{Use of Tech} Finding all roots Use Newton’s method to find al...

Question

{Use of Tech} Finding all roots Use Newton’s method to find all the roots of the following functions. Use preliminary analysis and graphing to determine good initial approximations.

f(x) = x⁵/5 - x³/4 - 1/20

Accepted Answer

Hello. In this video, we are going to be Newton's method to approximate all of the roots for the given function. Now, the function given to us is F of X equal to X fourth power divided by 4 minus X2 divided by 2 minus 1/8, and we want to use or choose starting points by analyzing the graph. So, just to quickly rewrite our given function, let's go ahead and rewrite our function by using coefficients of fractions. So we can rewrite F of X to be F of X is equal to 1/4 multiplied by X the 4th power, minus 12 X2 minus 1/8. Now, what I've done here is I went ahead and I used a graphing utility to plot the graph of this function, which I recommend you do as well. Notice that between the values of 1 and 2 and 1 and -2 there seems to be a root between these two values. So for Newton's method, we want to pick a starting point between 1 and 2, and -2 to -1. But what is Newton's method stating exactly? Well, Newton's method is a recursive formula that is given as the following XN + 1 is equal to XN minus F of XN divided by F of XN. So what this is saying is that we pick a starting point XN, then the next value of this of the approximation is going to equal to the value XN, the starting point we choose, minus the ratio of the function with XN plugged in, divided by its derivative plugged with X and plugged in. So, what we want to do is we want to determine our F of X of N and its derivative. We will go ahead and use the original function given to us as F of X sub N. But now what we need to do is we need to calculate its derivative. We will go ahead and calculate the derivative of F of X by using the power rule, and the derivative of F of X, which is F of X, will be X cubed minus X. So, what we can do is we can go ahead and rewrite the recursive formula as the following. XN + 1 is equal to XN minus the original function, which is 1/4 X minus 12 X2. 1/8 is all divided by X cubed minus X. Now what we are going to do is we are going to use Newton's method to approximate the roots. Now there are two routes that we are going to approximate. Let's go ahead and start by approximating the root that is between the values of -2 and -1. So, what is a good starting point that we can choose to approximate this route? Well, since it lies between the values of -2 and -1, why don't we let our starting value be -1.5? So we are going to let X X not our initial value, X N equal to -1.5. Then, by using the recursive formula, we get the next term, X1 equal to the current term, which is -1.5 minus the ratio of functions with the current term plugged in, which is going to be 14th multiplied by -1.5 rates to the fourth power, minus 12 multiplied by -1.5 squared, minus 1/8, all divided by -1.5 cubed. Minus -1.5. This is going to give us an approximate X1 value of approximately 1.49163. Now, what we want to go ahead and do is we want to go ahead and repeat this process until we get two matching values of egg sub N. Now, if we repeat this process, then the next term, X2, will approximate to be negative 1.49156. And if we use that value X2, we get an approximate value of X3 to be negative 1.49156. Notice that the values of X2 and XX3 match. So that means that the approximation of the root on the left hand side of the graph is going to be -1.49156. So that is going to be the first solution to this problem. Now, we are going to repeat this process, but for the root between the values of 1 and 2. Let's go ahead and choose the initial value X subzero to be 1.5, which is the average between these two roots. Then by the recursive formula, X 1 is going to equal to 1.5 minus 1/4 multiplied by 1.5 minus 1/2 multiplied by 1.5 squared, minus 1 divided by 8, all divided by 1.5 cubed minus 1.5. This is going to give us an approximate X1 value of 1.49163. And again, if we take this value X1 and plug it into the recursive formula, we get X 2 is approximately equal to 1.49. 856. And if we take this value and plug it into the cursive formula, we get X3 is equal to 1.49856. Notice that X2 and X3 match with each other, and that means that the approximate value of the root on the right hand side is going to be positive 1.49856. And these are going to be the two values of the roots by using Newton's method. So I hope this video helps you in understanding on how to approach this problem, and I will go ahead and see you all in the next video.

{Use of Tech} Finding all roots Use Newton’s method to find all the roots of the following functions. Use preliminary analysis and graphing to determine good initial approximations.

f(x) = x⁵/5 - x³/4 - 1/20

Key Concepts

Newton's Method

Preliminary Analysis

Graphing Functions

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