Mean Value Theorem Calculator

Q: What does the Mean Value Theorem say?

If f is continuous on [a, b] and differentiable on (a, b), then there is at least one c in (a, b) where f'(c) = \frac{f(b) - f(a)}{b - a}. Somewhere, the instantaneous rate equals the average rate.

Q: What if the conditions fail?

Then there may be no such c. f(x) = |x| on [-1, 1] has average slope 0 but its slope is never 0, because f is not differentiable at 0. The calculator warns when it detects a break.

Q: What is Rolle’s theorem?

The special case f(a) = f(b): then the average slope is 0 and there is a c with f'(c) = 0, a horizontal tangent.

Find the point c where the tangent runs parallel to the secant, meaning the instantaneous rate equals the average rate. Conditions checked, steps shown, both lines drawn.

Function f(x)

Reads as

x^{3} - x

Try:

Answer

c = \frac{2\sqrt{3}}{3}

average slope $3$

Step by step

Check the conditions
f is continuous on the closed interval and differentiable inside it (no breaks found), so the theorem applies.
Average rate of change
Slope of the secant line.
$\frac{f(2) - f(0)}{2 - 0} = \frac{6 - 0}{2} = 3$
Differentiate
$f'(x) = 3 x^{2} - 1$
Solve f′(c) = average rate
$3 x^{2} - 1 = 3 \;\Rightarrow\; c = \frac{2\sqrt{3}}{3}$

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What the theorem guarantees

The amber segment joins the endpoints; its slope is the average rate of change. Slide a line with that slope across the graph and it must touch the curve somewhere in between: that touching point is $c$ . The theorem only promises that one exists; the calculator actually finds it by solving $f'(c) = \frac{f(b)-f(a)}{b-a}$ .

The integral version says the same about heights instead of slopes: a continuous function hits its average value somewhere on the interval.

Questions students ask

What does the Mean Value Theorem say?

If $f$ is continuous on $[a, b]$ and differentiable on $(a, b)$ , then there is at least one $c$ in $(a, b)$ where $f'(c) = \frac{f(b) - f(a)}{b - a}$ . Somewhere, the instantaneous rate equals the average rate.

What does it mean in plain words?

If you drive 120 km in 1.5 hours, your average speed is 80 km/h, so at some moment your speedometer read exactly 80 km/h. That moment is $c$ .

What if the conditions fail?

Then there may be no such $c$ . $f(x) = |x|$ on $[-1, 1]$ has average slope 0 but its slope is never 0, because $f$ is not differentiable at 0. The calculator warns when it detects a break.

What is Rolle’s theorem?

The special case $f(a) = f(b)$ : then the average slope is 0 and there is a $c$ with $f'(c) = 0$ , a horizontal tangent.