Calculator Inputs
Example Data Table
| Mode | Base Point | Target Point | f(base) | Gradient | Key Hessian Values | Approximate Result |
|---|---|---|---|---|---|---|
| 2 variables | (1, 2) | (1.15, 1.8) | 5.2 | (1.1, -0.4) | fxx=2, fyy=1.5, fxy=-0.3 | 5.49125 |
| 3 variables | (1, 2, 1) | (1.1, 2.2, 0.9) | 4.8 | (0.5, 1.2, -0.7) | fxx=1.8, fyy=2.1, fzz=1.2 | Approximation depends on mixed terms |
| Optimization check | (0, 0) | (0.2, -0.1) | 10 | (0, 0) | fxx=4, fyy=3, fxy=1 | Curvature driven value |
Formula Used
For a function of several variables, the second order Taylor approximation near point a is:
Here, ∇f(a) is the gradient vector. H(a) is the Hessian matrix. The calculator expands this expression into linear, square, and mixed-product terms.
How to Use This Calculator
- Select whether the problem has two or three variables.
- Enter the function value at the expansion point.
- Add the base point and target point coordinates.
- Enter first partial derivative values for the gradient.
- Enter second partial derivative values for the Hessian.
- Add a true value if you want error analysis.
- Press the calculate button and review the result above the form.
- Use CSV or PDF download for reports and records.
Understanding Quadratic Taylor Polynomial Approximation
Local Second Order Model
A quadratic Taylor polynomial is a local second order model. It describes how a multivariable function changes near a selected base point. The model uses the function value, first partial derivatives, and second partial derivatives. This calculator turns that information into a clear approximation.
Role of the Hessian
For two variables, the polynomial has constant, linear, pure square, and mixed product parts. For three variables, the same idea extends with z terms. The Hessian controls curvature. Its diagonal entries measure bending along each axis. Its off diagonal entries measure interaction between variables.
Where It Helps
This tool is useful when the exact function is difficult to evaluate repeatedly. It is also helpful when you already know derivative data from calculus, optimization, economics, physics, or engineering. You can test a nearby point and compare the estimated value with an optional true value.
Result Breakdown
The result panel explains every main component. It shows coordinate changes from the base point. It reports the linear contribution, the quadratic contribution, and the final second order estimate. When a true value is supplied, it also displays absolute error, relative error, and percentage error.
Two and Three Variable Support
The calculator accepts two or three variables. For a two variable problem, z inputs are ignored. For a three variable problem, all gradient and Hessian entries can be used. Mixed partial values should be entered once, because the Hessian is treated as symmetric.
Visual Interpretation
The graph gives a visual check. In two variable mode, it draws the quadratic surface around the expansion point. In three variable mode, it plots the approximation along the straight path from the base point to the target point. This keeps the display readable.
Accuracy Guidance
Good results require nearby target values. Taylor models are local approximations. Accuracy usually decreases as the target moves farther from the base point. Large curvature also increases error. Use the distance value to judge how far the target is from the expansion point.
Export Options
This page also supports export. Download the result table as a CSV file. Create a printable PDF summary for reports, lessons, or homework records. Use the example table to understand typical inputs before entering your own derivative data. These exports make review easier for students, teachers, analysts, and developers during final checks.
FAQs
1. What is a quadratic Taylor polynomial?
It is a second order approximation of a function near a selected point. It uses the function value, gradient, and Hessian.
2. Can this calculator handle three variables?
Yes. Select three variables and enter z coordinates, z gradient data, and z related Hessian values.
3. What is the Hessian matrix?
The Hessian is a square matrix of second partial derivatives. It measures curvature and variable interaction near the base point.
4. Why are mixed partial values needed?
Mixed partial values measure how two variables interact. They affect terms like ΔxΔy, ΔxΔz, and ΔyΔz.
5. Is the approximation always accurate?
No. It is most accurate near the expansion point. Error can grow when the target point is farther away.
6. What should I enter for true value?
Enter the exact function value at the target point if known. The calculator will then show error values.
7. Does the graph show the original function?
No. The graph shows the quadratic Taylor model created from your derivative data, not the original function.
8. Can I export the results?
Yes. Use the CSV button for spreadsheet data. Use the PDF button for a printable summary.