Q. 10 If f(x) is an nth-degree polynom... [FREE SOLUTION]

Chapter 8: Q. 10 (page 692)

If f(x) is an nth-degree polynomial and $P_{n} (x)$ is the nth Taylor polynomial for fat $x_{0}$ , what is the nth remainder $R_{n} (x)$ ? What is $R_{n + 1} (x)$ ?

Short Answer

Expert verified

Thus, the required remainders are $R_{n} (x) = \frac{f^{n + 1} (c)}{(n + 1)!} {(x - x_{0})}^{n + 1} a n d R_{n + 1} (x) = \frac{f^{n + 2} (c)}{(n + 2)!} {(x - x_{0})}^{n + 2}$

Step by step solution

Step 1. Given Information

The given data isIf f(x) is an nth-degree polynomial and $P_{n} (x)$ is the nth Taylor polynomial for fat $x_{0}$

Step 2. Explanation

Consider a function fthat can be differentiated (n+1) times in some open interval I that contains the point $x_{0}$ and $R_{n} (x)$ be the n remainder for f at $x = x_{0}$

Hence, for each point $x 鈭� I$ , there is at least one c between $x_{0} a n d x$ such that,

$R_{n} (x) = \frac{f^{n + 1} (c)}{(n + 1)!} {(x - x_{0})}^{n + 1} Now, (n + 1) t h remainder is, R_{n + 1} (x) = \frac{f^{n + 1 + 1} (c)}{(n + 1 + 1)!} {(x - x_{0})}^{n + 1 + 1} R_{n + 1} (x) = \frac{f^{n + 2} (c)}{(n + 2)!} {(x - x_{0})}^{n + 2}$

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If f(x) is an nth-degree polynomial and $P_{n} (x)$ is the nth Taylor polynomial for fat $x_{0}$ , what is the nth remainder $R_{n} (x)$ ? What is $R_{n + 1} (x)$ ?

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Explanation

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91影视

If f(x) is an nth-degree polynomial and Pn(x) is the nth Taylor polynomial for fat x0, what is the nth remainder Rn(x)? What is Rn+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

Pure Maths

Mechanics Maths

Logic and Functions

Decision Maths

Geometry

Applied Mathematics

Study anywhere. Anytime. Across all devices.

If f(x) is an nth-degree polynomial and $P_{n} (x)$ is the nth Taylor polynomial for fat $x_{0}$ , what is the nth remainder $R_{n} (x)$ ? What is $R_{n + 1} (x)$ ?