Solve the problem.
-Find an upper bound for $| E_ S |$ in estimating $\int _ { 1 } ^ { 3 } \left( 2 x ^ { 4 } - 2 x \right) d x$ with $n = 6$ steps.
A) $\frac { 32 } { 81 }$
B) $\frac { 8 } { 1215 }$
C) $\frac { 4 } { 1215 }$
D) 1

Question

Quizplus · Accepted Answer

Using the formula for the error bound in the Trapezoidal Rule, we have:
$$|E_S| \leq \frac{(b-a)^3}{12n^2}\max_{a\leq x\leq b}|f''(x)|$$
In this case, $a=1$, $b=3$, $n=6$, $f(x)=2x^4-2x$, and $f''(x)=24x^2-2$. To find the maximum value of $f''(x)$, we set its derivative to zero and solve for $x$:
$$\frac{d}{dx}(24x^2-2) = 48x = 0 \Rightarrow x=0$$
However, $x=0$ is outside the interval $[1,3]$, so we check the endpoints and critical points inside the interval:
\begin{align*}
f''(1) &= 24(1)^2-2 = 22\
f''(3) &= 24(3)^2-2 = 214
\end{align*}
Hence, we have $\max_{1\leq x\leq 3}|f''(x)| = 214$, and plugging in the values, we get:
$$|E_S| \leq \frac{(3-1)^3}{12(6)^2}(214) = \frac{8}{1215}$$
Therefore, the best choice is B.

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