Chapter 1: Q43P (page 32)

$f (x) = c o t x, a = \frac{π}{2}$

Short Answer

Expert verified

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

$\cot (x) = (x - \frac{π}{2}) - \frac{{(x - \frac{π}{2})}^{3}}{3} - \frac{2 {(x - \frac{π}{2})}^{5}}{15} - \dots$

Step by step solution

Given Information

The given function

$f (x) = c o t x$

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 cotxin required form

Rewrite and use $\cot (\frac{π}{2} + x) = - \tan (x)$

$\cot (x) = \cot (\frac{π}{2} + (x - \frac{π}{2}))$

$\cot (\frac{π}{2} + (x - \frac{π}{2})) = - \tan (x - \frac{π}{2})$

Find the Taylor's series forcotx .

Use the Maclaurin series of $\tan x$ and replace $x$ by $(x - \frac{π}{2})$ ,

$\tan (x) = x + \frac{x^{3}}{3} + \frac{2 x^{5}}{15} + \dots$

$- \tan (x) = - x - \frac{x^{3}}{3} - \frac{2 x^{5}}{15} - \dots$

$- \tan (x - \frac{π}{2}) = - (x - \frac{π}{2}) - \frac{{(x - \frac{π}{2})}^{3}}{3} - \frac{2 {(x - \frac{π}{2})}^{5}}{15} - \dots$

$\cot (x) = (x - \frac{π}{2}) - \frac{{(x - \frac{π}{2})}^{3}}{3} - \frac{2 {(x - \frac{π}{2})}^{5}}{15} - \dots$

Write first few terms of Taylor’s series about $a = \frac{π}{2}$ ,

role="math" localid="1657419575245" $\cot (x) = (x - \frac{π}{2}) - \frac{{(x - \frac{π}{2})}^{3}}{3} - \frac{2 {(x - \frac{π}{2})}^{5}}{15} - \dots$

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

$\cot (x) = (x - \frac{π}{2}) - \frac{{(x - \frac{π}{2})}^{3}}{3} - \frac{2 {(x - \frac{π}{2})}^{5}}{15} - \dots$

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$f (x) = c o t x, a = \frac{π}{2}$

Short Answer

Step by step solution

Given Information

Definition of Taylor’s series

Rewrite cotxin required form

Find the Taylor's series forcotx .

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f(x)=cotx,a=π2

Short Answer

Step by step solution

Given Information

Definition of Taylor’s series

Rewrite cotxin required form

Find the Taylor's series forcotx .

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Engineering Physics

Quantum Physics

Fields in Physics

Atoms and Radioactivity

Translational Dynamics

Modern Physics

Study anywhere. Anytime. Across all devices.

$f (x) = c o t x, a = \frac{π}{2}$