Newton's Method Essay

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Newton’s Method
Abstract
This topic covers the use of Newton’s Method in finding the zeroes of functions that are either extremely difficult or cannot otherwise be found using simple algebraic means. Practical applications of Newton’s Method will be demonstrated using manual means as well as a process employing the aid of MathCad software.

Introduction
Depending on its complexity, finding the roots (or zeroes) of a function can either be a really simple task on one end, or a hair-pulling mental marathon on the other. For example, consider the following function: f(x) = x – 1
Finding the root of this function can easily be done by first letting f(x) = 0, and then solving for “x.” f(x) = x – 1 = 0 then, x = 1
Many people could have probably found the root of the function above mentally without having to resort to fancy calculators or their mathematically-inclined friends. If everything were this simple, we would not be in this class. Unfortunately, as functions become more complex, finding the roots utilizing the method above becomes extremely difficult. Consider the following polynomial: f(x) = 4x3 + 2x + 1
While possible, the task of finding the roots of this function using only pen and paper could be enough to deter some from pursuing any degree related to mathematics. Imagine how high our blood pressures would rise when presented with a 15th, 38th, or 91st degree polynomial. Fortunately for us, calculus provides a way for us to find the roots of any given function using a graph and Newton’s Method, which is a process that repeatedly calculates approximate roots until these approximations converge into one value. When used with computer software such as MathCad, this method makes the task of root-finding a lot more tolerable and less time-consuming. Of course, learning an easier way to accomplish a difficult mathematical task may open the floodgates to more, increasingly difficult problems. You have been warned.
The General Idea
The concept behind Newton’s Method is fairly simple. The process basically requires us to first estimate where the root is and to use this estimation as a starting point. We then input this “x” value into a linear equation to find a better approximation of the root. This value is then inputted back into the same linear equation to achieve a more accurate approximation. The idea is to rinse and repeat, solving for the next “x” again and again until the resulting values converge, thus revealing the best approximation for the root. Repeated calculations are known as iterations, and Newton’s Method is one that calculates root approximations.
To obtain Newton’s Formula, we essentially use the linear equation of a tangent line to the curve at the point (xn, f(xn)), let it equal to zero, and symbolically solve for “x.”

Replacing numbered subscripts with “n” allows us to apply this formula to any number of iterations, eliminating the need to rewrite the equation with different values. With this formula, MathCad has what it needs to relieve us of any extensive thinking. In order to emphasize this point, we will first manually apply the formula to a function, demonstrating the “hard way” of doing things before moving on to what makes more sense with the resources available.
The “Hard Way”
Consider the function, f(x) = x7 + 4x3 + 12. Since Newton’s Formula calls for the use of the function’s derivative, we find that f ꞌ(x) = 7x6 + 12x2. Now, we graph the original function to find a starting “x” value.

Using the eyeball test, we can estimate the value at which the function crosses the axis to be around x ≈ -1.5. This is the estimation we will use for our first iteration. The resulting value will be used in the second iteration and so on, until the results converge.

So, we have the option of either doing this the frustratingly, tedious way and manually solve each iteration or, since we have access to MathCad, we
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