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Taylor Series Calculator

Expand any function as a Taylor polynomial up to order 4 about any center point.

Taylor polynomial (order 4) about a
1 + -0.5(x − a)^2 + 0.0417(x − a)^4
Coefficients cₖ = f⁽ᵏ⁾(a) ÷ k! computed via numerical derivatives — values are approximate and capped at order 4. Variable: x.

How it works

  1. 1Enter a function f(x) using standard notation — e.g. cos(x), exp(x), or x^3 + 2*x.
  2. 2Set the center a (the point of expansion) and the order (1–4).
  3. 3Each coefficient cₖ = f⁽ᵏ⁾(a) ÷ k! is computed from finite-difference derivatives and assembled into the polynomial in powers of (x − a).

Use cases

  • Approximating transcendental functions (sin, cos, eˣ) near a point for hand calculations.
  • Comparing low- and high-order expansions to build intuition about convergence.
  • Seeing how derivatives govern the local shape of a function in calculus courses.

Frequently asked questions

What is a Taylor series?

It expresses a smooth function as a sum of terms from its derivatives at a point a: f(x) = Σ f⁽ᵏ⁾(a) ÷ k! · (x − a)ᵏ. This tool returns the polynomial truncated at your chosen order.

Why are the coefficients approximate?

Derivatives are estimated with finite-difference stencils rather than symbolic differentiation, so coefficients are accurate to a few decimals for well-behaved functions and capped at order 4.

What is a Maclaurin series?

A Maclaurin series is a Taylor series centered at a = 0. Set the center to 0 to obtain it.

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