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`What is the formula for the Lagrange error bound when approximating a function using a Taylor polynomial?`

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Answer from Sia

Posted 4 months ago

Solution by Steps

step 1

The Lagrange error bound for a Taylor polynomial approximation is given by the formula:
$R_n(x) = \frac{M}{(n+1)!} |x - c|^{n+1}$
where $M$ is an upper bound on the absolute value of the $(n+1)$th derivative of the function on the interval between $c$ and $x$

step 2

Identify the degree $n$ of the Taylor polynomial and the point $c$ around which the polynomial is centered

step 3

Determine the $(n+1)$th derivative of the function, $f^{(n+1)}(x)$

step 4

Find an upper bound $M$ for $|f^{(n+1)}(x)|$ on the interval between $c$ and $x$

step 5

Substitute $M$, $n$, $x$, and $c$ into the Lagrange error bound formula to find the error bound $R_n(x)$

Answer

The Lagrange error bound formula is $R_n(x) = \frac{M}{(n+1)!} |x - c|^{n+1}$.

Key Concept

Lagrange Error Bound

Explanation

The Lagrange error bound provides an estimate of the error when approximating a function using a Taylor polynomial. It depends on the $(n+1)$th derivative of the function and the distance between the point of approximation and the center of the Taylor series.

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