B Just for fun: Cubic Graph Plot

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Desmos.com is highlighted as an effective online graphing tool for experimenting with the Newton-Raphson method, particularly in solving cubic equations. The discussion emphasizes the importance of accurately calculating x-intercepts and turning points on cubic graphs, suggesting that additional iterations may improve precision. A specific example is provided for finding square roots using the positive x-intercept of the function f(x)=x^2-a, with a comparison to a more accurate numeric value. Users are encouraged to explore the provided links for practical applications of these concepts. Overall, the thread showcases the utility of Desmos for mathematical experimentation and visualization.
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Cubic Graph plot on Desmos
Desmos.com is a great online graphing utility which I'm sure is familiar to many PF users. I wanted to experiment with the Newton-Raphson method using it so chose solution of cubic graphs as an example. The graph shows a variable cubic on which all turning points and intercepts are calculated and shown. x-intercepts by Newton-Raphson.

I'm sure readers will find parameter values where the x-intercepts are not that accurate - in that case add a couple more "g(g(.." iterations!

https://www.desmos.com/calculator/lejtdd8fws
 
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Here's another use of Newton-Raphson whereby the square root of a number is determined by finding the positive x-intercept of ##f(x)=x^2-a##. The intercept is marked on the graph but the side panel shows a more accurate numeric value of the root against parameter d where ##d=\sqrt{a}##. The start value for determination of the square root is the square root of the nearest square greater than a.

https://www.desmos.com/calculator/qbyev8tuqz
 
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