Mean Value Theorem Calculator

Work out mean value theorem instantly with clear inputs, formula shown and shareable results.

About Mean Value Theorem Calculator

The mean value theorem guarantees that a function continuous on [a, b] and differentiable inside has some interior point where the instantaneous rate equals the average rate over the interval.

Formula

Mean value theorem

f'(c) = (f(b) − f(a)) / (b − a) for some c in (a, b)

Worked example

With the default values below, Mean Value Theorem Calculator returns Point c where f'(c) equals the secant slope of 2.64575131. Change any input above and every figure here is recalculated in your browser.

Inputs used

Polynomial coefficients, highest power first: 1, 0, 0, 0
Interval start a: 1
Interval end b: 4

Results

Point c where f'(c) equals the secant slope: 2.64575131
Secant slope (f(b) − f(a))/(b − a): 21.00000000
f'(c) — must match the secant: 21.00000000
f(c): 18.52025918

Step by step

1Values used
2Mean value theorem
3Point c where f'(c) equals the secant slope= 2.64575131
4Secant slope (f(b) − f(a))/(b − a)= 21.00000000
5f'(c) — must match the secant= 21.00000000
6f(c)= 18.52025918

Assumptions and limits

The result is derived only from the 3 inputs listed above. Anything not among them is not part of the calculation.
In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
Figures are rounded for display to 8 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

For f(x) = x³ on [1, 4]?

The secant slope is (64 − 1)/3 = 21, and 3c² = 21 gives c = √7 ≈ 2.6457513.

What does it mean physically?

If you average 60 mph over a journey, at some instant your speedometer must have read exactly 60 mph.