Find the smallest n such that the error estimate in the approximation of the definite integral is less than 0.00001 using the Trapezoidal Rule. Use a graphing utility to estimate the maximum of the absolute value of the second derivative.
A) 208
B) 189
C) 196
D) 201
E) 188

Question

Find the smallest n such that the error estimate in the approximation of the definite integral  is less than 0.00001 using the Trapezoidal Rule. Use a graphing utility to estimate the maximum of the absolute value of the second derivative.
A) 208 
B) 189 
C) 196 
D) 201 
E) 188

Quizplus · Accepted Answer

To find the smallest value of $n$, we use the error formula for the Trapezoidal Rule:
$$
E_T \leq \frac{1}{12}(	ext{Length of Interval})^3 	ext{max}(|f''(x)|),
$$
where the length of interval$=1$ and $	ext{max}(|f''(x)|)$ is the maximum absolute value of the second derivative of $f(x)$ on $[0,1]$.

Using a graphing utility, we can estimate that the maximum of the absolute value of the second derivative occurs around $x=0.5$, and its value is approximately $7.2$. Therefore, we have $	ext{max}(|f''(x)|)=7.2$.

Plugging this value into the error formula and solving for $n$, we get:
\begin{align*} 
0.00001 &\geq \frac{1}{12}(1/n)^3 \cdot 7.2 \
n^3 &\geq \frac{1}{0.00001 \cdot 12 \cdot 7.2} \
n &\geq 196.07...
\end{align*}
Therefore, the smallest value of $n$ that satisfies the error condition is $n=196$, and the closest choice is $\boxed{	extbf{(C) } 196}$.

Find the Smallest N Such That the Error Estimate in the Approximation