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Lagrange Error Bound Calculator

Calculate the maximum error in a Taylor polynomial approximation using the Lagrange remainder theorem.

Reviewed by Christopher FloiedPublished Updated

This free online lagrange error bound calculator provides instant results with no signup required. All calculations run directly in your browser — your data is never sent to a server. Enter your values below and see results update in real time as you type. Perfect for everyday calculations, homework, or professional use.

Maximum absolute value of the (n+1)th derivative on the interval

Point where the polynomial is evaluated

Center of the Taylor expansion

Results

Lagrange Error Bound

2.17013889e-5

How to Use This Calculator

1

Enter your input values

Fill in all required input fields for the Lagrange Error Bound Calculator. Most fields include unit selectors so you can work in your preferred unit system — metric or imperial, whichever matches your problem.

2

Review your inputs

Double-check that all values are correct and that you have selected the right units for each field. Incorrect units are the most common source of calculation errors and can produce results that are off by factors of 2, 10, or more.

3

Read the results

The Lagrange Error Bound Calculator instantly computes the output and displays results with units clearly labeled. All calculations happen in your browser — no loading time and no data sent to a server.

4

Explore parameter sensitivity

Try adjusting individual input values to see how the output changes. This is a quick and effective way to develop intuition about how different parameters influence the result and to identify which inputs have the largest effect.

When to Use This Calculator

  • •Use the Lagrange Error Bound Calculator when you need a quick mathematical result without writing out all the steps manually, saving time on repetitive calculations.
  • •Use it to verify hand calculations on tests or assignments and catch arithmetic mistakes.
  • •Use it when teaching or explaining mathematical concepts to others, demonstrating how changing inputs affects the result.
  • •Use it to explore the behavior of mathematical functions across a range of inputs.

About Lagrange Error Bound Calculator

The Lagrange error bound (also called the Taylor remainder bound) tells you the maximum possible error when you approximate a function by its nth-degree Taylor polynomial. If you know an upper bound M for the absolute value of the (n+1)th derivative of f on the interval between the center a and evaluation point x, then the error is at most M|x − a|^(n+1)/(n+1)!. This is one of the most important results in numerical analysis and calculus, as it lets you determine in advance how many terms of a Taylor series you need for a desired accuracy. AP Calculus BC students frequently encounter this on exams. This calculator computes the bound directly from the given parameters.

The Math Behind It

Taylor's theorem states that f(x) = Pₙ(x) + Rₙ(x), where Pₙ is the nth Taylor polynomial centered at a and Rₙ is the remainder. The Lagrange form of the remainder is Rₙ(x) = f^(n+1)(c) × (x−a)^(n+1) / (n+1)! for some c between a and x. Since c is unknown, we bound |Rₙ(x)| ≤ M × |x−a|^(n+1) / (n+1)! where M = max|f^(n+1)(t)| for t in [a, x]. For functions with bounded derivatives (like sin, cos, eˣ), M is easy to determine. The bound guarantees the error decreases as n increases (provided |x − a| < radius of convergence), because the factorial in the denominator eventually dominates any polynomial growth in the numerator. This is why Taylor series converge and why polynomial approximations become increasingly accurate.

Formula Reference

Lagrange Error Bound

|Rₙ(x)| ≤ M × |x − a|^(n+1) / (n+1)!

Variables: M = max|f^(n+1)(c)| on [a,x], n = polynomial degree

Worked Examples

Example 1: Error Bound for sin(0.5) with 5th-Degree Taylor Polynomial

Approximate sin(0.5) using a 5th-degree Taylor polynomial centered at 0.

Step 1:For sin(x), all derivatives are bounded by 1, so M = 1
Step 2:|R₅(0.5)| ≤ 1 × |0.5|⁶ / 6!
Step 3:= 0.015625 / 720

|R₅(0.5)| ≤ 0.0000217 — the error is at most about 2.17 × 10⁻⁵.

Common Mistakes & Tips

  • !Using the nth derivative instead of the (n+1)th derivative for the bound M.
  • !Forgetting to take the absolute value of (x − a).
  • !Using n! instead of (n+1)! in the denominator.
  • !Not finding a valid upper bound M — it must hold for all c between a and x.

Related Concepts

Sum of Series

Taylor series are infinite series; the error bound tells you when to truncate.

Exponential Function

eˣ is commonly approximated by Taylor polynomials.

Frequently Asked Questions

How do I find M for common functions?

For sin(x) and cos(x), all derivatives are bounded by 1, so M = 1. For eˣ on [0, x], M = eˣ (or e for [0,1]). For polynomials, derivatives beyond a certain order are zero.

Does the Lagrange error bound give the exact error?

No, it gives an upper bound. The actual error is usually much smaller than the bound. The bound guarantees the error does not exceed this value.