Taylor Approximation Calculator – Estimate f(x)

A Taylor polynomial approximates a function near a center using its derivatives, trading exactness for a finite sum you can evaluate by hand. Approximating sin x near 0: the second-order polynomial x − x³/6 at x = 0.3 gives 0.3 − 0.0045 = 0.2955, versus the true sin 0.3 = 0.29552 — an error under 2.1×10⁻⁵, comfortably bounded by the first omitted term 0.3⁵/120 ≈ 2.0×10⁻⁵ (the series alternates, so the bound holds). For eˣ near 0 with four terms, e¹ ≈ 1+1+0.5+0.1667 = 2.6667 vs 2.71828. The calculator selects the center (0 is the default Maclaurin case), the order n, and prints every term with its contribution, the partial sum, the true value when known, and the error estimate from the next term. Higher orders tighten fast for analytic functions but worsen near singularities — ln(1+x) at x = 0.9 warns of the convergence edge.

1. 1

   Choose function, center, order

   e.g. sin, center 0, order 3

2. 2

   List the terms

   Each (x−a)ᵏ/k! contribution is printed

3. 3

   Sum the polynomial

   0.2955 for sin at x = 0.3, order 3

4. 4

   Bound the error

   Next-term estimate vs the true 0.29552

• 1

  Alternating series: error is at most the first dropped term

• 2

  Maclaurin (center 0) is the standard quick tool

• 3

  Terms shrink factorially — order 5 often suffices near the center

• Skipping the factorial divisor on each term

• Evaluating far from the center where convergence is slow

• Ignoring the alternating-series error bound when reporting precision

• Confusing order (highest power) with term count (order + 1)