What should be the maximum value of $h$ so that the upper bound of the error of approximated value of $\int \dfrac {dx}{x^2}$ using Composite Trapezoidal Rule is within $10^{-3}$? The limit of integration is from $x=1$ to $x=2$.
My Attempt: $$\textrm {Error} \leq 10^{-3}$$ $$\dfrac {M_2 (b-a)^3}{12n^2} \leq \dfrac {1}{10^3}$$ Where $M_2=6$ is the maximum value of second order derivative in $[1,2]$. On solving that gives, $22.36068\leq n$. What should be the value of $n$ to be concluded from this?
So, we need $n\ge \left(\frac{6\cdot 10^3}{12}\right)^{\frac12}\approx 22.4$. What other restrictions are there on $n$? Just that it's an integer. The smallest integer $n$ satisfying the inequality is $23$. Then that gives $h=\frac1{23}$ for the answer to the original question.