Chapter 8: Q. 38 (page 692)

In Exercises 41–48 in Section 8.2, you were asked to find the fourth Taylor polynomial $P_{4} (x)$ for the specified function and the given value of x 0. Here give Lagrange’s form for the remainder role="math" localid="1650647764004" $R_{4} (x)$ .
$e^{x}, 1$

Short Answer

Expert verified

Ans: $R_{4} (x) = \frac{e^{c}}{120} (x - 1)^{5}$

Step by step solution

Step 1. Given information.

given,

$e^{x}, 1$

Step 2. Consider the given function,

The Lagrange's form for the remainder is $R_{n} (x) = \frac{f^{(n + 1)} (c)}{(n + 1)!} {(x - x_{0})}^{n + 1}$ , where c is between $x_{0} a n d x$ .

Since $f (x) = e^{x}$ , so we have $f^{(n + 1)} (c) = e^{c} for every n \geq 0$

Also, the series of Taylor, so use $x_{0} = 1$

Therefore,

Since $f^{(5)} (x) = e^{x} and x_{0} = 1 then$

role="math" localid="1650649900503" $R_{4} (x) = \frac{f^{5} (c)}{5!} (x - 1)^{5}$

That is,

role="math" localid="1650649942321" $R_{4} (x) = \frac{e^{c}}{120} (x - 1)^{5}$

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In Exercises 41–48 in Section 8.2, you were asked to find the fourth Taylor polynomial $P_{4} (x)$ for the specified function and the given value of x 0. Here give Lagrange’s form for the remainder role="math" localid="1650647764004" $R_{4} (x)$ .
$e^{x}, 1$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Consider the given function,

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In Exercises 41–48 in Section 8.2, you were asked to find the fourth Taylor polynomial P4(x)for the specified function and the given value of x 0. Here give Lagrange’s form for the remainder role="math" localid="1650647764004" R4(x). ex,1

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Consider the given function,

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Probability and Statistics

Discrete Mathematics

Geometry

Decision Maths

Logic and Functions

Study anywhere. Anytime. Across all devices.

In Exercises 41–48 in Section 8.2, you were asked to find the fourth Taylor polynomial $P_{4} (x)$ for the specified function and the given value of x 0. Here give Lagrange’s form for the remainder role="math" localid="1650647764004" $R_{4} (x)$ .
$e^{x}, 1$