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Problem 4 Medium Difficulty

For each initial approximation, determine graphically what happens if Newton's method is used for the function whose graph is shown.
(a) $ x_1 = 0 $ (b) $ x_1 = 1 $ (c) $ x_1 = 3 $
(d) $ x_1 = 4 $ (e) $ x_1 = 5 $

Answer

(a)Newton's method fails
(b)Newton's method fails
(c)Newton's method fails
(d)Newton's method fails
(e)Newton's method works

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Video Transcript

right? All right. Question or they only give us a graph. So I am going to attend to draw a graph. And here, my graph one. Really? All right. My grand Put something that So now, eh? Where I have experts, I have exit one equals zero. I'm gonna drop by tangent lines and see it. See that it's not converging at zero. And the method fails. See, because here we're estimating the real root. So we're actually looking for where crosses the x axis, and it doesn't cross the excesses there, so we would expect it to fail. The part B. We're looking at X equals one. That was one right here. You have horizontal and method fails, as we would expect it to fail her part. See, we're looking at X equals three. Yeah. Now, my graph isn't around to scale, so it would not, but basically, we're converging. And its field, that's a B X equals four. We have words out of line and nothing feel. And last five, we can see it crosses the x axis that we're not expecting it to fail. Um, I mean back purple. So I'm gonna five here looking at my worlds on line and they are all converging around six. So Newton's Method 16 and we have converges to six.