Solve the problem.
-Find an upper bound for $\left| \mathrm { E } _ { \mathrm { T } } \right|$ in estimating $\int _ { 4 } ^ { 6 } \frac { 1 } { ( \mathrm { x } - 1 ) ^ { 2 } } \mathrm { dx }$ with $\mathrm { n } = 7$ steps.
A) $\frac { 4 } { 30,625 }$
B) $\frac { 4 } { 11,907 }$
C) $\frac { 4 } { 3969 }$
D) $\frac { 4 } { 147 }$

Question

Quizplus · Accepted Answer

The error bound for the Trapezoidal Rule is given by $\left| \mathrm { E } _ { \mathrm { T } } \right| \leq \frac { ( b - a ) ^ { 3 } } { 12 n ^ { 2 } } \max \left| f''(x) \right|$, where $a$ and $b$ are the limits of integration, and $n$ is the number of steps. For the given function $f(x) = \frac { 1 } { ( x - 1 ) ^ { 2 } }$, the second derivative $f''(x) = \frac { 6 } { ( x - 1 ) ^ { 4 } }$. The maximum value of $f''(x)$ on the interval $[4,6]$ occurs at $x = 4$, giving $\max \left| f''(x) \right| = \frac { 6 } { 3 ^ { 4 } } = \frac { 6 } { 81 }$. Substituting $a = 4$, $b = 6$, $n = 7$, and $\max \left| f''(x) \right| = \frac { 6 } { 81 }$ into the error bound formula gives $\left| \mathrm { E } _ { \mathrm { T } } \right| \leq \frac { ( 6 - 4 ) ^ { 3 } } { 12 \cdot 7 ^ { 2 } } \cdot \frac { 6 } { 81 } = \frac { 8 } { 12 \cdot 49 } \cdot \frac { 6 } { 81 } = \frac { 1 } { 3 \cdot 49 } \cdot \frac { 1 } { 13.5 } = \frac { 4 } { 3969 }$.

Solve the Problem ∣ET∣\left| \mathrm { E } _ { \mathrm { T } } \right|∣ET​∣

Solve the Problem $\left| \mathrm { E } _ { \mathrm { T } } \right|$