Average Rate Of Change Calculator

What this calculator does

So, you have a function's value at two points and you want to know how fast it changed, on average, between them. This tool takes the two x-values and the two function values and returns the average rate of change over that interval.

You enter x1 and f(x1) for the first point, x2 and f(x2) for the second, and it gives the rate rounded to two decimal places.

How to use it

1. Enter x1 and f(x1), the first x and the function's value there.
2. Enter x2 and f(x2), the second x and its function value.
3. Press Calculate.

The two x-values should be different, since the rate compares the change in the function against the change in x, and the x-values need a gap between them to divide by.

Rate of change is slope for functions

Underneath, this is slope, the same rise over run you would use for a line. The difference is what it is measuring. Instead of the steepness of a straight line, it measures how quickly a function's output moves as its input moves, averaged across an interval. If the function tracked distance over time, the average rate of change would be the average speed. If it tracked a savings balance over months, it would be the average monthly change. Same arithmetic, but read as a rate rather than a slope.

The formula

The average rate of change of a function f between x1 and x2 is

( f(x2) - f(x1) ) / (x2 - x1)

The top is how much the function's value changed, the bottom is how much x changed, and the ratio is the change in the function per unit of x. It is exactly the rise-over-run of a line, with f(x1) and f(x2) playing the part of the y-values.

A worked example, step by step

Suppose a function has f(2) = 5 and f(6) = 17, and you want the average rate of change from x = 2 to x = 6.

• Change in the function: 17 - 5 = 12.
• Change in x: 6 - 2 = 4.
• Average rate of change: 12 / 4 = 3.

So across that interval the function rose by 3 units of output for every 1 unit of x, on average.

The secant line and the road to calculus

There is a nice geometric picture behind this number, and it is the first step into calculus. Draw the two points on the curve of the function and join them with a straight line. That line is called a secant line, and the average rate of change is exactly its slope. It captures the overall trend between the two points, even if the function wiggled up and down in between.

Now imagine sliding the second point closer and closer to the first, shrinking the interval toward zero. The secant line pivots until it just grazes the curve at a single point, becoming the tangent line, and its slope becomes the instantaneous rate of change at that point, which is the derivative. So the average rate of change you get here is the everyday, two-point version of one of the central ideas in calculus. The derivative is simply what this becomes when the gap closes to nothing.

Questions people ask

Is the average rate of change the same as slope?

Yes, the arithmetic is identical, rise over run. The difference is that it is read as how fast a function changes over an interval, rather than the steepness of a line.

What does the number mean?

How much the function's output changed per unit of input, on average, between the two points. For distance over time, that is average speed.

What is the secant line?

The straight line joining the two points on the function's curve. Its slope is the average rate of change between them.

How does this connect to the derivative?

Shrink the interval toward zero and the secant line becomes a tangent line. Its slope becomes the instantaneous rate of change, which is the derivative.

Why is it called an average rate?

Because it gives the overall change per unit of x across the whole interval, smoothing over any ups and downs the function had in between.

References

On the secant slope and the derivative. The average rate of change of a function between two points is the slope of the secant line joining them, and its limit as the points merge is the derivative.

1. Eric W. Weisstein, "Secant Line," from MathWorld, a Wolfram resource, on the secant line giving the average rate of change, whose limit is the derivative.
2. Christopher Stover and Eric W. Weisstein, "Line," from MathWorld, a Wolfram resource, on the line whose slope this rate of change represents.