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Home Practice Questions AP Calculus BC Series Question No. 4

4. Let \( f \) be a function having derivatives of all orders for all real numbers, and \( |f^{(4)}(x)| \le 3 \) for all \( x \) in the interval \( [0,2] \). If the third-degree Taylor polynomial for \( f \) about \( x=0 \) is used to approximate \( f \) on the interval \( [0,2] \), what is the Lagrange error bound for the maximum error on the interval \( [0,2] \)? {calculator}






Answer is: option1

\( 2 \)

Solution:

The Lagrange error is

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

Recall that \( M \) is the upper bound for all values of the \( (n+1) \)st derivative on the given interval \( [a,x] \) in this case.

At \( x=2,\; a=0,\; n=3,\; M=3 \):

\[ |R_3(2)| = \frac{3}{4!}|2-0|^4 = \frac{3\cdot16}{24} = 2 \]

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