Lagrange Error Bound Calculator — Taylor Polynomial Remainder
Given the maximum absolute value of the (n+1)th derivative (M), the evaluation point x, the series center a, and the polynomial degree n, this calculator gives the guaranteed upper bound on the approximation error.
The true error |f(x) − Pₙ(x)| is at most this value
Identify inputs
Compute |x − a|
Raise to (n+1) = 4
Compute (n+1)!
Apply formula: M · |x−a|^(n+1) / (n+1)!
- 1
|x − a|
|0.5 − 0| = 0.5 - 2
|x − a|^(n+1) = |x−a|^4
0.5^4 = 0.0625 - 3
(n+1)! = 4!
4! = 24 - 4
Error bound
M × |x−a|ⁿ⁺¹ ÷ (n+1)! = 1 × 0.0625 ÷ 24 = 0.00260417Worst-case upper bound on |f(x) − Pₙ(x)|.
How does this calculator work?
Lagrange error bound: |Rₙ(x)| ≤ M × |x−a|^(n+1) / (n+1)!. Enter the max absolute value of the (n+1)th derivative (M), the evaluation point x, the Taylor center a, and the polynomial degree n. The result is the worst-case error of the Taylor approximation at x.
Formula
How this is calculated
A Taylor polynomial Pₙ(x) approximates a smooth function f(x) near a center point a by summing n+1 terms built from the function's derivatives at a. The Lagrange form of the remainder Rₙ(x) = f(x) − Pₙ(x) quantifies exactly how wrong that approximation is. The Lagrange error bound theorem says the absolute error is bounded by M · |x − a|^(n+1) / (n+1)!, where M is any upper bound on the absolute value of the (n+1)th derivative of f on the interval between a and x.
To use this bound, you need to determine M yourself — for standard functions like sin, cos, or eˣ this is straightforward (e.g. for sin/cos all derivatives have absolute value ≤ 1; for eˣ on [0, b], M = eᵇ). You then plug M, the degree n, the evaluation point x, and the center a into the formula. A smaller |x − a| or a higher degree n dramatically shrinks the bound because factorial growth in (n+1)! dominates the power.
This bound is a worst-case guarantee. The actual error is often much smaller. The bound also assumes f has a continuous (n+1)th derivative on the entire interval from a to x.
Frequently asked questions
M is any number that is greater than or equal to the maximum of |f^(n+1)(t)| for all t between a and x. For common functions: sin and cos have all derivatives bounded by 1, so M = 1. For eˣ on [0, b], M = eᵇ. Use the largest possible value to get a safe (conservative) bound.
The (n+1)! in the denominator grows faster than the numerator |x−a|^(n+1) for small |x−a|, so adding more terms (higher n) dramatically reduces the bound. For example, going from n=3 to n=5 for x near a can cut the bound by several orders of magnitude.
No — the Lagrange bound is a worst-case ceiling. The true error |f(x) − Pₙ(x)| is often far smaller. The bound is most useful for proving that an approximation is accurate enough for a given application (e.g. computing tables, numerical algorithms).
Also known as
TG we-Calculate Editorial Team. (2026). Lagrange Error Bound Calculator — Taylor Polynomial Remainder [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/lagrange-error-bound-calculator
TG we-Calculate Editorial Team. "Lagrange Error Bound Calculator — Taylor Polynomial Remainder." TG we-Calculate. 2026. https://we-calculate.com/calculator/lagrange-error-bound-calculator.
TG we-Calculate Editorial Team, "Lagrange Error Bound Calculator — Taylor Polynomial Remainder," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/lagrange-error-bound-calculator
@misc{wecalculate_lagrange_error_bound_calculator, title = {Lagrange Error Bound Calculator — Taylor Polynomial Remainder}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/lagrange-error-bound-calculator}}, year = {2026}, note = {TG we-Calculate} }
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