Mean Rate of Change: Explanation & Examples

The plays a very central role in differential calculus, i.e. the derivation of functions difference quotient and the mean rate of change. The slope of non-linear functions is not so easy to read. In order to still be able to determine this from a differentiable function, we use the mean rate of change and the difference quotient. The topic can be assigned to the subject mathematics.

The difference quotient and the mean rate of change

We know that for a linear function, the slope is easy to read. It corresponds to the value of the coefficient m. This is more difficult for a non-linear function. With the help of difference quotient and the mean rate of change you can calculate the slope of a non-linear function. It’s actually not as difficult as it appears at first glance.

The slope of a function f(x) at the point corresponds to the slope of the tangent to the graph of f through the point .

The mean rate of change between the two points P and Q of a function is the slope of the secant s passing through those two points of the function. The slope of the secant is given as the mean rate of change over the interval.

The so-called difference quotient results for this gradient. The difference quotient can thus be interpreted geometrically as the gradient of the secant s through the graph points.

The so-called difference quotient results for the gradient:

sample task

The following example asks for the mean rate of change. This is often searched for when asking about average speed, average growth, etc. An interval is always specified, i.e. a certain period of time in which the growth is considered.

The growth of a flower can be described with f(x), i.e. y, indicates the height in cm and x the duration in weeks. How much does the flower grow in the period ?

First we calculate f(x) and f() by substituting x and into the function.

So the flower will grow about 1/4 inch in the first 5 weeks.

To repeat: when is a function differentiable?

A real function is differentiable if it is continuous at this point, i.e. if the graph of the function has no vertices there. Only then can a tangent be clearly laid at the point. The function has a unique derivative at this point.

When is a function continuous?

A function is continuous in an interval if you can plot the function «without discontinuities» or «without jumps».

With one of these options you can mathematically prove the differentiability of a function at the point:

  • The existence of left-sided differential quotient:

Here we approach the place from the left side.

  • The existence of right-sided differential quotient: Here we approach the point from the right side.
  • The equality of both limits
  • The continuity of f at the point

In general it can be said:

The rational functions, power functions, root functions, logarithmic functions, exponential functions, trigonometric functions are differentiable at every point of their maximum definition set.

Continuity and differentiability describe different properties of real functions. However, one can say:

If a function is differentiable at one point in its domain, then it is continuous there. But not every function that is continuous at one point in its definition set is also differentiable there. For example, the function f(x) = |x| continuous at x = 0, but not differentiable.

Difference quotient ≠ differential quotient

You’ve probably heard of the differential quotient. This sounds very similar to the difference quotient, but it is not the same. The difference quotient is related to the mean rate of change, while the derivative quotient is related to the local or instantaneous rate of change.

Here we summarize the most important thing about this topic for you: If the point Q gets closer and closer to the point P until it almost reaches it, the result is the instantaneous rate of change. For the slope of the tangent and thus the instantaneous rate of change, one obtains:

This limit is called differential quotient and corresponds to the 1st derivative at the point.

sample task

The following example task illustrates the difference between mean and instantaneous rate of change.

If x denotes the time in minutes (our period under consideration is between 3 and 10 minutes) since the beginning of the observation and y denotes the number of germs in the water (at minute 3 we have 210 germs and at minute 10 560 germs), then indicates the mean rate of change, by which number (f(x) – ()) the germs multiply in the period under consideration (x-) (then >0 and if they should decrease, <0 applies).

We get the mean rate of change by substituting the values ​​into the difference quotient:

In the period between 3 and 10 minutes after the start of observation, there are an average of 50 more germs per minute.

The instantaneous rate of change indicates by how much the number of germs is growing or shrinking at that point in time. To get this we use the differential quotient.

At the point in time, the number of germs per minute increases by 90.

Our tip for you

Check out our article on local rate of change or differential quotient and compare the two articles. In this way you will become aware of the differences between the difference quotient and the derivative quotient or the mean rate of change and the local rate of change and you will better understand the topic «mean rate of change». Actually, this topic is not that difficult!

Mean rate of change – the most important things at a glance

The mean rate of change describes how fast and how much something changes in a certain period. You can use it to calculate average speeds or average inclines, for example. You do this using the difference quotient.

You can think of the mean rate of change graphically as the secant slope between two points. This then shows you graphically the gradient or the average increase or decrease of a function in this interval.