Q. 24 Find the fourth Maclaurin polyno... [FREE SOLUTION]

91影视

Chapter 8: Q. 24 (page 680)

Find the fourth Maclaurin polynomial $P_{4} (x)$ for the specified function:
$\ln (1 + x)$ .

Short Answer

Expert verified

The fourth Maclaurin polynomial is,

$P_{4} (x) = x - \frac{1}{2} x^{2} + \frac{1}{3} x^{3} - \frac{1}{4} x^{4}$ .

Step by step solution

Step 1. Given Information.

The function is,

$\ln (1 + x)$ .

Step 2. Describing the polynomial.

Let $f (x) = \ln (1 + x)$ .

Since for any function $f$ with a derivative of order $4$ at $x = 0$ , the fourth Maclaurin polynomial is,

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

Step 3. Finding the fourth Maclaurin polynomial

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

$f (0) = \ln (1 + 0) = \ln (1) = 0$

Finding the derivatives of the function $f (x) = \ln (1 + x)$ .

$f' (x) = \frac{d (\ln (1 + x))}{d x} = \frac{1}{1 + x} f' (0) = \frac{1}{1 + 0} = 1$

Also,

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

Also,

$f''' (x) = \frac{d (\frac{- 1}{{(1 + x)}^{2}})}{d x} = - \frac{d (\frac{1}{{(1 + x)}^{2}})}{d x} = \frac{2}{{(1 + x)}^{3}} f''' (0) = \frac{2}{{(1 + 0)}^{3}} = 2$

Also,

$f'''' (x) = \frac{d (\frac{2}{{(1 + x)}^{3}})}{d x} = \frac{- 6}{{(1 + x)}^{4}} f'''' (0) = \frac{- 6}{{(1 + 0)}^{4}} = - 6$

Thus the fourth Maclaurin polynomial is,

$P_{4} (x) = 0 + 1 . x + \frac{(- 1)}{2!} x^{2} + \frac{2}{3!} x^{3} + \frac{(- 6)}{4!} x^{4} = x - \frac{x^{2}}{2} + \frac{x^{3}}{3} - \frac{x^{4}}{4}$

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Find the fourth Maclaurin polynomial $P_{4} (x)$ for the specified function:
$\ln (1 + x)$ .

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Describing the polynomial.

Step 3. Finding the fourth Maclaurin polynomial

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Pure Maths

Probability and Statistics

Logic and Functions

Geometry

Decision Maths

Study anywhere. Anytime. Across all devices.

91影视

Find the fourth Maclaurin polynomial P4(x)for the specified function:ln1+x.

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Describing the polynomial.

Step 3. Finding the fourth Maclaurin polynomial

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Pure Maths

Probability and Statistics

Logic and Functions

Geometry

Decision Maths

Study anywhere. Anytime. Across all devices.

Find the fourth Maclaurin polynomial $P_{4} (x)$ for the specified function:
$\ln (1 + x)$ .