Chapter 8: Q 9 (page 704)

Find third-order Maclaurin
or Taylor polynomial for the given function about the indicated
point.
$\tan^{- 1} x, x_{0} = 0$

Short Answer

Expert verified

The third-order Maclaurin series for the function $f (x) = \tan^{- 1} x$ is $P_{3} (x) = x - \frac{1}{3} x^{3}$

Step by step solution

Given information

The function is $f (x) = \tan^{- 1} x$

The third Maclaurin polynomial is given for any functionf with a derivative of order 3 at x=0

$P_{3} (x) = f (0) + f^{'} (0) x + \frac{f^{''} (0)}{2!} x^{2} + \frac{f^{'''} (0)}{3!} x^{3}$

First, determine the function's value as well as $f^{'} (x), f^{''} (x), f^{''} (x) a t x = 0$

Find the derivatives of the function

The value of the function $x = 0$ is,

localid="1653480203360" $f (0) = t a n^{- 1} 0 = 0$

The derivatives of the function $f (x) = \tan^{- 1} x$ are,

$f^{'} (x) = \frac{d}{d x} [\tan^{- 1} x] = \frac{1}{1 + x^{2}}$

When $x = 0$

$f^{'} (0) = \frac{1}{1 + 0^{2}}$

then also, localid="1653118971406" $f^{''} (x) = \frac{d}{d x} {(1 + x^{2})}^{- 1} = - {(1 + x^{2})}^{- 2} \cdot 2 x = - \frac{2 x}{1 + x^{2}}$

So,

$f^{''} (0) = - \frac{2 \cdot 0}{1 + 0^{2}} = 0$

Again,

$f^{'''} (x) = - 2 \frac{d}{d x} [x {(1 + x^{2})}^{- 2}] = - 2 [x \frac{d}{d x} {(1 + x^{2})}^{- 2} + {(1 + x^{2})}^{- 2} \frac{d}{d x} (x)] = - 2 [x \cdot - 2 {(1 + x^{2})}^{- 3} \cdot 2 x + {(1 + x^{2})}^{- 2} \cdot 1] = - 2 [- \frac{4 x^{2}}{{(1 + x^{2})}^{3}} + \frac{1}{{(1 + x^{2})}^{2}}]$

At $x = 0$

$f^{''} (0) = - 2 [\frac{4 {(0)}^{2}}{{(1 + 0^{2})}^{3}} + \frac{1}{{(1 + 0^{2})}^{2}}] = - 2$

Find the third-order Maclaurin series for the function.

The third-order Maclaurin series for the function $f (x) = \tan^{- 1} x$ is

$P_{3} (x) = 0 + 1 \cdot x + \frac{0}{2!} x^{2} + \frac{(- 2)}{3!} x^{3} P_{3} (x) = x - \frac{1}{3} x^{3}$

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Find third-order Maclaurin
or Taylor polynomial for the given function about the indicated
point.
$\tan^{- 1} x, x_{0} = 0$

Short Answer

Step by step solution

Given information

The third Maclaurin polynomial is given for any functionf with a derivative of order 3 at x=0

Find the derivatives of the function

Find the third-order Maclaurin series for the function.

One App. One Place for Learning.

Most popular questions from this chapter

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Find third-order Maclaurinor Taylor polynomial for the given function about the indicatedpoint.tan-1x,x0=0

Short Answer

Step by step solution

Given information

The third Maclaurin polynomial is given for any functionf with a derivative of order 3 at x=0

Find the derivatives of the function

Find the third-order Maclaurin series for the function.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Logic and Functions

Probability and Statistics

Applied Mathematics

Pure Maths

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

Find third-order Maclaurin
or Taylor polynomial for the given function about the indicated
point.
$\tan^{- 1} x, x_{0} = 0$