Advanced Multivariable Taylor Series Calculator

Formula used

For a two-variable function, the Taylor polynomial of order n around (a, b) is:

T_n(x, y) = Σ_i+j≤n [ ∂^i+jf(a,b) / (∂xⁱ∂y^j) ] · (x-a)ⁱ(y-b)^j / (i!j!)

The calculator evaluates the function numerically, estimates partial derivatives with central differences, then builds the polynomial coefficient by coefficient.

Accuracy improves when the target point stays near the expansion point and when the function is smooth in the selected region.

How to use this calculator

Enter a two-variable function using x and y.
Choose the Taylor order from 1 through 4.
Enter the expansion point (a, b).
Enter the target point where you want the approximation.
Set a small derivative step size, like 0.001.
Choose plot span and grid size for the graph.
Click Calculate Taylor Series.
Review the polynomial, coefficients, errors, and surface plot.

Function	Order	Expansion Point (a, b)	Target (x, y)	Taylor Approximation	Exact Value
exp(x+y)	2	(0, 0)	(0.20, 0.10)	1.345	1.349858808
log(1+x+y)	3	(0, 0)	(0.10, 0.15)	0.223958333	0.223143551
1/(1-x-y)	3	(0, 0)	(0.20, 0.10)	1.417	1.428571429

FAQs

1. What does this calculator compute?

It builds a two-variable Taylor polynomial around a chosen point. It also evaluates the approximation at a target point, compares it with the original function, and graphs both surfaces.

2. Which functions can I enter?

You can use x, y, numbers, parentheses, and functions like sin, cos, tan, exp, log, ln, sqrt, abs, sinh, cosh, tanh, asin, acos, atan, and log10.

3. Why must I use explicit multiplication?

The parser expects clear operators. Write x*y instead of xy, and write 2*x instead of 2x. This avoids ambiguity and reduces evaluation errors.

4. What does the derivative step size do?

The step size controls numerical differentiation. Smaller values can improve local accuracy, but extremely small values may amplify rounding noise. A value near 0.001 is a practical starting point.

5. Why does the approximation become worse far away?

Taylor polynomials are local models. They match a function well near the expansion point, but accuracy often declines as the target point moves farther away or nears singularities.

6. What order should I choose?

Start with order 2 or 3. Higher order usually captures more curvature, but it can also magnify numerical derivative noise. Compare the reported errors before trusting the result.

7. What if my function returns N/A?

N/A usually means the function could not be evaluated at the chosen point or inside the plotted region. Check domains for logarithms, square roots, and divisions.

8. What does the Plotly graph show?

The graph overlays the original surface and the Taylor approximation over a selected window. It helps you see where the approximation matches well and where the local model starts drifting.