Average rate of change: Interactive Visualization

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Average rate of change in this visualization

The average rate of change of a function $f$ over an interval $[a, b]$ is the change in its value divided by the change in its input: the difference quotient $\frac{\Delta y}{\Delta x}$ . On the graph it is the slope of the secant line through $(a, f(a))$ and $(b, f(b))$ . For a distance travelled over time it is the average speed. This page cuts $[a, b]$ into at most eight pieces and compares the whole with its pieces, for five functions.

Average rate of change over [a, b]

\frac{\Delta y}{\Delta x} = \frac{f(b) - f(a)}{b - a} = \frac{\sum_i \Delta x_i \, m_i}{\sum_i \Delta x_i}

The cuts $a = x_0 < x_1 < \dots < x_n = b$ make pieces of widths $\Delta x_i = x_i - x_{i-1}$ , each with its own rate $m_i = \Delta y_i / \Delta x_i$ , the slope of a thin yellow chord. Under the graph each piece is a bar: its width is $\Delta x_i$ , its height is $m_i$ , so its signed area is $\Delta x_i \, m_i = \Delta y_i$ , the number written inside it. The areas add up to $\Delta y$ , because in $\sum_i (f(x_i) - f(x_{i-1}))$ every value between the ends appears once with each sign and cancels. Spreading that total area evenly over $[a, b]$ gives the solid yellow line, at height $\Delta y / \Delta x$ : the rate over the whole interval is the pieces' rates averaged by width. The dashed line is their plain mean $\frac{1}{n} \sum_i m_i$ , which ignores the widths.

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Things to notice

A wide bar weighs more. Going 60 km at 60 km/h and coming back 60 km at 40 km/h takes 2.5 h: the average speed is 120 ÷ 2.5 = 48 km/h, not 50, because the slower leg lasts longer. (Average speed divides the distance travelled; average velocity divides the displacement, 0 for a round trip.) With pieces of equal width the two lines always coincide; they can also meet otherwise, as for a linear function, whose pieces all share one rate.
On pieces of equal width $h$ , the rates follow a pattern: for a linear function they are equal; for $ax^2 + bx + c$ each differs from the one before by $2ah$ ; for $b^x$ each is $b^h$ times the one before. Compare writes the differences and ratios between neighbouring bars.
The average rate of change is not the average value of $f$ , and it only sees the ends: $-x^2 + 4x$ climbs to 4 on $[0, 4]$ and comes back, so its rate there is 0. A positive whole rate allows a piece with a negative one.
Bars that rise from left to right say that the chords bend upward, not that the curve does everywhere: $x^2 + 0.1 \sin 2\pi x$ gives the bars 1, 3, 5, 7 on the cuts 0, 1, 2, 3, 4, yet bends downward at $x = \tfrac14$ , where $f'' = 2 - 0.4\pi^2 \approx -1.95$ .

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Controls

Drag the ends a and b or a cut, on the graph or at the bars' edges; they catch on multiples of ½. A cut dragged onto its neighbour merges with it. The digit wheel sets the number of pieces and cuts the interval evenly; Equal evens out the present pieces; Cuts at takes exact values, such as 1, 2.5 or 4/3.
Presentation mode opens a full-screen view for projectors (← → change the number of pieces; H hides the average rate of change).

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Average rate of change in this visualization

Things to notice

Controls

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