**Lecture 3.1:****Objective:** Determine the minimum degree $ (n) $ for the Taylor polynomial $ P_n(x) $ such that $ P_n(x) $ estimates $ f(x) = e^{2x} $ on the interval $[-2, 2]$, where $ P_n(x) $ is written at $ c = 1 $, with an accuracy of $ 10^{-11} $ (such that the exact error is no more than $ 10^{-11} $).### Key Points:1. **Function to Approximate:** \[ f(x) = e^{2x} \]2. **Interval of Interest:** \[ [-2, 2] \]3. **Center of the Taylor Polynomial:** \[ c = 1 \]4. **Desired Accuracy:** \[ 10^{-11} \]The task involves finding the smallest degree $ n $ of the Taylor polynomial $ P_n(x) $ such that it approximates the function $ f(x) = e^{2x} $ within the desired accuracy on the given interval. This process involves calculating the Taylor series expansion centered at $ c = 1 $ and ensuring that the error bound for the approximation is within $ 10^{-11} $. ### Transcription:#### Hint: Solve \[ \left| \frac{f^{(n+1)}(z)(x-1)^{n+1}}{(n+1)!} \right| \leq 10^{-11} \]- $ M? $- $ K? $### Description:In this mathematical expression, you are asked to solve an inequality involving derivatives and factorials. #### Breakdown of the Problem:1. $ f^{(n+1)}(z) $: Represents the $(n+1)^{th}$ derivative of some function $ f $ evaluated at some point $ z $.2. $ (x-1)^{n+1} $: This term is $(x-1)$ raised to the power of $ n+1 $.3. $(n+1)! $: The factorial of $ n+1 $.4. The absolute value of this fraction is required to be less than or equal to $ 10^{-11} $.#### Questions to address:- What is $ M? $- What is $ K? $These questions possibly relate to constants or bounds you may need to determine as part of solving the inequality.### Explanation:This type of problem is typically related to error estimation in numerical methods or series approximation, such as using Taylor series or Newton's method. The task likely involves finding values of $ M $ and $ K $ that satisfy the given inequality.

Answered: Image /qna-images/answer/d46e4f2a-5df9-4ce1-89cf-71c5588ba63d.jpg

Lec.3.1: Deter mme mmimum degree for Taylr polynomial, Pncel such that eacă) estimates 2メ * )= e D. on [-2,2J where Pn Cx) is written at cel, withee C=l; the accuracy of 10" Jexact error than 10 is no More

Lec.3.1: Deter mme mmimum degree for Taylr polynomial, Pncel such that eacă) estimates 2メ * )= e D. on [-2,2J where Pn Cx) is written at cel, withee C=l; the accuracy of 10" Jexact error than 10 is no More

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

**Lecture 3.1:**

**Objective:** Determine the minimum degree $ (n) $ for the Taylor polynomial $ P_n(x) $ such that $ P_n(x) $ estimates $ f(x) = e^{2x} $ on the interval $[-2, 2]$, where $ P_n(x) $ is written at $ c = 1 $, with an accuracy of $ 10^{-11} $ (such that the exact error is no more than $ 10^{-11} $).

### Key Points:
1. **Function to Approximate:**
\[
f(x) = e^{2x}
\]

2. **Interval of Interest:**
\[
[-2, 2]
\]

3. **Center of the Taylor Polynomial:**
\[
c = 1
\]

4. **Desired Accuracy:**
\[
10^{-11}
\]

The task involves finding the smallest degree $ n $ of the Taylor polynomial $ P_n(x) $ such that it approximates the function $ f(x) = e^{2x} $ within the desired accuracy on the given interval. This process involves calculating the Taylor series expansion centered at $ c = 1 $ and ensuring that the error bound for the approximation is within $ 10^{-11} $.

### Transcription:

#### Hint: Solve
\[ \left| \frac{f^{(n+1)}(z)(x-1)^{n+1}}{(n+1)!} \right| \leq 10^{-11} \]

- $ M? $
- $ K? $

### Description:

In this mathematical expression, you are asked to solve an inequality involving derivatives and factorials.

#### Breakdown of the Problem:

1. $ f^{(n+1)}(z) $: Represents the $(n+1)^{th}$ derivative of some function $ f $ evaluated at some point $ z $.
2. $ (x-1)^{n+1} $: This term is $(x-1)$ raised to the power of $ n+1 $.
3. $(n+1)! $: The factorial of $ n+1 $.
4. The absolute value of this fraction is required to be less than or equal to $ 10^{-11} $.

#### Questions to address:
- What is $ M? $
- What is $ K? $

These questions possibly relate to constants or bounds you may need to determine as part of solving the inequality.

### Explanation:

This type of problem is typically related to error estimation in numerical methods or series approximation, such as using Taylor series or Newton's method. The task likely involves finding values of $ M $ and $ K $ that satisfy the given inequality.

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