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Solve the given problems.By Newton's method, find the smallest positive root of the equation $\tan x=2 x .$ Verify the result by using the zero feature of a calculator.

Calculus 1 / AB

Chapter 27

Differentiation of Transcendental Functions

Section 4

Applications

Derivatives

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so the equation were given is negative. X minus one equals the inverse tangent of X. Um, and we're supposed to start at X zero, equals one and find ex of two. So the equation we're gonna use is, um, acts of n plus one equals access end minus F of X over the derivative of F of X. And to use this equation because this is finding the roots, we set, um, one side of this equation equal to zero. So you do that by putting all the components on one side. So the inverse tangent of X plus X plus one equals zero that we plug this in for, um, off of acts and I should say, x event here. Um, so access en Plus one. Actually, we're going to say x of one because we're starting with X zero equals Come. And I also messed that up. That's X zeros one. Um, certain doing X of one equals one minus the inverse tangent of X zero plus ex severe plus one over. Um, the derivative of the tangent is one of her X squared plus one. Um, plus one. Um and so you plug in only there were plug in, um, one for the other exes. Heroes. And you get except one equals, um, negative. 0.85 six. Ah, And, um, and to find, except to actually just copied and pasted the number I got here. So, um, these were not the same number of just what points that used. Um, So now, to find exit, too, we plug in x of one, um, into our equation one. So we get ex of two equals negative. 0.85 69 minus the inverse tangent. Uh, zero point negative. What should make that clear? Negatives. You. 85 69 Oh, minus zero point a bubba Bob plus one. And I should have been Not the whole doesn't point, but in that room. Over what Over negatives. Your point. A 56 I'm sward plus one plus one. And when you plug this in, um, you will get that this part here the, um f of X over f prime of X. That's equal to negative 0.3586 and you want this to be a smallest possible Um, this is not the best, um, number that we want. This is a little big, but it's good enough because we were asked to find an exit, too. Um, and so if you plug all those numbers in excess of two equals negative zero point for 98 three.

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