Complete Two Iterations of Newton's Method for the Function Using

question 158

Multiple Choice

Complete two iterations of Newton's Method for the function using the initial guess . Round all numerical values in your answer to four decimal places.

• A)
• \(\begin{array}{|c|c|c|c|c|c|}
• \hline n & x_{n} & f\left(x_{n}\right) & f^{\prime}\left(x_{n}\right) & \frac{f\left(x_{n}\right)}{f^{\prime}\left(x_{n}\right)} & x_{n}-\frac{f\left(x_{n}\right)}{f^{\prime}\left(x_{n}\right)} \\
• \hline 1 & 1.3 & 1.0000 & 0.0068 & -3.6021 & 4.9021 \\
• \hline 2 & 4.9021 & 0.2675 & -0.2675 & 0.0068 & 4.8953 \\
• \hline
• \end{array}\)
• B)
• \(\begin{array}{|c|c|c|c|c|c|}
• \hline n & x_{n} & f\left(x_{n}\right) & f^{\prime}\left(x_{n}\right) & \frac{f\left(x_{n}\right)}{f^{\prime}\left(x_{n}\right)} & x_{n}-\frac{f\left(x_{n}\right)}{f^{\prime}\left(x_{n}\right)} \\
• \hline 1 & 1.3 & 0.2675 & -0.9636 & -0.2776 & 1.5776 \\
• \hline 2 & 1.5776 & 0.9636 & 0.0068 & -0.2776 & 1.8552 \\
• \hline
• \end{array}\)
• C)
• \(\begin{array}{|c|c|c|c|c|c|}
• \hline n & x_{n} & f\left(x_{n}\right) & f^{\prime}\left(x_{n}\right) & \frac{f\left(x_{n}\right)}{f^{\prime}\left(x_{n}\right)} & x_{n}-\frac{f\left(x_{n}\right)}{f^{\prime}\left(x_{n}\right)} \\
• \hline 1 & 1.3 & 0.9636 & -0.2675 & -3.6021 & 4.9021 \\
• \hline 2 & 4.9021 & 1.0000 & 0.0068 & 146.6399 & -141.7378 \\
• \hline
• \end{array}\)
• D)
• \(\begin{array}{|c|c|c|c|c|c|}
• \hline n & x_{n} & f\left(x_{n}\right) & f^{\prime}\left(x_{n}\right) & \frac{f\left(x_{n}\right)}{f^{\prime}\left(x_{n}\right)} & x_{n}-\frac{f\left(x_{n}\right)}{f^{\prime}\left(x_{n}\right)} \\
• \hline 1 & 1.3 & 0.2675 & -0.9636 & -0.2776 & 1.5776 \\
• \hline 2 & 1.5776 & -0.0068 & -1.0000 & 0.0068 & 1.5708 \\
• \hline
• \end{array}\)
• E)
• \(\begin{array}{l}
• \begin{array} { | c | c | c | c | c | c | }
• \hline n & x _ { n } & f \left( x _ { n } \right) & f ^ { \prime } \left( x _ { n } \right) & \frac { f \left( x _ { n } \right) } { f ^ { \prime } \left( x _ { n } \right) } & x _ { n } - \frac { f \left( x _ { n } \right) } { f ^ { \prime } \left( x _ { n } \right) } \\
• \hline 1 & 1.3 & 0.9636 & - 0.9636 & 0.0068 & 1.2932 \\
• \hline 2 & 1.2932 & 1.0000 & - 1.0000 & - 3.6021 & 4.8953 \\
• \hline
• \end{array}
• \end{array}\)

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Definitions:

Random Number Intervals

Ranges generated through a stochastic process, used in simulations and statistical sampling to represent uncertain outcomes.

Random Numbers

Numbers generated in a sequence such that each number's appearance is without pattern and unpredictable, often used in statistical sampling and computer simulations.

Computer Simulation

A method that uses algorithms and computational models to replicate complex systems and study their behavior under various conditions.

Cumulative Probability Distribution

A cumulative probability distribution shows the probability that a random variable will take a value less than or equal to a specific value, summarizing the probabilities in a distribution up to a certain point.