Chapter 1: Q42P (page 32)

$f (x) = c o s x, a = π .$

Short Answer

Expert verified

The first few terms of Taylor’s series expansion are:

$\cos (x) = - 1 + \frac{(x - π)^{2}}{2} - \frac{(x - π)^{4}}{24} + \frac{(x - π)^{6}}{720} - \dots$

Step by step solution

Given Information

The given function.

$f (x) = \cos x$

Definition of Taylor’s series

The Taylor’s series is a power series that yields the expansion of a function in the vicinity of a point if the function is continuous, all of its derivatives exist, and the series converges.

Rewrite cosx in required form

Rewrite $\cos x$ and use $\cos (π + x) = - \cos (x)$ ,

$\cos (x) = \cos [π + (x - π)] \cos [π + (x - π)] = - \cos (x - π)$

Find the Taylor’s series forcosx

Use the Maclaurin series of $\cos x$ and replace $x$ by $(x - π)$ ,

$\cos (x) = 1 - \frac{x^{2}}{2} + \frac{x^{4}}{24} - \frac{x^{6}}{720} + \dots$

$- \cos (x) = - [1 - \frac{x^{2}}{2} + \frac{x^{4}}{24} - \frac{x^{6}}{720} + \dots]$

$- \cos (x) = - 1 + \frac{x^{2}}{2} - \frac{x^{4}}{24} + \frac{x^{6}}{720} - \dots$

$- \cos (x) = - 1 + \frac{(x - π)^{2}}{2} - \frac{(x - π)^{4}}{24} + \frac{(x - π)^{6}}{720} - \dots$

Write first few terms of Taylor’s series aboutlocalid="1657417586942" $a - π .$

localid="1657417074607" $\cos (x) = - 1 + \frac{(x - π)^{2}}{2} - \frac{(x - π)^{4}}{24} + \frac{(x - π)^{6}}{720} - \dots$

Hence, first few terms of Taylor’s series expansion are:

$\cos (x) = - 1 + \frac{(x - π)^{2}}{2} - \frac{(x - π)^{4}}{24} + \frac{(x - π)^{6}}{720} - \dots$

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$f (x) = c o s x, a = π .$

Short Answer

Step by step solution

Given Information

Definition of Taylor’s series

Rewrite cosx in required form

Find the Taylor’s series forcosx

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f(x)=cos x,a=π.

Short Answer

Step by step solution

Given Information

Definition of Taylor’s series

Rewrite cosx in required form

Find the Taylor’s series forcosx

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Mechanics and Materials

Turning Points in Physics

Circular Motion and Gravitation

Waves Physics

Astrophysics

Further Mechanics and Thermal Physics

Study anywhere. Anytime. Across all devices.

$f (x) = c o s x, a = π .$