Question 1(a) Find the fourth order Taylor polynomial for f(x) = re about z = 0.(b) Using the polynomial from part (a), find a Taylor polynomial expansion aboutI = 0 for te"dt and use it to approximate te"dt leaving your final solutioncorrect to 5 decimal places.

Answered: (a) Find the fourth order Taylor…

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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The nth order Taylor polynomial of a function $f (x)$ about $x = a$ is defined by $f (x) \approx P (x) = f (a) + \frac{f' (a)}{1!} (x - a) + \frac{f'' (a)}{2!} {(x - a)}^{2} + \frac{f''' (a)}{3!} {(x - a)}^{3} + \dots + \frac{f^{(n)} (a)}{n!} {(x - a)}^{n}$ .

Then, the fourth-order Taylor polynomial is as follows: $P (x) = f (a) + \frac{f' (a)}{1!} (x - a) + \frac{f'' (a)}{2!} {(x - a)}^{2} + \frac{f''' (a)}{3!} {(x - a)}^{3} + \frac{f^{(4)} (a)}{4!} {(x - a)}^{4}$ .

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