Math Problem Statement
Using Newton−Raphson method, derive a formula for finding the kth root of a positive number N and hence compute the value of (25)1/4.
Solution
To derive a formula using the Newton-Raphson method for finding the th root of a positive number , let's start with the general approach.
The Newton-Raphson method is used to find successively better approximations to the roots (or zeroes) of a real-valued function. The method is based on the formula:
For finding the th root of , we want to find such that . We can reformulate this as finding the root of the function:
The derivative of is:
Using the Newton-Raphson formula, we have:
Simplifying the right-hand side:
Combining the terms gives us:
Thus, the iterative formula to find the th root of using the Newton-Raphson method is:
Example: Compute
Here, and .
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Choose an initial guess, . Let's choose .
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Apply the iterative formula:
Let's perform a few iterations:
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Iteration 1:
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Iteration 2:
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Iteration 3:
Further iterations will yield results closer to the true value. After a sufficient number of iterations, we will approximate .
Would you like more details or have any questions?
Here are 5 questions you might want to explore next:
- How can the convergence of the Newton-Raphson method be proven?
- Can the Newton-Raphson method be applied to functions with multiple variables?
- How does the choice of initial guess affect the convergence of the Newton-Raphson method?
- What are some practical applications of the Newton-Raphson method in engineering?
- Can the Newton-Raphson method be used for functions that are not differentiable?
Tip: When choosing the initial guess for the Newton-Raphson method, select a value close to the expected root to ensure faster convergence.
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Math Problem Analysis
Mathematical Concepts
Newton-Raphson method
Root finding
Formulas
Newton-Raphson iterative formula for root finding
Theorems
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Suitable Grade Level
Advanced High School to College