Mean Value Theorem (MVT) Calculator
Find every c in (a, b) where the instantaneous rate equals the average rate of change.
Enter your values
Enter the function f(x) you are working with.
Enter the interval start a you are working with.
Enter the interval end b you are working with.
Results update instantly as you type — no submit needed. Your values are remembered on this device, and the shareable link reopens the calculator with exactly these numbers.
- Average rate of change
- 1
- f(a)
- -2
- f(b)
- 2
- c value(s) in (a, b)
- -1.154701, 1.154701
At each c the tangent line is parallel to the secant through (a, f(a)) and (b, f(b)).
Working
- 1Secant slope = (f(2) − f(-2)) / (2 − -2) = 1
- 2Solve f′(c) = secant slope for c inside the open interval.
- 3c = -1.154701, 1.154701
Quick answer
The mean value theorem says that for a function continuous on [a, b] and differentiable on (a, b) there is at least one c where f′(c) = (f(b) − f(a)) / (b − a). This calculator finds every such c.
f′(c) = [f(b) − f(a)] / (b − a)
At a glance
| What it does | Find every c in (a, b) where the instantaneous rate equals the average rate of change. |
|---|---|
| Category | Math |
| Inputs needed | Function f(x), Interval start a, Interval end b |
| Main output | Average rate of change |
| Formula | f′(c) = [f(b) − f(a)] / (b − a) |
| Cost | Free — no sign-up, no download |
How to use the Mean Value Theorem Calculator
- 1Enter a function that is continuous on the closed interval.
- 2Enter the endpoints a and b.
- 3Compare the secant slope with f′(c) at each returned c.
Inputs explained
- Function f(x)
- Enter the function f(x) you are working with.
- Interval start a
- Enter the interval start a you are working with.
- Interval end b
- Enter the interval end b you are working with.
Worked example
Using the values the calculator loads with:
Inputs
- Function f(x)x^3 - 3x
- Interval start a-2
- Interval end b2
Results
- Average rate of change1
- f(a)-2
- f(b)2
- c value(s) in (a, b)-1.154701, 1.154701
- Secant slope = (f(2) − f(-2)) / (2 − -2) = 1
- Solve f′(c) = secant slope for c inside the open interval.
- c = -1.154701, 1.154701
Frequently asked questions
What are the hypotheses of the mean value theorem?
f must be continuous on the closed interval [a, b] and differentiable on the open interval (a, b). If either fails, the conclusion can fail too.
Can there be more than one c?
Yes. The theorem guarantees at least one c, but many functions have several points where the tangent is parallel to the secant.
How is the MVT related to Rolle's theorem?
Rolle's theorem is the special case where f(a) = f(b), which makes the secant slope zero, so f′(c) = 0.
What do I need to enter into the Mean Value Theorem Calculator?
Just 3 values: function f(x), interval start a, interval end b. Every field starts with a realistic example, so you can change one number at a time and watch the result update instantly.
How does the Mean Value Theorem Calculator work out the answer?
The mean value theorem says that for a function continuous on [a, b] and differentiable on (a, b) there is at least one c where f′(c) = (f(b) − f(a)) / (b − a). This calculator finds every such c. It applies the formula f′(c) = [f(b) − f(a)] / (b − a) and shows the working so you can check each step by hand.
Results are estimates for general information. For medical, legal, structural or financial decisions, confirm with a qualified professional.