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# Of the infinitely many lines that are tangent to the curve $y = -\sin x$ and pass through the origin, there is one that has the largest slope. Use Newton's method to find the slope of that line correct to six decimal places.

## $y=-\cos (4.493409) x$

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questions. They're here. We are looking at that with X equal sign of vets therefore f frying equal. Negative, right is negative. Sign this equals negative co sign. No, thanks. What I wanted at the giving us crying of a what? He will Negative co sign. Hey, well, so far Slope are smooth, of course is by two minus by one over X two minus x one. So I would have a sign of a negative sign of a minus zero. Looking yours over a minus zero, which is negative. Sign of egg over a. And if I want a lying tangent two at a I'm gonna set this flow. Negative sign of a over a equal to my derivative. So I have negative co sign of a equals negative sign of a over a And then I'm gonna solve for a So I get a equals a holes sign over co sign, and that's actually get a positive sign. Overcoat size. A sign of a over co sign of a which, of course, is Qianjin of a Yeah, okay. And now we're used a Newton's nothing so g of X is gonna equal 10 of X. That's an X with eggs minus X. And then when I take the derivative of that I seek, it's where X minus one. So now I'm going to put into my graphing calculator this into by one and this into y two. And then I put Oh, and it's one ex my news g mine g over Prime, Which in my calculator waas Um why one over Why, too? So in why three I put X minus. Why one divide by y two. And then I got, uh, except one. He started with 4.5 from the grass. Thanks to you, I got four point or nine 36 14 That's a three like that. Or pointy or nine. 3410 four for we for nine, 34 03 And then you answer the question. I did f prime of four point or nine three or zero. And that gave me 0.217 23 Water

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