Q. 42 In Exercises 41鈥�48 find the fo... [FREE SOLUTION]

Chapter 8: Q. 42 (page 680)

In Exercises 41鈥�48 find the fourth Taylor polynomial $P_{4} (x)$ for the specified function and the given value of $x_{0}$ .
localid="1649476772910" $e^{x}, 1$

Short Answer

Expert verified

Ans: The fourth Taylor polynomial for the specified function is $= e [x + \frac{(x - 1)^{2}}{2} + \frac{(x - 1)^{3}}{6} + \frac{(x - 1)^{4}}{24}]$

Step by step solution

Step 1. Given information:

$e^{x}, 1$

Step 2. The fourth Taylor polynomial:

Since for any function $f$ with a derivative of order 4 at $x = 1$ , the fourth Taylor polynomial for $x = 1$ is given by

P_{4} (x) = f (1) + f^{'} (1) (x - 1) + \frac{f^{''} (1)}{2!} (x - 1)^{2} + \frac{f^{''} (1)}{3!} (x - 1)^{3} + \frac{f^{''''} (1)}{4!} (x - 1)^{4}

Therefore, first find the value of the function along with $f^{'} (x), f^{''} (x), f^{''} (x)$ and $f^{''''} (x)$ at $x = 1$

Step 3. Finding the fourth Taylor polynomial through derivative of order 4:

Thus, the value of the function at $x = 1$ is

$f (1) = e^{1} = e$

Therefore, all the derivatives of the function $f (x) = e^{x}$ is $e^{x}$ , so $f^{n} (1) = e$

Step 4. Substituting the derivative of order 4 in the fourth Taylor polynomial :

Therefore, the fourth Taylor polynomial for the function $f (x) = e^{x}$ is

$P_{4} (x) = e + e (x - 1) + \frac{e}{2!} (x - 1)^{2} + \frac{e}{3!} (x - 1)^{3} + \frac{e}{4!} (x - 1)^{4}$ localid="1649470393598" $= e [1 + x - 1 + \frac{(x - 1)^{2}}{2} + \frac{(x - 1)^{3}}{6} + \frac{(x - 1)^{4}}{24}] = e [x + \frac{(x - 1)^{2}}{2} + \frac{(x - 1)^{3}}{6} + \frac{(x - 1)^{4}}{24}]$

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

In Exercises 41鈥�48 find the fourth Taylor polynomial $P_{4} (x)$ for the specified function and the given value of $x_{0}$ .
localid="1649476772910" $e^{x}, 1$

Short Answer

Step by step solution

Step 1. Given information:

Step 2. The fourth Taylor polynomial:

Step 3. Finding the fourth Taylor polynomial through derivative of order 4:

Step 4. Substituting the derivative of order 4 in the fourth Taylor polynomial :

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Probability and Statistics

Mechanics Maths

Logic and Functions

Discrete Mathematics

Calculus

Study anywhere. Anytime. Across all devices.

91影视

In Exercises 41鈥�48 find the fourth Taylor polynomial P4xfor the specified function and the given value of x0.localid="1649476772910" ex,1

Short Answer

Step by step solution

Step 1. Given information:

Step 2. The fourth Taylor polynomial:

Step 3. Finding the fourth Taylor polynomial through derivative of order 4:

Step 4. Substituting the derivative of order 4 in the fourth Taylor polynomial :

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Probability and Statistics

Mechanics Maths

Logic and Functions

Discrete Mathematics

Calculus

Study anywhere. Anytime. Across all devices.

In Exercises 41鈥�48 find the fourth Taylor polynomial $P_{4} (x)$ for the specified function and the given value of $x_{0}$ .
localid="1649476772910" $e^{x}, 1$