{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) = x/6 - sec x on [0,8]

Accepted Answer

Hi everyone. Let's take a look at this practice problem. This problem says to use New's method to find all real roots of F of X, which is equal to x divided by 4 minus sequin of X, when they closed interval from 0 to 10. By first performing a preliminary analysis with graphene, if desired, to choose good starting approximations. Stop iterating when two successive approximations agree to 5 decimal places. So we're actually going to start off by creating a graph of our function F of X. And in doing so, we see that we actually only have 2 roots on the closed interval from 0 to 10. And so we're gonna need to approximate these roots using Newton's method. So recall for Newton's method that we have an iterative process where XN + 1 is equal to XN minus F of XN. Divided by F of X of N. So, here, in order to use this um approximation, we need to first calculate our derivative. So, we'll have F of X. is equal to the derivative with respect to X of the quantity of X divided by 4 minus sequent of X. In quantity. And so taking this derivative, this is going to be equal to 1 divided by 4. Minus sequent of X. Multiplied by tangent of X. And so we can rewrite X N + 1 as being equal to XN minus the quantity of X divided by 4 minus sequent of X. In quantity, divided by the quantity of 1 divided by 4 minus sequent of X multiplied by tangent of X. So now we need to choose two good starting points. So, we'll start off with the root that is contained between X equal to 5 and X equal to 6. So for this, a good starting value would be for X 0 to be equal to 5.5. And so now we're going to apply Newton's method to calculate X1. So for exub one, It's going to be equal to 5.5 minus the quantity of 5.5 divided by 4, minus sequent of 5.5. In quantity, divided by the quantity of 1 divided by 4 minus sequent of 5.5. Multiplied by tangent of 5.5. And so now we can evaluate this expression using our calculator. In doing so, this is going to be equal to. 5.52181. So, we will now use this to calculate our next value for X. So we'll have X of 2 is equal to 5.52181. Minus the quantity of 5.52181 divided by 4. Minus the sequent. of 5.5. 2181. In quantity, divided by the quantity of 1 divided by 4 minus the sequent. Of 5.52181, multiplied by the tangent of 5.5. 2181. And so again, we can evaluate this expression using our calculator. So this is going to be equal to. 5.5. 2243. Itterating again, we'll get X of 3 is equal to 5.5,2243 minus the quantity of 5.5, 2243. Divided by or minus sequent. Of 5.52243. In quantity, divided by the quantity of 1 divided by 4 minus sequent. Of 5.5. 2243. Multiplied by the tangent of 5.52243. And so again, when we use our calculator to evaluate this expression. This is going to be equal to. 5.5 22 Or 3. And so here we've repeated the value out to 5 decimal places, and so that means that our first route is located at 5.52243. So we're gonna have to repeat this process for the second route, which is somewhere between X equal to 7 and X equal to 8. So we'll choose a starting value of X0 to be equal to 7.25. And so, iterating will have X 1 is equal to 7.25. Minus the quantity of 7.25 divided by 4, minus sequent of 7.25. In quantity, divided by the quantity of 1 divided by 4 minus the sequent of 7.25. Multiplied by the tangent of 7.25. And when we evaluate this expression using our calculator, we get 7.27 246. So, using this value in our next iteration, we'll have X of 2 is equal to. 7.27246 minus the quantity of 7.27246 divided by 4, minus. The sequent Of 7. 27246. In quantity, divided by the quantity of 1 divided by 4 minus the sequent. Of 7.27246, multiplied by the tangent of 7.27. 246. Evaluating this expression in our calculator, we will get. 7.27151. So using that value for the next iteration, we'll have X of 3 is equal to 7.27151. Minus the quantity of 7.27151 divided by 4. Minus the sequent. Of 7.27151. In quantity, divided by the quantity of 1 divided by 4 minus the sequence of 7.27151. Multiplied by the tangent of 7. 27151. And again, evaluating this in our calculator, this is going to be equal to. 7.27151. So again we have the same value repeated out to five decimal places. So this is the second route that we were looking for. So we have a route at X equal to 7.27151. So these are two answers for this problem. 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) = x/6 - sec x on [0,8]

Key Concepts

Newton's Method

Preliminary Analysis

Graphing Functions

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