Solve the problem.
-Find an upper bound for $\left| E _ { S } \right|$ in estimating $\int _ { 2 } ^ { 4 } \frac { 1 } { x - 1 } d x$ with $n = 8$ steps.
A) $\frac { 1 } { 960 }$
B) $\frac { 1 } { 1920 }$
C) $\frac { 1 } { 466,560 }$
D) $\frac { 1 } { 30 }$

Question

Quizplus · Accepted Answer

The error bound for the Trapezoidal Rule is given by $\left| E _ { T } \right| \leq \frac { K ( b - a ) ^ { 3 } } { 12 n ^ { 2 } }$, where $K$ is the maximum value of the absolute value of the second derivative of the function on the interval $[a, b]$, $a$ and $b$ are the limits of integration, and $n$ is the number of steps. For the given integral, the function is $f(x) = \frac{1}{x-1}$, so the second derivative is $f''(x) = \frac{2}{(x-1)^3}$. The maximum value of $\left| f''(x) \right|$ on the interval $[2, 4]$ is at $x = 2$, which gives $K = \frac{2}{(2-1)^3} = 2$. Plugging into the error bound formula gives $\left| E _ { T } \right| \leq \frac { 2 ( 4 - 2 ) ^ { 3 } } { 12 \cdot 8 ^ { 2 } } = \frac { 1 } { 960 }$.

Solve the Problem ∣ES∣\left| E _ { S } \right|∣ES​∣ In Estimating

Solve the Problem $\left| E _ { S } \right|$ In Estimating