Find the smallest n such that the error estimate from the error formula in the approximation of the definite integral $\int _ { 0 } ^ { 2 } \sqrt { x + 7 } d x$ is less than $0.00001$ using the Trapezoidal Rule.
A) 49
B) 26
C) 73
D) 30
E) 15

Question

Quizplus · Accepted Answer

Using the Trapezoidal Rule, the error estimate is given by $$ |E_T| \leq \frac{(b-a)^3}{12n^2} \max_{a \leq x \leq b} |f''(x)| $$ where $b=2$, $a=0$, $f(x)=\sqrt{x+7}$, and $f''(x)=-\frac{3}{4}(x+7)^{-\frac{5}{2}}$. Then we have $$ \max_{a \leq x \leq b} |f''(x)|=\frac{27}{64} $$ Thus, we need to find the smallest integer $n$ satisfying $$ \frac{(b-a)^3}{12n^2} \max_{a \leq x \leq b} |f''(x)|< 0.00001 $$ which is equivalent to \begin{align*} &\frac{2^3}{12n^2} \cdot \frac{27}{64} < 0.00001 \ \Rightarrow \ & n^2 > \frac{2^5 \cdot 64}{27 \cdot 0.00001} \ \Rightarrow \ & n > 29.6345 \dots \end{align*} Thus, the smallest integer $n$ satisfying the inequality is $n=30$.

Find the Smallest N Such That the Error Estimate from the Error