Solve the problem.
-Estimate the minimum number of subintervals needed to approximate the integral $\int _ { 2 } ^ { 5 } \frac { 1 } { x - 1 } d x$ with an error of magnitude less than $10 ^ { - 4 }$ using Simpson's Rule.
A) 570
B) 24
C) 6
D) 48

Question

Quizplus · Accepted Answer

Simpson's Rule's error bound is given by $|E| \leq \frac{M(b-a)^5}{180n^4}$, where $M$ is the maximum value of the fourth derivative of the function on the interval $[a, b]$, and $n$ is the number of subintervals (which must be even). For the given integral, the function is $f(x) = \frac{1}{x-1}$. The fourth derivative of $f(x)$ is $f^{(4)}(x) = \frac{24}{(x-1)^5}$. The maximum value of $|f^{(4)}(x)|$ on the interval $[2, 5]$ is at $x = 2$, which gives $M = 24$. Plugging into the error formula with $a = 2$, $b = 5$, and the desired error less than $10^{-4}$, we solve for $n$. Rearranging the error formula and solving for $n$ gives $n \approx 24$, ensuring $n$ is even and meets the error requirement.

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