AP Calculus BC 10.12 Lagrange Error Bound Study Notes - New Syllabus

Lagrange Error Bound

The Lagrange Error Bound (or remainder estimate) gives an upper bound on the error when a Taylor polynomial \( P_n(x) \) is used to approximate a function \( f(x) \) near \( x = a \). It quantifies how close the polynomial approximation is to the actual function.

Formula:

If \( f \) is \( (n+1) \) times differentiable on an interval containing \( a \) and \( x \), then the remainder term is

$ R_n(x) = \dfrac{f^{(n+1)}(c)}{(n+1)!} (x-a)^{n+1} $ for some \( c \) between \( a \) and \( x \).

The Lagrange Error Bound is found by taking a bound \( M \) on \( |f^{(n+1)}(t)| \) for all \( t \) between \( a \) and \( x \):

$ |R_n(x)| \le \dfrac{M}{(n+1)!} |x-a|^{n+1} $

Interpretation:

• \( M \) is the maximum absolute value of the \((n+1)\)th derivative over the interval.
• The bound tells us the worst-case error for approximating \( f(x) \) by \( P_n(x) \).
• Smaller \( |x-a| \) and larger \( n \) usually make the error smaller.

Purpose:

This bound is used to ensure that the approximation meets a desired accuracy. In AP Calculus, it often appears in questions asking how many terms are needed to guarantee the error is less than a given tolerance.

Alternating Series Error Bound and Taylor Polynomials

If the Taylor series for a function is an alternating series (terms alternate in sign) and the absolute value of the terms decreases to zero, then the Alternating Series Error Bound applies to the Taylor polynomial approximation.

Theorem:

Let $ S = \sum_{n=0}^{\infty} (-1)^n b_n $ be an alternating series with \( b_{n+1} \le b_n \) and \( \lim_{n\to\infty} b_n = 0 \).

If \( S_N \) is the sum of the first \( N \) terms (the \( N \)th-degree Taylor polynomial), then the remainder (error) satisfies: $ |R_N| \le b_{N+1} $

In Taylor Polynomial Context:

• The remainder \( R_N(x) \) is the difference between the actual function value \( f(x) \) and the \( N \)th-degree Taylor polynomial \( P_N(x) \).
• If the Taylor series is alternating and decreasing in absolute value for \( x \) in the interval, then: $ | f(x) – P_N(x) | \le \text{magnitude of next term} $
• This gives a quick and easy way to estimate how accurate the approximation is, without computing higher derivatives or using the full Lagrange error bound.