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# Linear approximation example (old)

Video transcript

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.