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Approximate the 4th root of 75 using Newtowns Method

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Approximate 754\sqrt[4]{75} using the Newton Raphson method

Newton's method is an iterative technique used to find successively better approximations to the roots of a real-valued function. To approximate the fourth root of 75 using Newton's method, we start by defining the function f(x)=x4−75f(x) = x^4 - 75 . Our goal is to find the root of this function, which corresponds to the value of x that makes f(x) = 0. Starting with an initial guess, we apply the iteration formula xn+1=xn−f(xn)f′(xn)x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} , where f'(x) is the derivative of f(x). For our function, f′(x)=4x3f'(x) = 4x^3 . By repeating this process, each iteration brings us closer to the actual fourth root of 75. This method showcases the power of calculus in solving complex numerical problems through approximation.

Posted by grwgreg 4 months ago

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