Average Rate of Change Calculator
Enter a function or a table of data and two x-values and get the average rate of change between them — the slope of the secant line — with Δy and Δx, the secant line's equation, a plain-English sentence with your units (metres per second, dollars per year), a table showing how the rate approaches the instantaneous rate as the interval shrinks, and the Mean Value Theorem point. Free, no sign-up.
Use + − * / ^ and brackets. Functions: sin, cos, tan, exp, ln, log (base 10), sqrt, abs and more. Constants: pi and e.
A number or constant such as 1, −2.5 or pi/2.
Results
4
Average rate of change
8
Change in y, Δy
2
Change in x, Δx
1
y at the start
9
y at the end
What it means
From x = 1 to x = 3, y increases by 8 while x changes by 2: on average 4 per unit of x.
Secant line
The average rate of change is the slope of the secant line through (a, f(a)) and (b, f(b)).
Shrinking the interval
| h | Interval | Average rate |
|---|---|---|
| 1 | [1, 2] | 3 |
| 0.1 | [1, 1.1] | 2.1 |
| 0.01 | [1, 1.01] | 2.01 |
| 0.001 | [1, 1.001] | 2.001 |
| 0.0001 | [1, 1.0001] | 2.0001 |
The average rate over [a, a + h] approaches the instantaneous rate f′(1) = 2 as h → 0.
Mean Value Theorem
There is a point c between a and b where the instantaneous rate f′(c) equals this average rate: c ≈ 2.
Graph
- y = f(x)
- Secant line (slope 4)
- Tangent parallel to the secant at c
How to Use the Average Rate of Change Calculator
- 1. Choose function or table. For a function type f(x), such as
x^2orsin(x). For data, paste one “x, y” pair per line; the x-values can be in any order. - 2. Enter the interval. Give the start a and the end b. For a table, both must be x-values that appear in your data.
- 3. Add units if you like. Type the unit of y (metres) and of x (second) to get the answer as a sentence with “metres per second”.
- 4. Read the result. You get the rate, Δy and Δx, the secant line, and a graph. For a function you also see the limit table and the Mean Value Theorem point.
The Formula
The average rate of change of f over [a, b] is (f(b) − f(a)) / (b − a) — the change in output divided by the change in input, or rise over run. Geometrically it is the slope of the secant line through the points (a, f(a)) and (b, f(b)). It is negative when f falls overall, zero when f ends where it began, and it can be computed over any interval, even one where f rises and falls in between.
Worked Examples
A function. For f(x) = x² on [1, 3]: f(1) = 1 and f(3) = 9, so the rate is (9 − 1)/(3 − 1) = 4, and the secant line is y = 4x − 3. The instantaneous rate at x = 1 is f′(1) = 2, and the average rates over [1, 1 + h] are 3 (h = 1), 2.1 (h = 0.1) and 2.01 (h = 0.01): they close in on 2 as h shrinks. The Mean Value Theorem guarantees a point c in (1, 3) with f′(c) = 4; here f′(x) = 2x, so c = 2.
A table. A ball dropped from rest has fallen 0, 4.9, 19.6, 44.1 and 78.4 metres after 0, 1, 2, 3 and 4 seconds. Between t = 1 and t = 3 the distance rises from 4.9 to 44.1, so the average speed is (44.1 − 4.9)/(3 − 1) = 19.6 metres per second. The rates between consecutive seconds are 4.9, 14.7, 24.5 and 34.3 m/s, climbing steadily because the ball is accelerating.
Average Versus Instantaneous Rate
The average rate describes a whole interval; the instantaneous rate is the derivative, the limit of the average rate as the interval shrinks to a point. Where the average rates settle toward one number, that number is f′(a). If you only have a table of data you cannot take that limit, but the smaller the gap between measurements the better the average rate estimates the instantaneous one. Do not confuse the average rate of change with the average value of a function, (1/(b − a))∫ f(x)dx, which is the mean height of the graph rather than its mean slope.
Frequently Asked Questions
- How do you find the average rate of change?
- Subtract the function values at the two ends and divide by the difference of the inputs: (f(b) − f(a))/(b − a). For a table, read the y-values at the two x-values and do the same.
- What does a negative average rate of change mean?
- That the output decreased overall across the interval: the secant line slopes downward. It says nothing about what happened in between, since the function may have risen first and then fallen further.
- How is it different from the slope of a line?
- For a straight line the average rate of change is the same over every interval and equals the slope. For a curve it depends on the interval chosen, which is why the idea of instantaneous rate (the derivative) is needed.
- How do I include units?
- The units of the rate are the units of y divided by the units of x. If y is distance in metres and x is time in seconds, the rate is in metres per second. Type the two units into the calculator to get the sentence written out for you.
- What is the Mean Value Theorem in this context?
- If f is continuous on [a, b] and differentiable between, then at some point c in (a, b) the instantaneous rate f′(c) equals the average rate over [a, b]: the tangent at c is parallel to the secant. The calculator finds such points numerically and draws one tangent.
- Can I use it for non-evenly spaced data?
- Yes. The rate between any two x-values you pick is exact for your data, and the consecutive-rates table handles unequal gaps because each row divides by its own Δx.