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Newton's Method Calculator

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Example
Created on 2024-06-20Asked by Olivia Johnson (Solvelet student)
Use Newton's method to approximate the root of the function f(x)=x2−2 f(x) = x^2 - 2 starting from x0=1 x_0 = 1 .

Solution

To approximate the root of the function f(x)=x2−2 f(x) = x^2 - 2 using Newton's method starting from x0=1 x_0 = 1 , follow these steps: 1. Newton's method formula is: xn+1=xn−f(xn)f′(xn). x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}. 2. Calculate f(x) f(x) and its derivative f′(x) f'(x) : f(x)=x2−2, f(x) = x^2 - 2, f′(x)=2x. f'(x) = 2x. 3. Starting from x0=1 x_0 = 1 : x1=x0−f(x0)f′(x0)=1−12−22⋅1=1−−12=1+0.5=1.5. x_1 = x_0 - \frac{f(x_0)}{f'(x_0)} = 1 - \frac{1^2 - 2}{2 \cdot 1} = 1 - \frac{-1}{2} = 1 + 0.5 = 1.5. 4. Repeat the process for x1=1.5 x_1 = 1.5 : x2=x1−f(x1)f′(x1)=1.5−1.52−22⋅1.5=1.5−2.25−23=1.5−0.253≈1.4167. x_2 = x_1 - \frac{f(x_1)}{f'(x_1)} = 1.5 - \frac{1.5^2 - 2}{2 \cdot 1.5} = 1.5 - \frac{2.25 - 2}{3} = 1.5 - \frac{0.25}{3} \approx 1.4167. Therefore, after two iterations, the approximate root is x≈1.4167 x \approx 1.4167 . Solved on Solvelet with Basic AI Model
Some of the related questions asked by Abigail Hall on Solvelet
1. Use Newton's method to approximate the root of the equation f(x)=x3−2x2−5=0 f(x) = x^3 - 2x^2 - 5 = 0 starting with the initial guess x0=2 x_0 = 2 .2. Apply Newton's method to find the root of the function f(x)=ex−x−2=0f(x) = e^x - x - 2 = 0 with an initial guess x0=1x_0 = 1.
DefinitionActually, what we known in Newton's Method and others names (Newton-Raphson Method) is not powerful tool for digging or hunting out a solution for a real-valued function. From an initial guess x0​, the method uses the formula xn+1​=xn​−f′(x) f(x)​ and converges to a root. Answer: For instance, using the formula repeatedly to approximate the root of f(x)=x2−2 starting with x0​=1 with ever increasing accuracy.
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