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.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

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

Decision Maths

Discrete Mathematics

Theoretical and Mathematical Physics

Calculus

Geometry

Logic and Functions

Study anywhere. Anytime. Across all devices.

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

Decision Maths

Discrete Mathematics

Theoretical and Mathematical Physics

Calculus

Geometry

Logic and Functions

Study anywhere. Anytime. Across all devices.

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