Evaluate partial sums, mixed derivatives, and nearby behavior. Compare polynomial estimates against actual function values. Build confidence in multivariable series convergence with practical diagnostics.
| Function | Center (a, b) | Point (x, y) | Order | Tolerance | Expected Use |
|---|---|---|---|---|---|
| exp(x + y) | (0, 0) | (0.2, 0.1) | 4 | 0.0001 | Test rapid local convergence near the origin. |
| ln(1 + x + y) | (0, 0) | (0.2, 0.1) | 5 | 0.00001 | Check convergence before approaching the boundary. |
| 1 / (1 - x - y) | (0.1, 0.1) | (0.2, 0.15) | 4 | 0.0001 | Observe how singular lines affect convergence speed. |
Multivariable Taylor polynomial:
Tn(x, y) = Σi + j ≤ n [∂i+jf(a, b) / (i!j!)] (x - a)i(y - b)j
Absolute error: |f(x, y) - Tn(x, y)|
Relative error: |f(x, y) - Tn(x, y)| / |f(x, y)|
Distance from center: ρ = √[(x - a)2 + (y - b)2]
Derivative estimation: Central finite differences are used for mixed partials.
This page checks practical numerical convergence by comparing successive polynomial orders against the exact function value at the chosen point.
Multivariable Taylor series helps describe local behavior of smooth functions. The method expands a function near a chosen center. It uses partial derivatives and mixed derivatives. The resulting polynomial often gives fast local estimates. Convergence matters because not every point behaves equally well.
A multivariable Taylor approximation is built from values at the center. Each higher order term adds local curvature information. Linear terms track slope. Quadratic terms track bending. Cubic and quartic terms capture richer interaction. Mixed terms show how variables influence each other together.
Convergence usually improves when the evaluation point stays close to the center. Distance matters. Nearby points often produce smaller remainder terms. Higher order polynomials also reduce local error for analytic functions. The key test compares successive approximations with the exact value. Stable error reduction suggests practical convergence.
This calculator estimates multivariable Taylor polynomials up to a chosen order. It evaluates several common functions of two variables. It computes order by order approximations. It also reports absolute error, relative error, and a convergence verdict. A remainder estimate is taken from the final difference between the exact function and the polynomial value.
The tool also checks a simple radius style diagnostic for functions with nearby singular sets. For logarithmic, square root, and rational forms, singular lines limit safe expansion zones. If the point lies near that boundary, convergence may slow or fail. Entire functions such as exponential or trigonometric combinations usually allow broader local expansion.
Use the calculator to study approximation quality, compare centers, and choose suitable orders. Smaller finite difference steps can sharpen derivative estimates, but very tiny steps may introduce roundoff noise. Moderate orders often balance speed and stability. Always inspect the error table before trusting the approximation. Good convergence is numerical evidence, not a blind guarantee.
The generated convergence plot makes the trend easier to read. Falling error bars across orders indicate improving local agreement. Flat or rising error warns that the center, order, or function domain needs attention. This practical workflow supports calculus study, numerical analysis practice, engineering modeling, and scientific computing where local series behavior guides approximation decisions and error control. It also helps explain mixed derivative sensitivity in two-variable surfaces.
It measures how well a multivariable Taylor polynomial matches the selected function at a chosen point. It compares several orders and reports approximation error, relative error, and a simple convergence verdict.
Convergence can fail when the evaluation point is too far from the center, when the function is near a singular boundary, or when numerical derivative estimation becomes unstable.
Mixed derivatives capture interaction between variables. Without them, the polynomial misses coupling behavior. For two variable functions, these terms are essential for realistic local approximation.
The tolerance gives a target error level. If the final absolute error falls below that value, the page labels the approximation as convergent within tolerance.
The step controls numerical derivative estimation. A very large step reduces precision. A very small step can amplify roundoff error. Moderate values often work best.
Not always. Higher order terms usually help near the center, but numerical noise or proximity to singularities can weaken improvement. The convergence table shows the actual behavior.
For selected functions, the page estimates the distance from the center to a nearby singular line or boundary. This gives a simple local warning about possible convergence limits.
Yes. It is useful for calculus exercises, numerical analysis practice, and local approximation experiments. It also helps compare centers, orders, and function behavior quickly.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.