Notation to describe the average rate of change in a interval?

Question

Instead of writing "The average rate of change where $x \in\left[0,10\right]$ is..", are there any mathematical notations to describe it even shorter? Or how would you do it if you where to give an answer to a task like this?

Thanks for all help.

Answer 1

The average rate of change on an interval $[a,b]$ is the change in $y$ divided by the change in $x$ over that interval. This is simply the slope! $$ \frac{\Delta y}{\Delta x} = \frac{y_2-y_1}{x_2-x_1}$$ So for your example over $[0,10]$, you might write $$ \frac{f(10)-f(0)}{10-0} = \frac{f(10)-f(0)}{10}$$

Answer 2

Yes, it is possible to do so

Background-

If you would have had Calculus, it would have been easy to see that-

Rate of change of $f$ at $x_0 \in [ a, b] = \frac{df}{dx} _{(x_0)}$ and,averageof $g$ over that interval is $<g> = \frac{1}{b-a}\cdot\int_a^bg(x) \ dx$.

Using the fact that $\int \frac{df}{dx} dx = f(x) + C , C \in \mathbb{R}$ so that $\int_a^b \frac{df}{dx}\cdot dx = f(b) - f(a)$, we have -

$$<\frac{df}{dx}>= \frac{f(b)-f(a)}{b-a}$$, which is the required averge of the rate of change.

It is slightly difficult to see if you don't know calculus, but we can see that -

The rate of change of a function at a point is the slope of the tangent to its graph at that point (See wikipedia, for a gentler introduction this. Why? And what would be the average rate of change?

I leave the first question for you and proceed to the realvant other. The rate of change of $f(x)$ w.r.t $x$ is the ratio of change in the value of $f$ (say, $f(x+h) - f(x)$)corresponding to a small change in the value of $x$ (say, $h$ going from $x$ to $x+h$, to the change in $x$. Now, if the change is measured over a large (finite and non-zero) interval $[a,b]$, the rate of change (actually, the average rate of change) over the interval will be -

$\frac{f(b)-f(a)}{b-a}$, as required.

Final answer-

SUbstituting the given values, the answer is - $\frac{f(10)-f(0)}{10}$.

If you simply want a notation, it can be - $$<\frac{df}{dx}>_{x=0}^{x=10}$$.