1a O Consider the task of approaimatingSaedu using the Compositetrapezoidal rule. Howshoulddla igeorder ton and hbe chaveninthatatmost g 001?ensureerror Is

The error term in the composite trapezoidal rule is h212b-af''ξ , where h=b-an Given f(x)=e-3x2…

Answered: @ Consider the task of approximating…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Related questions

Question

O Consider the task of approaimating
Saedu using the Composite
trapezoidal rule. How
shouldd
la ige
order to
n and h
be chaven
in
that
at
most g 001?
ensure
error Is

Expert Solution

Step 1

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

Given $f (x) = e^{- 3 x^{2}}$

Differentiate both sides with respect to $x$ to obtain $f' (x)$

$f' (x) = - 6 x e^{- 3 x^{2}}$

Differentiate both sides with respect to $x$ to obtain $f'' (x)$

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

To obtain the maximum value of $f'' (x)$ 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 $f''' (x)$

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

Equate the third derivative to zero.

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