Lagrange Error Bound Calculator
Calculate the maximum error when approximating functions with Taylor polynomials
Calculate Lagrange Error Bound
Load Example Function
sin(x) approximated around x = π/6 ≈ 0.524
Number of terms in the Taylor polynomial
Point where you're approximating the function
Center of the Taylor polynomial (use 0 for Maclaurin series)
Maximum of |f^(n+1)(z)| for z between a and x
Lagrange Error Bound Result
Formula: e_max = M × |x - a|^(n+1) / (n+1)!
Interpretation: The actual error |R_n(x)| ≤ 8.333e-8
Relative error: ≤ 0.0001%
Step-by-Step Calculation
Formula Setup
Step 1: Calculate |x - a|
x - a = 0.1 - 0 = 0.100000
|x - a| = |0.100000| = 0.100000
Step 2: Calculate |x - a|^(n+1)
n + 1 = 4 + 1 = 5
|x - a|^5 = 0.100000^5
= 1.000000e-5
Step 3: Calculate (n+1)!
(n + 1)! = 5!
= 1 × 2 × 3 × 4 × 5
= 120
Step 4: Final Calculation
Numerator = M × |x - a|^(n+1)
= 1 × 1.000e-5
= 1.000000e-5
e_max = 1.000e-5 / 120
= 8.333333e-8
Error Analysis
Interpretation
- • The actual error is guaranteed to be less than 8.333e-8
- • Using more terms (higher n) will reduce the error bound
- • Points closer to the center (a) have smaller error bounds
Lagrange Error Bound Formula
Common Function Derivatives
Calculation Tips
Find M by analyzing the (n+1)th derivative on the interval [a,x]
Use calculus to find max/min or evaluate at endpoints
Higher degree polynomials give smaller error bounds
Points closer to center have smaller errors
Use Maclaurin series (a=0) for many common functions
Understanding the Lagrange Error Bound
What is the Lagrange Error Bound?
The Lagrange error bound provides an upper limit on the error when approximating a function using a Taylor polynomial. It tells us the maximum possible difference between the actual function value and the polynomial approximation.
Taylor Series Background
A Taylor series represents a function as an infinite sum of terms calculated from the function's derivatives at a single point. When we truncate this series to n terms, we get a Taylor polynomial that approximates the function.
Taylor Polynomial
Key Applications
- 📊Numerical analysis and computational mathematics
- 🔬Scientific computing and error estimation
- 📐Engineering approximations and modeling
- 💻Algorithm development and optimization
- 📈Mathematical analysis and proof verification
How to Find M
Step 1: Find the (n+1)th derivative of your function f(x)
Step 2: Determine the interval [a,x] where you need the maximum
Step 3: Find the maximum absolute value of the derivative on this interval