@ Consider the task of approximating 3x2 using the Compoute trapezoidal rule. How large should n and h be chaven in order to ensure errors IJ at thaat most 0,001?

The error term in the composite trapezoidal rule is  $\frac{h^2}{12}\left(b-a\right)f\text{'}\text{'}\left(\xi \right)$ , where $h=\frac{b-a}{n}$

Given $f\left(x\right)=e^{-3x^2}$

Differentiate both sides with respect to $x$ to obtain $f\text{'}\left(x\right)$

$f\text{'}\left(x\right)=-6x{e}^{-3x^2}$

Differentiate both sides with respect to $x$ to obtain $f\text{'}\text{'}\left(x\right)$

$\begin{array}{rcl}f\text{'}\text{'}\left(x\right)& =& -6[e^{-3x^2}+x(-6x\left)e^{-3x^2}\right]\\ & =& (36x^2-6)e^{-3x^2}\end{array}$

To obtain the maximum value of $f\text{'}\text{'}\left(x\right)$ in the interval [0,1], obtain the third derivative and equate it to zero to obtain the roots between 0 and 1.

Differentiate both sides with respect to $x$ to obtain $f\text{'}\text{'}\text{'}\left(x\right)$

$\begin{array}{rcl}f\text{'}\text{'}\text{'}\left(x\right)& =& [72x{e}^{-3x^2}+(36x^2-6\left)\right(-6x{e}^{-3x^2}]\\ & =& 72x{e}^{-3x^2}-216x^3{e}^{-3x^2}+36x{e}^{-3x^2}\\ & =& (108x-216x^3)e^{-3x^2}\\ & =& 108x{e}^{-3x^2}(1-2x^2)\end{array}$

Equate the third derivative to zero.

$\begin{array}{rcl}108x{e}^{-3x^2}(1-2x^2)& =& 0\\ x\left(1-2x^2\right)& =& 0\\ 1-2x^2& =& 0 or x=0\\ 1& =& 2x^2\\ \frac{1}{2}& =& x^2\\ \pm \frac{1}{\sqrt{2}}& =& x \end{array}$