Chapter 8: Q. 42 (page 680)

In Exercises 41–48 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" $= e [1 + x - 1 + \frac{(x - 1)^{2}}{2} + \frac{(x - 1)^{3}}{6} + \frac{(x - 1)^{4}}{24}] = e [x + \frac{(x - 1)^{2}}{2} + \frac{(x - 1)^{3}}{6} + \frac{(x - 1)^{4}}{24}]$

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In Exercises 41–48 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 :

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In Exercises 41–48 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

Mechanics Maths

Logic and Functions

Discrete Mathematics

Theoretical and Mathematical Physics

Geometry

Probability and Statistics

Study anywhere. Anytime. Across all devices.

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