{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) = cos x - x/7

Accepted Answer

Hi, everyone, let's take a look at this practice problem. This problem says use Newton method to find all roots of F of X, which is equal to cosine of X minus X divided by 2, by first using analysis and graphing to choose good starting values. Now, this problem wants us to find all the roots of function F of X using Newton's method. So, we call for Newton's method that we can iterate using the expression XN + 1 is equal to XN. Minus F of exaben. Divided by F of X N. So, in order to use this iteration method, we need to first calculate our first derivative of our function F of X. We'll have F of X. is equal to the derivative with respect to X. Of the quantity of cosine of x. Minus X divided by 2 in quantity. So taking this derivative, this is going to be equal to. Recall that the derivative with respect to X of cosine of X is minus sine of X. And then I'll have minus the derivative with respect to X of X divided by 2, which is 1 divided by 2. That means that our iteration process is going to be X N + 1 is equal to XN minus the quantity of cosine of XN. Minus X N divided by 2 in quantity, divided by the quantity of minus sine of XN. -1 divided by 2 in quantity. So now what we want to do is actually graph our function of F of X to get a good starting value. So when we graph our function, we get something similar to what we see on the screen. And here we see that our function F of X is crossing the x-axis once, and it is occurring between the values of X equals 1 and X equal to 2. So we're going to choose a starting value of X equals to 1.5. So that means that we're going to have X zero Equal to 1.5. So we're now going to use this value to calculate X of 1. So we'll have X 1 is equal to. X sub zero, which is 1.5. Minus the quantity of cosine of 1.5. -1.5 divided by 2 in quantity, divided by the quantity of minus sign. Of 1.5. -1 divided by 2 in quantity. And we can evaluate this expression in our calculator. And when we do, we get 1.04640. So we'll now use this value to calculate our next x value. So we'll have X 2 is equal to 1.04640 minus the quantity of cosine of 1.04640. Minus 1.04640 divided by 2 in quantity, divided by the quantity of minus sign. Of 1.04640 minus 1 divided by 2 in quantity. And again, evaluating this expression in our calculator. We will get 1.02992. So again, we'll use this to calculate our next X value, so we'll have X of 3 is equal to. 1.02992. Minus the quantity of cosine of 1.02992 minus. 1.02992, divided by 2 in quantity, divided by the quantity of minus sign. Of 1.02992 minus 1 divided by 2 in quantity. Evaluating this expression, we will get. 1.02987. And performing another iteration. We will have X4. is equal to. 1.02987. Minus the quantity of cosine. Of 1.02987. -1.02987. Divided by 2 in quantity, divided by. The quantity of minus sign. Of 1.02987. -1 divided by 2 in quantity. And evaluating this expression, this is going to be equal to 1.02987. So here we've repeated the same value for X out to five decimal places in two successive um iterations. So that means that our final answer to this problem is going to be 1.02987. So I hope that this explanation has been useful, and I'll see you 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) = cos x - x/7

Key Concepts

Newton's Method

Preliminary Analysis

Graphing Functions

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