Taylor Polynomial Simulator

Compare a function with its Maclaurin polynomial approximation

Function

Order

Taylor Polynomial Formula

Taylor polynomial	P_n(x) = sum f^(k)(0)x^k / k!
Center	0 for Maclaurin series
Order	highest power kept in the polynomial

How to Read the Simulator

The blue curve is the true function.
The red curve is the Taylor polynomial approximation.
Higher order usually improves accuracy near x = 0.
The approximation may become poor far from the center.

Example: sin(x) is approximated by x - x^3/3! + x^5/5! - ... . Increasing the order adds more terms to the red curve.

Frequently Asked Questions

What is a Taylor polynomial?▼

A Taylor polynomial approximates a function near a chosen center using derivatives of the function at that center.

What is a Maclaurin polynomial?▼

A Maclaurin polynomial is a Taylor polynomial centered at x = 0.

Why does a higher order improve the approximation?▼

Higher order terms capture more curvature and local behavior of the original function.

Where are Taylor polynomials useful?▼

They are used for approximation, numerical methods, calculus, physics, and engineering models.

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