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Use Newton's method to approximate a root of the equation 4x² + 7x² + 2 = 0 as follows. Let x₁ = 3 be the initial approximation. x1 The second approximation 2 is and the third approximation x3 is (Although these are approximations of the root, enter exact expressions for each approximation.)

Use Newton's method to approximate a root of the equation 4x² + 7x² + 2 = 0 as follows. Let x₁ = 3 be the initial approximation. x1 The second approximation 2 is and the third approximation x3 is (Although these are approximations of the root, enter exact expressions for each approximation.)

Advanced Engineering Mathematics
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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**Using Newton's Method for Root Approximation** Consider the equation \(4x^7 + 7x^4 + 2 = 0\). We will use Newton's method to approximate a root, starting with an initial approximation. 1. **Initial Approximation:** - Given: \(x_1 = 3\) 2. **Second Approximation (\(x_2\)):** - Calculate the next approximation using the formula for Newton's method. 3. **Third Approximation (\(x_3\)):** - Further refine the approximation to better approach the root. Please enter the exact expressions for each approximation. These calculations are intended to improve accuracy iteratively.
Transcribed Image Text:**Using Newton's Method for Root Approximation** Consider the equation \(4x^7 + 7x^4 + 2 = 0\). We will use Newton's method to approximate a root, starting with an initial approximation. 1. **Initial Approximation:** - Given: \(x_1 = 3\) 2. **Second Approximation (\(x_2\)):** - Calculate the next approximation using the formula for Newton's method. 3. **Third Approximation (\(x_3\)):** - Further refine the approximation to better approach the root. Please enter the exact expressions for each approximation. These calculations are intended to improve accuracy iteratively.
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