2) Let's make a similar estimation rule for - If we consider the function f(x) = - a then when f(x) = 0 we have that x =! a) Use Newton's method to give the formula for getting a better approximation given an initial approximation. b) What would be a quick approximation for the following: 1 1 1 1 1 1 52' 23' 98' 103'997' 1006 c) Using those initial approximations use 1 iteration of the rule to get a better approximation.
2) Let's make a similar estimation rule for - If we consider the function f(x) = - a then when f(x) = 0 we have that x =! a) Use Newton's method to give the formula for getting a better approximation given an initial approximation. b) What would be a quick approximation for the following: 1 1 1 1 1 1 52' 23' 98' 103'997' 1006 c) Using those initial approximations use 1 iteration of the rule to get a better approximation.
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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![Certainly! Below is a transcription of the text as it would appear on an educational website:
---
## Approximations Explained with Calculus
### 2) Let's make a similar estimation rule for \(\frac{1}{a}\)
If we consider the function \(f(x) = \frac{1}{x} - a\), then when \(f(x) = 0\), we have that \(x = \frac{1}{a}\).
#### a) Use Newton’s method to give the formula for getting a better approximation given an initial approximation.
#### b) What would be a quick approximation for the following:
\[
\frac{1}{52}, \quad \frac{1}{23}, \quad \frac{1}{98}, \quad \frac{1}{103}, \quad \frac{1}{997}, \quad \frac{1}{1006}
\]
#### c) Using those initial approximations, use one iteration of the rule to get a better approximation.
#### d) Are there cases where a quick estimation rule can be put into words?
---
This content focuses on deriving a method for estimating reciprocal values using calculus, specifically by employing Newton’s method for improvement of initial approximations.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fff17d012-70ad-4782-80ac-5077bdfc24e1%2Fcf212a84-89d8-4832-8162-454ef1e894de%2Fongqa5_processed.png&w=3840&q=75)
Transcribed Image Text:Certainly! Below is a transcription of the text as it would appear on an educational website:
---
## Approximations Explained with Calculus
### 2) Let's make a similar estimation rule for \(\frac{1}{a}\)
If we consider the function \(f(x) = \frac{1}{x} - a\), then when \(f(x) = 0\), we have that \(x = \frac{1}{a}\).
#### a) Use Newton’s method to give the formula for getting a better approximation given an initial approximation.
#### b) What would be a quick approximation for the following:
\[
\frac{1}{52}, \quad \frac{1}{23}, \quad \frac{1}{98}, \quad \frac{1}{103}, \quad \frac{1}{997}, \quad \frac{1}{1006}
\]
#### c) Using those initial approximations, use one iteration of the rule to get a better approximation.
#### d) Are there cases where a quick estimation rule can be put into words?
---
This content focuses on deriving a method for estimating reciprocal values using calculus, specifically by employing Newton’s method for improvement of initial approximations.
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