Quadratic Taylor Polynomial Multivariable Calculator

Calculator Inputs

Number of variables

Decimal precision

Third derivative bound M

Variable 1 label

Variable 2 label

Variable 3 label

Base coordinate a1

Base coordinate a2

Base coordinate a3

Target coordinate p1

Target coordinate p2

Target coordinate p3

Function value f(a)

Gradient component f1

Gradient component f2

Gradient component f3

Hessian h11

Hessian h22

Hessian h33

Mixed Hessian h12

Mixed Hessian h13

Mixed Hessian h23

Third variable entries are ignored when two variables are selected.

Formula Used

For a base point a and target point x, let h = x - a.

T2(x) = f(a) + ∇f(a) · h + 1/2 hᵀ H(a) h

If a valid third derivative bound M is entered, the calculator shows this estimate:

Remainder bound ≈ M ||h||³ / 6

How to Use This Calculator

Select two or three variables.
Enter the base point and the target point.
Add the function value, gradient, and Hessian entries at the base point.
Set decimal precision and an optional third derivative bound.
Press calculate, then review the result above the form.
Use the CSV or PDF buttons for downloads.

Example Data Table

Function	Base point	Target point	f(a)	Gradient at a	Hessian entries	Expected estimate
sin(x) + cos(y) + xy	(0, 0)	(0.2, -0.1)	1	(1, 0)	h11 = 0, h22 = -1, h12 = 1	1.175
x² + y² + z²	(1, 1, 1)	(1.1, 0.9, 1.2)	3	(2, 2, 2)	h11 = h22 = h33 = 2	3.26

About the Calculator

A quadratic Taylor polynomial gives a local model for a smooth function. The model uses one function value, first partial derivatives, and second partial derivatives. For many variables, the second order part comes from the Hessian matrix. This calculator keeps those parts visible. It is useful for calculus, optimization, physics modeling, economics, and engineering estimates.

Why It Matters

A multivariable function can curve in several directions at once. A linear approximation only follows the tangent plane. A quadratic approximation also follows local bending. That extra term often gives a better estimate near the chosen base point. It can also reveal whether the function looks bowl shaped, hill shaped, saddle shaped, or uncertain near that point.

What You Enter

Enter the function value at the base point. Then enter each coordinate of the base point and the target point. The calculator forms the displacement vector. Next enter the gradient values at the base point. Finally enter the Hessian entries. For a smooth function, mixed partial derivatives match, so one cross term is enough for each pair of variables. You may enter two or three variables.

What You Get

The tool returns the displacement vector, linear contribution, quadratic contribution, final polynomial estimate, and curvature summary. It also builds a readable polynomial form. When you add a third derivative bound, it estimates a possible remainder size with a cubic norm bound. This does not prove exact error unless the bound is valid on the whole local region.

Best Practice

Choose a base point close to the target point. Smaller steps usually improve the approximation. Use exact derivative values when possible. Round only the final result when accuracy matters. Check units before mixing inputs from applied problems. Export the result when you need records for homework, reports, or repeated comparisons.

Advanced Use

The Hessian test gives helpful clues, but it is local. It depends on the entered second derivatives. For two variables, the determinant separates several common cases. For three variables, leading minors give a quick definiteness check. Use the message as guidance, not as a full proof. If the Hessian changes quickly, compare several nearby points and use the remainder option. This habit makes numerical work easier to audit later.

FAQs

What is a quadratic Taylor polynomial?

It is a second order local approximation. It uses the function value, gradient, and Hessian at a chosen base point.

Can this calculator use three variables?

Yes. Choose three variables, then fill the third coordinate, third gradient component, and related Hessian entries.

What is the Hessian matrix?

The Hessian is the matrix of second partial derivatives. It controls the quadratic bending part of the approximation.

Why are mixed Hessian entries entered once?

For smooth functions, mixed partial derivatives are equal. The calculator assumes a symmetric Hessian and reuses each mixed entry.

What does the remainder bound mean?

It is an estimated maximum error based on a third derivative bound. It is useful only when that bound is valid nearby.

Does this replace symbolic differentiation?

No. You still need correct derivative values. This tool evaluates the Taylor model after those values are entered.

When should I use a nearby base point?

Use a base point close to the target. Taylor approximations usually improve when the displacement is small.

Can I export the answer?

Yes. After calculating, use the CSV or PDF buttons shown in the result section above the form.