Taylor Polynomial Calculator
Build the order-n Taylor polynomial of a function about a center c, evaluate it at a point x, and measure how close it is to the exact value.
Function f(x)
Approximation of f(x) from the truncated Taylor series
How does this calculator work?
A Taylor polynomial approximates a function near a center c using Pₙ(x) = Σ f⁽ᵏ⁾(c)(x − c)ᵏ⁄k!. Choose the function, center, evaluation point, and order; the tool returns the approximation, the exact value, and the absolute error, plus a plot comparing both curves so you can see how the fit degrades away from c.
Formula
How this is calculated
The Taylor polynomial approximates a smooth function near a chosen center c using its derivatives at that point. Each term adds f⁽ᵏ⁾(c)·(x − c)ᵏ⁄k!, so the constant term reproduces f(c), the linear term matches the slope, the quadratic term matches curvature, and so on up to order n. When c = 0 this is the Maclaurin series.
You pick a function, the center c, the evaluation point x, and the order n. The derivatives f⁽ᵏ⁾(c) are computed numerically with k-th central finite differences, then divided by k! and multiplied by powers of (x − c) and summed. The calculator reports the approximation Pₙ(x), the exact value f(x), and the absolute error |f(x) − Pₙ(x)|.
Accuracy improves as n grows and as x moves closer to c, since the remainder behaves like (x − c)ⁿ⁺¹. For ln(1 + x) the domain requires x > −1, and 1/(1 − x) has a singularity at x = 1, whose Taylor series only converges for |x − c| within its radius of convergence. Because derivatives are estimated by finite differences, very high orders may accumulate small rounding noise.
Frequently asked questions
A Maclaurin series is simply a Taylor series centered at c = 0. Set the center to 0 here to reproduce it.
The Taylor remainder scales with (x − c)ⁿ⁺¹, so points far from the center need a higher order n to stay accurate, and some functions only converge within a finite radius.
Each f⁽ᵏ⁾(c) is approximated with k-th order central finite differences, then combined with the factorial and power terms to form the polynomial.
Also known as
TG we-Calculate Editorial Team. (2026). Taylor Polynomial Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/taylor-polynomial-calculator
TG we-Calculate Editorial Team. "Taylor Polynomial Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/taylor-polynomial-calculator.
TG we-Calculate Editorial Team, "Taylor Polynomial Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/taylor-polynomial-calculator
@misc{wecalculate_taylor_polynomial_calculator, title = {Taylor Polynomial Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/taylor-polynomial-calculator}}, year = {2026}, note = {TG we-Calculate} }
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