Newton's Method is used to find the root of an equation provided that the function f[x] is equal to zero. Newton Method is an equation created before the days of calculators and was used to find approximate roots to numbers. The roots of the function are where the function crosses the x axis. The basic principle behind Newton's Method is that the root can be found by subtracting the function divided by its derivative from the initial guess of the root. Newtons Method worked well because an initial guess was given to put into the equation. This is important because a wrong initial guess may give you the wrong root for the function. With Mathematica, a program for Newton's method can be produced and a graph of the function can be made. From the graph, the a good initial guess can be made.
Although Newton's Method works to find roots for many functions, it does have its disadvantages. The root sometimes cannot be found by using Newton's Method. The reason it sometimes cannot be found is because when the function is equal to zero, there is no slope to the tangent line.
As seen in experimentation's, it is important to select an initial guess close to the root because some functions have multiple roots. Failure to choose an initial value that is close to the root could result in finding a the wrong root or wasting a lot of time doing multiple iterations while getting close to the actual root.
On some occasions, the program cannot find a root to an initial guess that is placed into the program. In some instances Mathmatica could not find the root to the function, like if it is a parabola with its vertex is placed right on the y axis with its roots an equal distance away in both directions. In a case like this, the computer could not decide which root to work towards so it gave an indeterminate answer.
Although Newton's Method does have its disadvantages, it is very effective for finding the roots...