Rates of Change

The average rate of change

For a function $y=f(x)$ between distinct inputs $x_1$ and $x_2$, the average rate of change is

Here $\Delta$, read “delta”, means the final value minus the initial value. The numerator and denominator must use the same endpoint order. Reversing both differences leaves the quotient unchanged; reversing only one reverses its sign incorrectly.

The units are output units per input unit. If height is measured in centimetres and time in days, the rate is centimetres per day. If both variables are abstract numbers, explain which output is changing with respect to which input.

The OpenStax treatment of rates of change connects this quotient to the slope of a secant line. The calculation applies even when the function itself is not linear.

Reading changes from a table

Consider these hypothetical observations of water volume:

Between zero and two minutes, the average rate is

Between two and five minutes, it is

The different time gaps matter. The raw changes, $-8$ L and $-12$ L, are not equal, but the changes per minute are equal. These observations are consistent with the rule $V(t)=50-4t$. A small table alone does not establish that the rate stayed constant between observations. If that emptying rule continues, the tank reaches zero at $50/4=12.5$ min. The physical model stops there; extending it to negative water volumes would be meaningless.

In this model, $V(5)/5=6$ L/min is not the rate of change. That quotient divides the remaining volume by elapsed time. It ignores the starting 50 L.

Positive, negative and zero rates

A positive average rate means the final output exceeds the initial output when the input interval is read from smaller to larger. A negative average rate means the final output is smaller. Zero means the two endpoint outputs agree.

These statements concern the endpoints. A zero average rate does not prove the quantity stayed constant between them. For example, $f(x)=x^2$ has equal values at $x=-2$ and $x=2$, so its average rate on that interval is zero. Nevertheless, its value falls to zero and then rises again.

Similarly, a positive average rate can hide a temporary decrease. To establish that a function increases throughout an interval, more information than its endpoints is needed.

Secant slopes for a nonlinear function

For $f(x)=x^2$, the average rate from $x=1$ to $x=3$ is

From $x=3$ to $x=5$, it is

The intervals have equal width, but the second output change is twice as large. The curve does not have one constant slope. Each average rate is the slope of the straight line through the relevant pair of points.

A line’s displayed angle is not a substitute for calculating slope. Rescaling one axis changes the apparent steepness. Use the numbered coordinates and their units, especially when comparing different graphs. Gradient or Slope of a Line develops the coordinate calculation in detail.

Average velocity and average speed

Position and distance travelled are different quantities. Suppose an object moves along a straight path from position 0 m to position 6 m in two seconds, then returns to 0 m over the next three seconds.

Its net displacement is $0-0=0$ m, so its average velocity over the full five seconds is

The total distance travelled is $6+6=12$ m. Its average speed is therefore

Average velocity retains direction through the signed position change. Average speed uses total distance and is non-negative. Neither calculation requires the object to have moved at a constant rate during either leg.

Absolute change and relative change

Suppose a hypothetical index increases from 50 to 65 over three years. Its total change is 15 index points, and its average change is 5 index points per year.

Relative change uses the starting value as a reference:

That is a 30% increase over the full interval. Dividing 30% by three gives an average increase of 10% of the original value per year, not a compound annual growth rate. A repeated 10% increase would instead give

which is not 65. Keep additive rates and multiplicative growth models separate. See Percentage and Exponential Functions for those different comparisons.

From an interval to an instant

Average change uses two distinct inputs. To investigate the rate at one input $a$, examine progressively smaller non-zero intervals:

For $f(x)=x^2$ at $a=2$, this becomes

As $h$ approaches zero, the quotient approaches 4. This limiting value is the derivative at 2. It is not found by substituting $h=0$ into the original fraction, which would give an undefined quotient. Introduction to Differential Calculus develops the limiting process; this page’s principal tool remains the average rate over an interval.

Choosing the right calculation

Identify the changing quantity, its reference variable, both endpoint values and the interval. Subtract before dividing, and state units. Then ask whether the result describes net change, total activity, or change relative to a baseline.

A repeated unit price may support a proportional model. A constant slope supports a linear model but not necessarily direct variation: $V(t)=50-4t$ has a non-zero starting value. A varying average rate suggests that one linear rule may not represent the entire interval.

Related topics

Use Direct Variation to distinguish a constant ratio from a constant slope. Dimensional Analysis checks the units in rate calculations, and Differentiation studies rates at individual inputs.