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Home Practice Questions AP Calculus BC Series Question No. 3

3. Let \( f \) be a function having 5 derivatives on the interval \( [2,2.9] \) and assume that \( |f^{(5)}(x)| \le 0.8 \) for all \( x \) in the interval \( [2,2.9] \). If the fourth-degree Taylor polynomial for \( f \) about \( x=2 \) is used to approximate \( f \) on the interval \( [2,2.9] \), what is the Lagrange error bound for the maximum error on the interval \( [2,2.9] \)? {calculator}






Answer is: option1

\( 0.004 \)

Solution:

The Lagrange bound on the error in this 4th degree Taylor polynomial is

\[ \frac{M}{(n+1)!}|x-a|^{\,n+1} \]

At \( x=2.9,\; a=2,\; n=4,\; M=0.8 \), we have

\[ \frac{0.8}{5!}|2.9-2|^5 = \frac{0.8}{120}(0.9)^5 \approx 0.004 \]

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