1. a) Use commonly known Taylor series to find the Taylor series of f(x) = cos(x²)centered at a =0. Show how you obtained the Taylor series and express your answerin summation notationb) Use the Taylor series found in part a) to represent the integral cos(x²)dx as aninfinite series.express the series in summation notation.c) Approximate the value of the integral with an error less than 10-5. Justify thatyour approximation satisfies the error bound.

Answered: 1. a) Use commonly known Taylor series…

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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Question

1. a) Use commonly known Taylor series to find the Taylor series of f(x) = cos(x²)
centered at a =
0. Show how you obtained the Taylor series and express your answer
in summation notation
b) Use the Taylor series found in part a) to represent the integral cos(x²)dx as an
infinite series.
express the series in summation notation.
c) Approximate the value of the integral with an error less than 10-5. Justify that
your approximation satisfies the error bound.

Expert Solution

Step 1

part (a)

given

$f (x) = \cos (x^{2})$

to find

taylor series of $f (x) = \cos (x^{2})$ centered at $a = 0$

solution

as we know

taylor series of function $f (x)$ centered at $a$ is defined as

$f (x) = f (a) + \frac{f^{'} (a)}{1!} (x - a) + \frac{f^{''} (a)}{2!} {(x - a)}^{2} + \frac{f^{'''} (a)}{3!} {(x - a)}^{3} + \frac{f^{''''} (a)}{4!} {(x - a)}^{4} + . . . . .$

according to question

$a = 0$

series becomes

$f (x) = f (0) + \frac{f^{'} (0)}{1!} (x) + \frac{f^{''} (0)}{2!} {(x)}^{2} + \frac{f^{'''} (0)}{3!} {(x)}^{3} + \frac{f^{''''} (0)}{4!} {(x)}^{4} + . . . . . . . . (i)$

Step 2

now as $f (x) = \cos (x^{2})$

put $x = 0$

$f (0) = \cos (0) = 1$

differentiate $f (x)$ with respect to $x$

$f^{'} (x) = - \sin (x^{2}) \times 2 x \Rightarrow f^{'} (0) = - \sin (0) \times 0 = 0$

differentiate $f^{'} (x)$ again with respect to $x$

we will use product rule

$f^{''} (x) = - 2 [\sin (x^{2}) + x \cos (x^{2}) \times 2 x] \Rightarrow f^{''} (0) = - 2 [\sin (0) + 0 \cos (0) \times 0] = 0$

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