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Problem 39 Hard Difficulty

Use Newton's method to find the coordinates, correct to six decimal places, of the point on the parabola $ y = (x - 1)^2 $ that is closest to the origin.

Answer

$(0.410245,0.347810)$

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Video Transcript

All right, Question 39. I'm gonna start with the distance formula because I want to find the distance, the smoke closest distance from that point to the equation. And they give me X and f of X. So my distance, of course. Close, X minus x so x weird, less X minus one, Which is my f of X, my dear. Um, in here that would be squared, actually have you the fourth, because it would be squared. So that is to the fourth power, which means p weird equal to X where no X minus one to the fore erratic sign Come into play later on. We'll actually solve Were the equation at the end then. So for right now, I'm gonna let my f of X equal be So now I have f crying and I'm gonna take the derivative of the square. So I have two x mine its core X minus one to the third power times the derivative of the inside, which is just one. And then I have a double prime equals two minus well X minus one squared. So I'm gonna put tch um uh c in my graphing calculator, I'm gonna put this equation in my wife one. This is my wife to my wife three. And then my f n plus one equals X minus and crying over F double prime. This is gonna be my why four, Which is really my thanks. Minus y two over. Why three in my graphing calculator. So now I plug in, uh, ex of one into my woman, uh, and you ask. I changed it to ask, and I love you. Wait five. And I look at wife, of course. And then x two gives me white four. So then I plug in 0.4, which gives me no movie at my wife toward my graphing calculator, which is the equation X minus f prime over f Devil prime and accept. Read gives me point or wait. We wore one hero, 1 to 7 except four equals except five which equal white for one. The road to were five. So now I plug in or 102 were five into my function, which is in my wife one position. And that gives me why to eight 92 Remember, that equals c squared. I want to take the square root of that to find my answer. Those square roots. Let me this square root, huh? 89 2734 Yeah. 153 78 or one. So my answer. Um, one. The road to buy equals 0.53 781