Solve the problem.
-The polynomial $1 + 8 x + 28 x ^ { 2 }$ is used to approximate $f ( x ) = ( 1 + x ) ^ { 8 }$ on the interval $- 0.01 \leq x \leq 0.01$. Use the Remainder Estimation Theorem to estimate the maximum absolute error.
A) $\approx 3.642 \times 10 ^ { - 5 }$
B) $\approx 5.886 \times 10 ^ { - 5 }$
C) $\approx 5.886 \times 10 ^ { - 4 }$

Question

Quizplus · Accepted Answer

The Remainder Estimation Theorem states that if we approximate a function by a polynomial of degree $n$, then the error of the approximation is given by $| R_n ( x ) | \leq \frac{M}{(n+1)!} |(x-x_0)|^{n+1}$, where $M$ is the maximum value of the $(n+1)$-th derivative of the function on the interval, and $x_0$ is the center of the approximation (in this case, $x_0=0$).

For $f(x)=(1+x)^8$, the $(n+1)$-th derivative is $f^{(n+1)}(x)=8(8-1)\dots(8-n)x^{n-1}$, which has a maximum absolute value of $8^ { n+1 }$.

We approximate $f(x)$ by the polynomial $1+8x+28x^2$ of degree 2. Thus, the maximum absolute error is given by 
\begin{align*}
|R_2(x)|&\leq \frac{8^3}{3!}|x|^3\
&=\frac{512}{6}\cdot (0.01)^3\
&\approx 5.886 	imes 10^{-5}.
\end{align*}

Therefore, the best choice is $\boxed{	ext{B}}$.

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