Chapter 8: Q. 26 (page 692)

In Exercises 23–32 we ask you to give Lagrange’s form for the corresponding remainder, $R_{4} (x)$
$\ln (1 + x)$

Short Answer

Expert verified

The required answer is $R_{4} (x) = \frac{1}{5 {(1 + c)}^{5}} x^{5}$

Step by step solution

Step 1. Given Information

The given function is $f (x) = \ln (1 + x)$

Step 2. Explanation

Using the Lagrange form for the remainder, we have,

$R_{n} (x) = \frac{f^{n + 1} (c)}{(n + 1)!} x^{n + 1} R_{4} (x) = \frac{f^{5} (c)}{5!} x^{5}$

Now, we will find the fifth derivative of the function,

$f^{1} (x) = \frac{1}{1 + x} f^{2} (x) = - \frac{1}{{(1 + x)}^{2}} f^{3} (x) = \frac{2}{{(1 + x)}^{3}} f^{4} (x) = - \frac{6}{{(1 + x)}^{4}} f^{5} (x) = \frac{24}{{(1 + x)}^{5}}$

Thus, we get,

$R_{4} (x) = \frac{\frac{24}{{(1 + c)}^{5}}}{5!} x^{5} R_{4} (x) = \frac{24}{120 {(1 + c)}^{5}} x^{5} R_{4} (x) = \frac{1}{5 {(1 + c)}^{5}} x^{5}$

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.

In Exercises 23–32 we ask you to give Lagrange’s form for the corresponding remainder, $R_{4} (x)$
$\ln (1 + x)$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Explanation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Theoretical and Mathematical Physics

Mechanics Maths

Logic and Functions

Decision Maths

Applied Mathematics

Study anywhere. Anytime. Across all devices.

In Exercises 23–32 we ask you to give Lagrange’s form for the corresponding remainder, R4(x)ln(1+x)

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Explanation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Theoretical and Mathematical Physics

Mechanics Maths

Logic and Functions

Decision Maths

Applied Mathematics

Study anywhere. Anytime. Across all devices.

In Exercises 23–32 we ask you to give Lagrange’s form for the corresponding remainder, $R_{4} (x)$
$\ln (1 + x)$