Average Rate of Change Calculator
Connect two points. Measure the change. See how a secant becomes a tangent.
Function & Interval Inputs
Compare how much the output changes with how much the input changes. The two x-values must differ.
Rate & Secant Summary
Function modeOutput units per input unit · over [1, 4]
Your rise-over-run triangle
Positive slopeThe dashed grey line is the tangent at A: y = 2x + 1.
Bring the points closer. Discover the derivative.
Explore · function modeKeep A fixed at a = 1. Reduce h = b − a to bring B toward A.
| h | b = 1 + h | Secant slope |
|---|---|---|
| 3 | 4 | 5 |
| 1 | 2 | 3 |
| 0.1 | 1.1 | 2.1 |
| 0.01 | 1.01 | 2.01 |
For this function: m = 2a + h → 2a = f′(a). At h = 0 the difference quotient is undefined; its limit is 2.
Formula & Worked Example
Find the output change:
Δy = 18 − 3 = 15
Find the input change:
Δx = 4 − 1 = 3
Divide rise by run:
m = 15 / 3 = 5
y − 3 = 5(x − 1), so y = 5x − 2. Both A and B lie on this line.
Frequently Asked Questions
What if both x-values are the same?
The average rate is undefined because the input change is zero. This remains true if both output values are also equal.
Does a negative rate mean the function always decreases?
No. A negative endpoint rate means a net decrease across the interval; the function can still increase on parts of that interval.
Can I reverse the two endpoints?
Yes. Reversing both points reverses both differences, leaving their quotient unchanged.
Is average rate the same as the derivative?
No. Average rate is a secant slope across an interval. The derivative is a limit at a single point, when that limit exists.
Can two points determine a unique curve?
No. Infinitely many curves can pass through the same two points. Point mode draws only their secant, not an assumed underlying function.
What units does the result have?
Output units divided by input units. If the output is position and the input is time, this is average velocity, not average speed.
Sources & References
Independent educational references · No institutional endorsement or expert review claimed.
Connect two points. Measure the change. See how a secant becomes a tangent.