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# Suppose the tangent line to the curve $y = f(x)$ at the point $(2, 5)$ has the equation $y = 9 - 2x$. If Newton's method is used to locate a root of the equation $f(x) = 0$ and the initial approximation is $x_1 = 2$, find the second approximation $x_2$.

## $$\frac{9}{2}$$

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All right, listen. This problem So you have a point to calm If I was so special about it. Well, it's a part off. Nine months to X. What's so special about this one? Well, you have a graph f of X, and I don't know what the graph looks like at all, but I know that too. Come on, five is going to be a point on it. And the reason why I know that is because why equals to nine minus two X is a line that contains right here that is Taligent to ffx. So the graph that looks like this is tangent to the function f. So if f looks something like that, it could be one possible case. It could also look something like this. That's another option. So if I want to figure out where the where the route is going to be based on the Newton's approximation, I would say that all right, here is my approximated route. I'm not really satisfy it by it, because if I plugged it into the original function, it'll be right here. My next guest is going to be located right there. Oh, and it went to two again, and it went there again and it went there again, and it's actually not going to be the best guess possible. But that was just simply because I don't know what FedEx looks like. Okay. However, I at least know that my next guest is going to be located exactly right there. So that's what my ex to looks like because they call this point x one. So what's x two? It simply the X intercept off this equation right there. So X is equal to nine over to is going to be the position off x two, and this is how we answer this question.

University of California, Berkeley

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