Multivariable Taylor Series Calculator

Calculator input

Preset function

Function f(x,y)

Use x and y as variables. Use explicit multiplication like x*y. Supported functions include sin, cos, tan, exp, log, sqrt, abs, and pow.

Expansion point a

Expansion point b

Taylor order

Evaluate at x

Evaluate at y

Derivative step size h

Graph x-min

Graph x-max

Graph y-min

Graph y-max

Graph grid points

Formula used

For a function of two variables expanded around the point (a,b), the calculator evaluates the truncated Taylor series up to order N:

T_N(x,y) = Σ_n=0..N Σ_i=0..n [ ∂ⁿf(a,b) / (∂xⁱ ∂y^n-i) ] · (x-a)ⁱ(y-b)^n-i / [ i!(n-i)! ]

Exact error at the chosen point is computed as:

Error = f(x,y) - T_N(x,y)

Numerical partial derivatives are estimated using recursive central differences. Smaller step sizes may improve local precision, but extremely small values can amplify floating-point noise.

How to use this calculator

Choose a preset function or enter your own function in terms of x and y.
Set the expansion point (a,b). This is where all derivatives are measured.
Select the Taylor order. Higher orders usually improve local accuracy.
Enter the target point where you want the approximation evaluated.
Adjust the derivative step size h if your function is sensitive.
Set graph ranges and grid density to compare the exact and approximate surfaces.
Press the calculate button to see the polynomial, values, errors, derivatives, and plot.
Use the export buttons to save a CSV summary or a PDF report.

Example data table

Function	Center (a,b)	Order	Evaluation point	Exact value	Taylor value	Absolute error
exp(x + y)	(0, 0)	3	(0.20, 0.10)	1.349859	1.349500	0.000359
sin(x) * cos(y)	(0, 0)	3	(0.30, 0.20)	0.289629	0.289500	0.000129
log(1 + x + y)	(0, 0)	3	(0.15, 0.10)	0.223144	0.223958	0.000814

Frequently asked questions

1) What is a multivariable Taylor series?

It is a polynomial approximation of a function near a chosen point. The approximation uses partial derivatives, including mixed derivatives, to match local behavior in both variables.

2) Why does the expansion point matter?

Accuracy is highest near the expansion point because the polynomial is built from derivatives measured there. Moving far away can increase error quickly, especially for low-order approximations.

3) What do mixed partial derivatives represent?

Mixed partial derivatives describe how changes in one variable interact with changes in the other. They are essential when the surface bends jointly in x and y rather than independently.

4) Should I always choose the highest order?

Not always. Higher orders may improve local accuracy, but they also require more derivative calculations and can become numerically noisy for difficult expressions or poor step sizes.

5) What does the derivative step size h control?

It controls the spacing used by central differences. A moderate value often works best. Too large reduces accuracy, while too small may magnify rounding errors.

6) Why might the exact function value fail to compute?

Some expressions are undefined on parts of the graph, such as log of a nonpositive number or division by zero. Choose ranges and points inside the valid domain.

7) Can I enter my own function?

Yes. Use x and y, standard operators, and supported functions like sin, cos, exp, log, sqrt, abs, and pow. Write multiplication explicitly, such as x*y.

8) How does the graph help?

The graph lets you compare the original surface with the Taylor surface over a selected region. Matching shapes indicate a strong local approximation near the expansion point.