Linear approximation example (old)

Consider the function F of X is equal to the natural log of X squared. Let L of X be the local linearization of F at A is equal to E. What is a rule for L of X? Let's just make sure we understand what they're asking for, and I've copied and pasted this exact exercise on my little scratchpad so we can work through it. So, let's just visualize what f of x looks like. And what l of x is. What the local linearization, if we are centering it at x equals e. If we're saying that there's the set, when they say a that's just a convention for what are we approximating around. What the locality, what the x value that we are approximating around. So lets just, let me grab some points here just to visualize f of x. So let's say this is 1, this is 2, this is, actually let me spread them out a little bit more. So this is 1, this is 2, this is 3. F of 1 is gonna be 0. The natural log of 1 squared is just 0. And let's see, we could think of F of E, and that's actually where we're centering our linearization. We're gonna think about values in that locality. So E is roughly right over here. E is right, roughly right over here. And F of E is equal to natural log of E squared is, well, that's just going to be 2. So this is going to be, you're gonna have 1 and 2 and so, this is also going to be a point on the graph and it's going to look something like this. Gonna look something, something like that. That's what, that's the graph of Y is equal to F of X. y is equal to f of x. Now, what do they mean when they say the local, the local linearizaton of f at a, is equal to e. Well the convention here, is they use the letter a when they're talking about, when they're talking about local linearization. Say okay, this is what we're approximating around. So this our a value right over here. So we're essentially just gonna think about the slope of the tangent line at that point, or this point E comma F of E. So the L of X is going to describe, L of X is going to describe this line right over here. So this is, this right over here, is Y is equal to L of x, but we write it in a way that is well suited for local linearization, for approximating the value of the function around, well, for x values around e. So what do we mean by that? Well, the general way of describing L of x say that L of x is equal to the function evaluated. At that point that you're centering around at a, at a, so it's going this, it's going to be this value right over here. Plus the slope of the tangent line today, plus the slope of the tangent line today times how far, times the difference between the x value that you're approximating and a. And once again why does this make sense? Well let's say that you're trying to approximate half of this value right over here. So the function evaluated there would be right over there, but we want to use the local linearization, so we essentially want to say, what is l of this x value? So we want to figure out what is this right over here. Well you start at f of a, which is this. And then you multiply, or to that, you add the slope times the difference between these two values. X minus a is this distance. You multiply that times the slope, that's gonna give you your change in y. So your change in y, plus this y right over here, is going to give you the y value of that. Right over there. So this could be useful for approximating, let's say you want to approximate what, what the, what the natural log of e plus 0.2 squared is, and so this could, could be, could be valuable. But let's just work through this for this particular case. We know that, so we're saying that a is equal to e. So let's replace, this should be e, this should be e, and this should be e. And now what is f of e? Well this is this right over here is the natural log of e squared and now whats f-prime of e? Well let's just calculate that down here. So f-prime of x we can just use the chains rules. It's the derivative of the natural log of x squared with respect to x squared, so that's going to be 1 over x squared, times the derivative of x squared with respect to x. So times 2x and so this is going to be equal to 2 over x. So f prime of e is going to be two over e, and then times x minus e so that's, that's our rule, our definition for l of x and lets see which one of these lets see which one of these oh and of course we can simplify the natural log of e squared. The power that have to raise e to get to e squared, well this is of course equal to 2. So 2 plus 2 over e times x minus e. So 2 plus 2 over e times x minus e. It's that choice right over there. But just to make it clear, how this could be useful, let's say that we're trying to approximate, we're trying to approximate f of e plus 0.1. So, we're trying to approximate that. Now, like if I just put e plus 0.1 and I squared and I have to figure out the natural log of that without a calculator, that seems hard to me. But now we can use a local linearization. This is going to be approximately equal to 2 plus, 2 over e, times, well what's e plus 0.1, minus e. Well it's just gonna be 0, 0.1. And I guess you probably still need a calculator to figure this out, or you could just leave it here, anyway this is a pretty good approximation. For what the natural log of e plus 0.1 squared is going to be equal to. And let's just, let's just check our answer. So, it is that one right there. And we got it right.