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 $x \in I$ , 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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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

Discrete Mathematics

Calculus

Statistics

Mechanics Maths

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Geometry

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 $R_{n + 1} (x)$ ?