Lagrange Error Bound Calculator

Your details

The degree of the Taylor polynomial you are using to approximate the function. Must be a non-negative integer.
The point at which you want to evaluate the approximation error. This is where you are estimating the function.
The point around which the Taylor series is expanded, also called the center or the point of expansion.
The maximum absolute value of the (n+1)th derivative of the function on the interval between a and x. For bounded functions such as sin and cos this is at most 1.
Lagrange error boundGood accuracy
0.00032796

Maximum possible error |f(x) - P_n(x)|

|x - a|0.5236
n + 15
(n+1)!120
M x |x-a|^(n+1)0.03935484
0.00032796
Excellent<0.000001Good0.000001-0.001Moderate0.001-0.01Low0.01-0.1Poor0.1+

Maximum error is 0.00032796 using a degree-4 Taylor polynomial.

  • Your degree-4 Taylor polynomial approximates the function with a maximum error of 0.00032796.
  • The evaluation point x = 1.0472 is 0.5236 units away from the center a = 0.5236. Closer evaluation points always give tighter error bounds.
  • With M = 1, this is consistent with functions whose derivatives are bounded by 1, such as sin(x) or cos(x).
  • Increasing the polynomial degree to 5 would reduce the bound to approximately 2.862e-5, assuming the same M.

Next stepThe approximation is already quite accurate. This error bound confirms the Taylor polynomial is a reliable stand-in for the true function near x = 1.0472.

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