Mean Value Theorem Calculator
Enter a function and an interval [a, b] to find the point c guaranteed by the mean value theorem.
The mean value theorem says that for a function continuous on [a, b] and differentiable on (a, b), there is at least one point c where the instantaneous slope fโฒ(c) equals the average slope (f(b) โ f(a))/(b โ a). This calculator computes that average slope and solves fโฒ(c) equal to it.
How to Use This Calculator
Enter your function of x and the interval endpoints a and b. The calculator finds the average rate of change across the interval, then solves for the point c inside the interval where the instantaneous rate of change โ the derivative โ equals that average.
The theorem guarantees at least one such c exists when the function is continuous on the closed interval and differentiable on the open interval.
How the Calculation Works
First the calculator computes the average slope โ the slope of the straight line (secant) joining the endpoints. Then it differentiates the function and solves the equation fโฒ(x) equals that average slope.
Any solutions that fall inside the open interval (a, b) are the values of c the theorem promises. Geometrically, c is where the tangent line is parallel to the secant line across the interval.
average slope = (f(b) โ f(a)) รท (b โ a); solve fโฒ(c) = average slope
Worked Example
For f(x) = xยฒ on [0, 2], the average slope is (f(2) โ f(0))/(2 โ 0) = (4 โ 0)/2 = 2. The derivative is fโฒ(x) = 2x, so solve 2x = 2.
That gives x = 1, which lies in (0, 2). So c = 1 is the point where the tangent slope equals the average slope of 2.
Assumptions and Limitations
- Requires the function to be continuous on [a, b] and differentiable on (a, b)
- There may be more than one valid c; all in (a, b) are reported
- Uses x as the variable and ^ for powers
- Only values strictly inside the interval count
Who Uses a Mean Value Theorem Calculator
Calculus students use this to find the point c where a function's tangent slope equals its average slope over an interval. Run it for homework and to build intuition for the theorem. Enter the function and interval to get the value of c.
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Open calculator โFrequently Asked Questions
What is the mean value theorem?
It states that for a function continuous on [a, b] and differentiable on (a, b), there is at least one point c where the derivative equals the average rate of change across the interval.
How do you find c in the mean value theorem?
Compute the average slope (f(b) โ f(a))/(b โ a), then solve fโฒ(c) equal to it. Any solution inside the interval (a, b) is a valid c.
What does c represent geometrically?
It is the x-value where the tangent line to the curve is parallel to the secant line joining the two endpoints of the interval.
Can there be more than one c?
Yes. Depending on the function, several points in the interval can have a tangent slope equal to the average slope. The theorem guarantees at least one.
What are the conditions for the theorem?
The function must be continuous on the closed interval [a, b] and differentiable on the open interval (a, b). If either fails, the theorem may not apply.
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