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Irina18 [472]
4 years ago
14

Calculate two iterations of Newton's Method for the function using the given initial guess. (Round your answers to four decimal

places.) f(x) = x2 − 8, x1 = 2
Mathematics
1 answer:
pickupchik [31]4 years ago
7 0

Answer:

The first and second iteration of Newton's Method are 3 and \frac{11}{6}.

Step-by-step explanation:

The Newton's Method is a multi-step numerical method for continuous diffentiable function of the form f(x) = 0 based on the following formula:

x_{i+1} = x_{i} -\frac{f(x_{i})}{f'(x_{i})}

Where:

x_{i} - i-th Approximation, dimensionless.

x_{i+1} - (i+1)-th Approximation, dimensionless.

f(x_{i}) - Function evaluated at i-th Approximation, dimensionless.

f'(x_{i}) - First derivative evaluated at (i+1)-th Approximation, dimensionless.

Let be f(x) = x^{2}-8 and f'(x) = 2\cdot x, the resultant expression is:

x_{i+1} = x_{i} -\frac{x_{i}^{2}-8}{2\cdot x_{i}}

First iteration: (x_{1} = 2)

x_{2} = 2-\frac{2^{2}-8}{2\cdot (2)}

x_{2} = 2 + \frac{4}{4}

x_{2} = 3

Second iteration: (x_{2} = 3)

x_{3} = 3-\frac{3^{2}-8}{2\cdot (3)}

x_{3} = 2 - \frac{1}{6}

x_{3} = \frac{11}{6}

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