Determine the maximum error guaranteed by Taylor's Theorem with Remainder when the fifth-degree Taylor polynomial is used to approximate $e ^ { - x }$ in the interval $[ 0,1 ]$ centered at 0. Round your answer to five decimal places.
A)0.02139
B)0.00339
C)0.00833
D)0.00139
E)0.05139

Question

Quizplus · Accepted Answer

The maximum error guaranteed by Taylor's Theorem with Remainder for a function $f(x)$ approximated by its $n$th-degree Taylor polynomial $P_n(x)$ around a point $a$ is given by $\frac{M}{(n+1)!}|x-a|^{n+1}$, where $M$ is an upper bound on the absolute value of the $(n+1)$th derivative of $f(x)$ on the interval of interest. For $f(x) = e^{-x}$, all derivatives are of the form $\pm e^{-x}$, so the maximum value of these derivatives on the interval $[0,1]$ is 1 (since $e^{-x}$ decreases as $x$ increases and its maximum value in this interval is at $x=0$, which is $e^0 = 1$). Using a fifth-degree Taylor polynomial means $n=5$, so we are interested in the error term associated with the 6th derivative, which also has a maximum of 1 in absolute value on the interval. Therefore, the maximum error is $\frac{1}{6!}|x-0|^6 = \frac{1}{720}$ (since $|x-0|^6 \leq 1^6 = 1$ for $x \in [0,1]$). Calculating $\frac{1}{720}$ gives approximately 0.00139.

Determine the Maximum Error Guaranteed by Taylor's Theorem with Remainder e−xe ^ { - x }e−x

Determine the Maximum Error Guaranteed by Taylor's Theorem with Remainder $e ^ { - x }$