10) Use Newton's method to derive the ancient and long standing divide and average method for finding square roots by hand. It is also known as the Babylonian method. √ can be approximated by iterating x¡+1 using an initial guess xo. Because √ is the positive solution to the equation x² - r = 0, derive the iterating formula above by simplifying the Newton's method formula using ƒ(x) = x² – r. Note that since r is a constant, d/dx(r) = 0 . - 2 1 +
10) Use Newton's method to derive the ancient and long standing divide and average method for finding square roots by hand. It is also known as the Babylonian method. √ can be approximated by iterating x¡+1 using an initial guess xo. Because √ is the positive solution to the equation x² - r = 0, derive the iterating formula above by simplifying the Newton's method formula using ƒ(x) = x² – r. Note that since r is a constant, d/dx(r) = 0 . - 2 1 +
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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Refer to image and show how to derive!
![**Newton's Method for Finding Square Roots**
**Overview:**
Newton's method can be used to derive the ancient and long-standing divide and average method for finding square roots manually. This technique is also known as the Babylonian method.
**Iterative Formula:**
The square root of \( r \) can be approximated by iterating the formula:
\[ x_{i+1} = \frac{1}{2} \left( x_i + \frac{r}{x_i} \right) \]
This process begins with an initial guess \( x_0 \).
**Understanding the Formula:**
Since \( \sqrt{r} \) is the positive solution to the equation \( x^2 - r = 0 \), we can derive the iterative formula by simplifying Newton's method formula using the function \( f(x) = x^2 - r \). It is important to note that because \( r \) is a constant, the derivative with respect to \( x \), \( d/dx(r) \), equals zero.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F787e229a-2368-453d-a111-0058be6bb6d9%2F3603011a-0056-41a0-ba4b-6c9a756160d2%2F3x1iwsp_processed.png&w=3840&q=75)
Transcribed Image Text:**Newton's Method for Finding Square Roots**
**Overview:**
Newton's method can be used to derive the ancient and long-standing divide and average method for finding square roots manually. This technique is also known as the Babylonian method.
**Iterative Formula:**
The square root of \( r \) can be approximated by iterating the formula:
\[ x_{i+1} = \frac{1}{2} \left( x_i + \frac{r}{x_i} \right) \]
This process begins with an initial guess \( x_0 \).
**Understanding the Formula:**
Since \( \sqrt{r} \) is the positive solution to the equation \( x^2 - r = 0 \), we can derive the iterative formula by simplifying Newton's method formula using the function \( f(x) = x^2 - r \). It is important to note that because \( r \) is a constant, the derivative with respect to \( x \), \( d/dx(r) \), equals zero.
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