{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) = x²(x - 100) + 1

Accepted Answer

Hi everyone, let's take a look at this practice problem. This problem says to use Newton's method to find all roots of F of X. Which is equal to X cube multiplied by the quantity of X minus 50 in quantity plus 2, by first using analysis and graphing to choose good starting values. Stop iterating when two successive approximations agree to 5 decimal places. So, we need to find all the roots of our function F of X using Newton's method. So recall for Newton's method that we have the iterative process of X N + 1. Which is equal to X N. Minus F of X of N. Divided by F of X7N. So, in order to use Newton's method, I'm going to need to calculate our first derivative of our function F of X. So I'll have F of X. is equal to the derivative with respect to X. Of the quantity, and here in our function F of X, I'm going to multiply out our first two terms, so we'll have X to the 4th, minus 50, multiplied by X cubed. Plus 2 In quantity. And using our constant multiple rule, as well as our power rule and constant rule for derivatives, this is going to be equal to. Or execute Minus 150 X squad. So that means that our iteration process or Newton's method is going to be X N + 1 is equal to XN. Minus the quantity of X of n cued multiplied by the quantity of x n minus 50 in quantity plus 2 in quantity divided by the quantity of 4 x n cubed minus 150. X N squared. So now what we need to do is graph our function. And use it to to determine a good starting value. So when we graph our function, we see that our function crosses the x axis only once, so we only have one route, and it crosses between X equals to 0.3 and X equal to 0.4. So we're going to choose a starting value of x0 is equal to. 0.35. And so what we're going to do now is just repeat Newton's method until we get a repeating value for X of N. Out to 5 decimal places. So we'll use our initial starting value of 0.35 to calculate X1 X1, and that's going to be equal to 0.35 minus the quantity of 0.35. Cubed, multiplied by the quantity of 0.35 minus 50 in quantity plus 2 in quantity. Divided by The quantity of 4 multiplied by 0.35 cubed. -150, multiplied by 0.35 squared in quantity. So, evaluating this expression with our calculators, we will get 0.34293. So we'll now use this value in our next iteration. So we'll calculate X 2 is equal to 0.34293. Minus quantity of 0.34293. But Multiplying the quantity Of 0.34293. 50 in quantity. Plus 2 In quantity, divided by the quantity of 4 multiplied by 0.34293. Cubed minus 150 multiplying. 0.34293 squared. In quantity. So again, evaluating this expression in our calculator, we will get 0.34. 278. So again, we're going to repeat the iteration process, and we will calculate X 3. And that's gonna be equal to. 0.34278 minus the quantity of 0.34,278 cubed. Multiplying the quantity of 0.34278 minus 50 in quantity plus 2 in quantity, divided by The quantity of 4 multiplied by 0.34278 cubed. -150, multiplied by 0.34. 278 squared in quantity. So again, evaluating this expression in our calculator, this is going to be equal to. 0.34278. So here we repeated the same value for X in two successive iterations out to 5 decimal places. That means that our final answer for this problem is going to be equal to 0.34278. 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) = x²(x - 100) + 1

Key Concepts

Newton's Method

Preliminary Analysis

Graphing Functions

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