Linearization calculus examples1/20/2024 ![]() ![]() Is going to be negative two down here, so this is equal Negative one over negative one minus one squared and so this So f prime of negative one is equal to, I could just write this is, negative, all right like this way, One with respect to x is zero, so we could say times one here if we like or we could just not write that 'cause it don't change the value. Respect of x is one, the derivative of negative Minus one with respect to x, well that's just going to be one, right. We're gonna multiply that times the derivative of x One times x minus one to the negative two and then Well that's just going to be, I'm just gonna use the power rule here, it's gonna be negative One of the negative one with respect to x minus one, Respect to x is equal to, so the derivative of x minus We can use the power rule and a little bit of a chain rule, so the derivative of f with Just write f of x again, so I'm gonna write it as x minus one to the negative one power, that makes it a little bit clearer that ![]() Slope of the tangent line and that's where the derivative is useful. Other point on the line and that point is going toīe your slope of your line. Look, if x one and y one are on the line, the slope between any ![]() Y one over x minus x one is equal to b because this comes as straight out of the idea. This point slope form like this sometimes, y minus So x one comma y one sits on that line some place. The slope times x minus the corresponding x one, Minus some y that sits on that line is equal to In terms of point slope where you could say, y Y is equal to mx plus b where m is a slope andī is the y intercept. So in order to find theĮquation of the tangent line, the equation of a line is Is the best approximation and all of your choices are, are lines, well essentially, they'reĪsking you to find the equation of the tangent So when people say, hey,įind the linear approximation of f around x equals negative one or they say, what is the following Or at least in this example is a very good linear approximation. X equals negative one, what's a decent, it isĪ, as good as you can get for a linear approximation Going to look something, something like that and as we can see, as we get further and further from, from x equals negative one, the approximation gets worse and worse but if we stay around And what we're essentially going to do is we're gonna approximate it with the equation of the tangent line. There, let me use a better color, so it's right over thereĪnd what we wanna do is approximate it with a line around that. Negative, is negative 1/2, which sticks us right over So what do we mean by that? Well let's look at this graph over here, on this curve, when x isĮqual to negative one, f of negative one is I wanna find a linearĪpproximation, approximation of f, of f, around, and you need to know where you're going to be approximating it, around x equals negative one. Let me write this down, I wanna find an approximation for, actually meant to be clear, I wanna find a linear approximation so I'm gonna approximate it with a line. It with a linear function especially around a certain value, and so what we're going to do is, we wanna find an approximation, F of x is equal to one over x minus one, this is its graph or at ![]() So there are situations where you have some type of a function, this is clearly a nonlinear function. ![]()
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