Lagrange Error Bound Calculator
Estimate error in Taylor series approximations
How to Use This Calculator
Enter Maximum Derivative
Input M, the maximum absolute value of the (n+1)th derivative on the interval between a and x.
Enter Degree
Input n, the degree of the Taylor polynomial being used for approximation.
Enter x and Center Point
Input the point x where you're evaluating the function and the center point a of the Taylor series.
Calculate
Click "Calculate Error Bound" to see the maximum possible error.
Formula
|R_n(x)| ≤ M × |x - a|^(n+1) / (n+1)!
Where:
- R_n(x) = remainder (error) term
- M = maximum value of |f^(n+1)(z)| for z between a and x
- n = degree of Taylor polynomial
- a = center point of Taylor series
- x = point of evaluation
- (n+1)! = factorial of (n+1)
Example: Error bound for e^x at x=1 using 3rd degree polynomial centered at 0
If |f⁽⁴⁾(z)| ≤ e for z in [0,1], then M = e ≈ 2.718, n = 3, x = 1, a = 0
|R₃(1)| ≤ e × |1-0|⁴ / 4! = e × 1 / 24 ≈ 0.113
About Lagrange Error Bound Calculator
The Lagrange error bound (also called the Taylor remainder theorem) provides an upper bound for the error when approximating a function using a Taylor polynomial. It's essential for understanding the accuracy of polynomial approximations.
Key Concepts
- Taylor Polynomial: Polynomial approximation of a function
- Remainder Term: The difference between the function and its polynomial approximation
- Error Bound: Maximum possible value of the remainder
Applications
- Estimating accuracy of numerical approximations
- Determining required polynomial degree for desired accuracy
- Understanding convergence of Taylor series
Frequently Asked Questions
What is the Lagrange error bound?
The Lagrange error bound gives an upper limit on how much error exists when approximating a function with a Taylor polynomial. It uses the maximum value of a higher-order derivative.
How do I find M (the maximum derivative)?
Find the (n+1)th derivative of your function, then determine its maximum absolute value on the interval between the center point a and the evaluation point x.
What if I don't know the maximum derivative?
You need to estimate or calculate it. For many functions, you can find an upper bound by analyzing the derivative on the relevant interval.
Does a smaller error bound mean better approximation?
Yes! A smaller error bound indicates the Taylor polynomial is closer to the actual function value at that point.
Can the error be negative?
The error bound gives the maximum absolute error. The actual error can be positive or negative, but its absolute value won't exceed the bound.