Lagrange Error Bound for Taylor Polynomials

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AP Calculus BC › Lagrange Error Bound for Taylor Polynomials

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Let $P_{5}(x)$ be the fifth-degree Taylor polynomial approximation for , centered at .

What is the Lagrange error of the polynomial approximation to ?

$\frac{1}{7!}$

CORRECT

$\frac{1}{5!}$

$\frac{1}{6!}$

The fifth degree Taylor polynomial approximating centered at is:

$sin(x) \approx x - \frac{x^3}{3!} + \frac{x^5}{5!}$

The Lagrange error is the absolute value of the next term in the sequence, which is equal to $\left |\frac{x^7}{7!} \right |$ .

We need only evaluate this at and thus we obtain $\frac{1^7}{7!} = \frac{1}{7!}$