Calculator Input
Use polynomial coefficients only. This design keeps continuity and differentiability assumptions valid for the theorem.
Formula Used
Mean Value Theorem: If a function is continuous on [a,b] and differentiable on (a,b), then at least one interior point c satisfies:
f'(c) = ( f(b) - f(a) ) / ( b - a )
This calculator first computes the average rate of change across the interval. It then builds the equation f'(x) - average slope = 0 and searches numerically for all valid interior solutions.
Because the input is polynomial, continuity and differentiability are automatically satisfied on every real interval.
How to Use This Calculator
- Choose the polynomial degree from 0 through 5.
- Enter the coefficients for the visible terms.
- Provide the interval values a and b with a < b.
- Adjust scan samples, decimal precision, and tolerance if needed.
- Click the calculate button to show the result above the form.
- Review the average slope, derivative, secant equation, and interior theorem points.
- Use the chart to compare the curve against the secant line visually.
- Download the summary as CSV or PDF when required.
Example Data Table
| Example | Function | Interval | Average Slope | Example c |
|---|---|---|---|---|
| 1 | x^3 - 3x + 1 | [0, 2] | 1 | 1.1547 |
| 2 | x^2 + 4x + 2 | [1, 5] | 10 | 3 |
| 3 | 2x^4 - x^2 + 3 | [-1, 2] | 10 | 1.1358 |
Frequently Asked Questions
1) What does this calculator actually find?
It computes the interval’s average rate of change and then finds interior point values c where the instantaneous rate, f′(c), matches that average slope.
2) Why does the tool use polynomial input?
Polynomial input keeps the theorem assumptions reliable and makes derivative evaluation fast. It also avoids symbolic parsing errors that often appear with arbitrary text expressions.
3) Can there be more than one valid c value?
Yes. Some functions meet the theorem at multiple interior points. The calculator scans the interval and lists every distinct numerical solution it detects.
4) What happens for linear or constant functions?
For a linear or constant polynomial, the derivative equals the interval’s average slope everywhere. In that situation, every interior point c satisfies the theorem.
5) Why might numerical roots depend on tolerance?
Root detection uses scanning and refinement. Tighter tolerance improves accuracy, while more samples improve the chance of finding closely spaced solutions inside the interval.
6) Does the graph prove the theorem visually?
It illustrates the idea clearly. The secant line represents the average slope, while highlighted curve points show where the tangent slope matches that same rate.
7) When is the Mean Value Theorem applicable?
The function must be continuous on the closed interval and differentiable on the open interval. Polynomials satisfy both conditions for all real numbers.
8) Can I export the computed summary?
Yes. The page includes CSV export for spreadsheets and PDF export for reports, making it easy to document theorem steps and results.