Recall that the error in the Trapezoidal Rule approximation with $ n $ subdivisions ($ T_n $) to approximate the integral \[\int_a^b f(x) \, dx\]obeys the error bound \[|E_T| \leq \frac{K(b-a)^3}{12n^2}\]where \[|f''(K)| \leq K \text{ on } [a, b].\](a) Set up, but do NOT evaluate, the sum $ T_5 $ to approximate the integral \[\int_0^1 e^{x^2} \, dx.\]

Trapezoidal Rule:∫abgxdx≈Tn=∆x2gx0+2gx1+2gx2+...+2gxn-1+gxnwhere ∆x=b-an,xi=a+i∆xn is the number of…

Answered: Recall that the error in the…

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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Recall that the error in the Trapezoidal Rule approximation with $ n $ subdivisions ($ T_n $) to approximate the integral

\[
\int_a^b f(x) \, dx
\]

obeys the error bound

\[
|E_T| \leq \frac{K(b-a)^3}{12n^2}
\]

where

\[
|f''(K)| \leq K \text{ on } [a, b].
\]

(a) Set up, but do NOT evaluate, the sum $ T_5 $ to approximate the integral

\[
\int_0^1 e^{x^2} \, dx.
\]

Expert Solution

Step 1

$T r a p e z o i d a l R u l e : \int_{a}^{b} g (x) d x \approx T_{n} = \frac{∆ x}{2} (g (x_{0}) + 2 g (x_{1}) + 2 g (x_{2}) + . . . + 2 g (x_{n - 1}) + g (x_{n})) w h e r e ∆ x = \frac{b - a}{n}, x_{i} = a + i ∆ x n i s t h e n u m b e r o f i n t e r v a l s$

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