Calculator inputs
Use one responsive form area. The fields arrange into three columns on large screens, two on medium screens, and one on mobile.
Formula used
Function mode: The calculator uses the central difference formula, f''(x) ≈ [f(x+h) - 2f(x) + f(x-h)] / h², which estimates curvature from nearby values.
Polynomial mode: Each term is differentiated twice. For axⁿ, the second derivative becomes n(n-1)axⁿ⁻².
Dataset mode: With equally spaced points, y''(xᵢ) ≈ [yᵢ₊₁ - 2yᵢ + yᵢ₋₁] / h², giving curvature directly from tabulated values.
How to use this calculator
- Choose a mode: function, polynomial, or dataset.
- Enter the expression, coefficients, or x,y values.
- Set the evaluation point and decimal precision.
- Provide a small step size for function mode.
- Pick a middle row for dataset mode.
- Press the calculate button to show results below the header.
- Review curvature, method, formula, and sample table.
- Export the result as CSV or PDF when needed.
Example data table
Example function: f(x) = x³ - 4x² + 2x + 1 at x = 2, using h = 0.01.
| Point | x value | Function value | Meaning |
|---|---|---|---|
| x - h | 1.99 | 0.999499 | Left sample for curvature estimate |
| x | 2.00 | 1.000000 | Center evaluation point |
| x + h | 2.01 | 1.000899 | Right sample for curvature estimate |
| Estimated f''(2) | — | ≈ 4.000000 | Matches the exact second derivative |
Frequently asked questions
1. What does the second derivative measure?
It measures how the first derivative changes. In graphs, it describes curvature and helps identify concavity, bending, and possible inflection behavior near a point.
2. Why are there three input modes?
Different courses and tasks use different inputs. This page handles explicit functions, direct polynomial coefficients, and equally spaced tables from experiments or worksheets.
3. When should I use a smaller step size?
Use a smaller h when the function is smooth and well-behaved near the point. Very tiny values can increase rounding error, so test a few sizes.
4. Does polynomial mode give an exact answer?
Yes. Polynomial mode differentiates coefficients directly, so the second derivative is exact apart from display rounding in the final printed value.
5. Why must dataset points be equally spaced?
The three-point finite difference formula used here assumes constant spacing around the selected point. Unequal spacing needs a different approximation formula.
6. Can this help identify concavity?
Yes. Positive results suggest concave up behavior, while negative results suggest concave down behavior at the chosen location.
7. What functions can I type?
You can enter x, numbers, parentheses, powers, and common functions like sin, cos, tan, log, exp, sqrt, and abs.
8. What do the export buttons save?
CSV saves the visible result table for spreadsheets. PDF creates a clean result summary with method details, values, and interpretation.