question 4 (3 points)
use one iteration of newtons method to approximate a solution of
\x^2 - 2 = 0\
starting with $x_0 = 1.5$.
enter $x_1$ rounded to four decimal places.

Question

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Newtons Method", "Derivative of Polynomials" ], "new_concepts": [], "current_concepts": [ "Newtons Method", "Derivative of Polynomials" ] } ## Define the function and its derivative Using the *Newtons Method* and *Derivative of Polynomials* knowledge points $$ \begin{aligned} f(x) &= x^2 - 2 \ f'(x) &= 2x \end{aligned} $$ ## State Newton's Method formula Using the *Newtons Method* knowledge point $$ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} $$ ## Calculate the first iteration Using the *Newtons Method* knowledge point $$ \begin{aligned} x_1 &= x_0 - \frac{x_0^2 - 2}{2x_0} \ x_1 &= 1.5 - \frac{1.5^2 - 2}{2(1.5)} \ x_1 &= 1.5 - \frac{2.25 - 2}{3} \ x_1 &= 1.5 - \frac{0.25}{3} \ x_1 &= 1.5 - 0.083333... \ x_1 &\approx 1.4167 \end{aligned} $$ Use one iteration of Newton's Method to approximate a solution of $x^2 - 2 = 0$ starting with $x_0 = 1.5$. Enter $x_1$ rounded to four decimal places. 1.4167 { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Calculus", "Newtons Method" ] }

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Question

Explanation:

Define the function and its derivative

State Newton's Method formula

Calculate the first iteration

Answer: