Mastering the Lagrange Error Bound for Reliable Function Approximations

The Lagrange Error Bound provides an upper limit for the error when approximating a function \( f(x) \) with a Taylor polynomial. Given a Taylor series centered at \( a \), the error bound for approximating \( f(x) \) by an \( n \)-degree Taylor polynomial \( P_n(x) \) is represented by \( R_n(x) \), the remainder term:

\( [
|R_n(x)| \leq \frac{M |x – a|^{n+1}}{(n+1)!}
] \)

Here, \( M \) is the maximum value of the absolute value of the \( (n+1) \)-th derivative of \( f(x) \) on the interval between \( a \) and \( x \). This bound quantifies the potential error, helping to determine how closely \( P_n(x) \) approximates \( f(x) \) at a specific point. In practice, the Lagrange Error Bound is essential in calculus and numerical methods, enabling mathematicians to control approximation errors effectively, especially in fields requiring high precision, such as engineering and physics.

Consider approximating \( f(x) = e^x \) at \( x = 0.5 \) using the second-degree Taylor polynomial centered at \( a = 0 \):

\( [
P_2(x) = 1 + x + \frac{x^2}{2}
] \)

To find the error, apply the Lagrange Error Bound. Here, the third derivative of \( e^x \) is \( e^x \), and the maximum value on \([0, 0.5]\) is \( e^{0.5} \approx 1.65 \).

Using \( M = 1.65 \), \( n = 2 \), and \( x = 0.5 \), we get:

\( [
|R_2(0.5)| \leq \frac{1.65 \cdot (0.5)^3}{3!} \approx 0.034
] \)

Thus, the error in approximating \( e^{0.5} \) with \( P_2(0.5) \) is at most \( 0.034 \), providing a reliable accuracy check for this approximation.