U2S1V05 Linear approximation.txt

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In our last video, we discussed how we can use tangent lines
as approximations to functions.
If you have the value of a function at a point,
and the value of the function's derivative at the same point,
then you can find the tangent line at that point.
And we know that the tangent line
is going to be really close to the function near that point.
This is called linear approximation,
because we're using the tangent line.
In the example we had, you think about the function
was the square root function.
And you can calculate that g of 100 equals 10,
so that's the height of this point.
And the derivative of g at x equals 100, that was 0.05.
So that's the slope of the tangent line.
And we wanted to estimate g of 104.
So 104, that means we've shifted our x value by 4.
So that's our delta x.
And our tangent line has gone up by 4 times the slope.
The slope as 0.05, so 0.2.
And that's what we estimate the change in the function's value
to be, or delta g.
The initial value of the function was 10,
it changes by roughly 0.2, so we're
going to get an approximation of 10.2 for g of 104.
Let's talk about doing this in general.
I'll erase some of this stuff, and let's say
that the point where we want an estimate is
x, and the point where we're starting is x0.
That means that the height here is g of x0.
And the slope, I'm going to denoted it by m, but m,
of course, is the derivative g prime of x0.
What we've been saying is that we can approximate
delta g by this vertical distance right here,
and that's just the slope, m, times the horizontal distance,
delta x.
Let me write that over here.
Delta g is approximately equal to m times delta x.
And let's just play with this a little bit.
We've got, on the left, g of x minus g of x0,
and that's approximately equal to-- now our m,
we said, was g prime at x0, and delta x is x minus x0.
Moving some things around, we have
g of x is approximately equal to g prime of x0 times
x minus x0 plus g of x0.
So here, on the left, we have just the formula
for the graph of the function.
And on the right, well, that's the formula for a line.
What line is it?
Well, it's our tangent line.
So we have that the graph is very close to the tangent line.
And this is all true when x is near x0.

Now, this all stems from the definition of the derivative.
We know that our m, our slope, is
defined to be the limit as delta x goes
to 0 of delta g over delta x.
And so another way of interpreting that
is that m is going to be very close to delta g over delta x
when delta x is small, when it's near 0.
And just rearranging, we get m times delta
x is approximately delta g.
And that's this over here that we started with.
So it all comes together.
This is linear approximation.
So we've got an example for you to try,
and then we want to start thinking
about just how good are these linear approximations anyway.
Are they too big?
Are they too small?
How can we tell?
So take some time and think about those,
and I'll talk to you in a little bit.