiculous information and cannot serve any useful purpose. We cannot apply this result to find f(2) and conclude that f(2) = f(1) = 0, whereas actually f(2) = -3. Therefore, always calculate the average change rate for a very close interval.
Let us see how mathematicians got over this point. They introduced the concept of instantaneous change rate taking the interval infinitely small and finding the limit when the width of the interval tends to 0. This is called the derivative of the function in the topic of calculus.
As a matter of fact, the anomaly of average change rate of a function has led into the introduction of the topic of calculus.
However, still the concept of average change rate of a function is used in certain studies like mean value theorem. The concept of mean value theorem is used in many practical applications.
Definition of Rate of Change
A function varies in accordance with the variation of the variable. The nature of variation of the function depends on the relation of the function. The rate of change of a function, shortly described as rate of change or change rate is the ratio of the change in values of the function and the change in the values of the variable in that interval. To be more explicit, if f(x) is a function of x, and f(a), f(b) are the values of the function when x changes from a to b, the change rate is, [f(b) – f(a)]/[b – a]. This is called as the rate of change formula.
We must consider one fact when we talk of change rates of functions of different types. Excepting for linear functions, the change rate will not be constant. That is, for any function other than linear, the change rate from points a to b will not necessarily be same as the change rate for another set of points from c to d.
Also the change rate in a given interval may not be same even in a sub interval within the main interval. Therefore, what we really calculate from the definition is the average rate of change and the formula [f(b) – f(a)]/[b – a] will only be an average rate of change formula.
One can now easily predict that the average rate change of a function will be more meaningful only if the considered interval is as short as possible. For example, if f(x) = (x – 3)2 – 4, then as per the formula, the average change rate of the function in the interval [1, 5] is [{(5 – 3)2 – 4} – {(1 – 3)2 – 4}]/[5 – 1] = 0.
It is ridiculous information and cannot serve any useful purpose. We cannot apply this result to find f(2) and conclude that f(2) = f(1) = 0, whereas actually f(2) = -3. Therefore, always calculate the average change rate for a very close interval.
Let us see how mathematicians got over this point. They introduced the concept of instantaneous change rate taking the interval infinitely small and finding the limit when the width of the interval tends to 0. This is called the derivative of the function in the topic of calculus.
As a matter of fact, the anomaly of average change rate of a function has led into the introduction of the topic of calculus.
However, still the concept of average change rate of a function is used in certain studies like mean value theorem. The concept of mean value theorem is used in many practical applications.