Local linearity and differentiability

- [Instructor] What we're
going to do in this video

is explore the relationship
between local linearity

at a point and
differentiability at a point.

So local linearity is this
idea that if we zoom in

sufficiently on a point,

that even a non-linear
function that is differentiable

at that point will actually look linear.

So let me show some examples of that.

So let's say we had y

is equal to

x squared.

So that's that there, clearly
a non-linear function.

But we can zoom in on a point,

and if we zoom sufficiently in,

we will see that it looks roughly linear.

So let's say we wanna zoom in
on the point one comma one,

so let's do that.

So zooming in on the point one comma one,

already it is looking
roughly linear at that point.

And this property of local linearity

is very helpful

when trying to approximate
a function around a point.

So for example, we could figure out,

we could take the derivative
at the point one one,

use that as the slope of our tangent line,

find the equation of the tangent line,

and use that equation
to approximate values

of our function around

x equals one.

And you might not need to do that

for y is equal to x squared,

but it could actually be very very useful

for a more complex function.

But the big takeaway here,

at the point one one,

it is displaying this
idea of local linearity,

and it is also
differentiable at that point.

Now let's look at another example

of a point on a function

where we aren't differentiable,

and we also don't see the local linearity.

So for example,

let's do the

absolute value of x,

and let me shift it over a little bit

just so that we don't overlap as much.

Alright, so the absolute
value of x minus one.

It actually is differentiable

as long as we're not at
this corner right over here,

as long as we're not at
the point one comma zero.

For any other x value,
it is differentiable,

but right at x equals one,

we've talked in other videos

how we aren't differentiable there.

And then we can use this
local linearity idea

to test it as well.

And once again, this is
not rigorous mathematics,

but it is to give you an intuition.

No matter how far we zoom in,

we still see this sharp corner.

It would be hard to construct
the only tangent line,

a unique line, that goes through
this point one comma zero.

I can construct an actual
infinite number of lines

that go through one comma zero

but that do not go through
the rest of the curve.

And so notice,

wherever you see a hard
corner like we're seeing

at one comma zero in this
absolute value function,

that's a pretty good indication

that we are not going to be differentiable

at that point.

Now let's zoom out a little bit,

and let's take another function.

Let's take a function where

the differentiability or the
lack of differentiability

is not because of a corner,

but it's because as we zoom in,

it starts to look linear,

but it starts to look
like a vertical line.

So a good example of that would be

square root of

let's say

four minus x squared.

So that's the top half of
a circle of radius two.

And let's focus on the
point two comma zero.

Because right over there,

we actually are not differentiable,

and if we zoom in far enough,

we see right at two comma zero

that we are approaching

what looks like a vertical line.

So once again,

we would not be differentiable
at two comma zero.

Now another thing I wanna point out,

all of these, you really
didn't have to zoom in too much

to appreciate that hey I got a corner here

on this absolute value function,

or at two comma zero, or
at negative two comma zero,

something a little bit
stranger than normal

is happening there, so maybe
I'm not differentiable.

But there are some functions
that we don't see as typically

in a algebra or precalculus
or calculus class,

but it can look like a hard corner

from a zoomed out perspective,

but as we zoom in once again
we'll see the local linearity,

and they are also
differentiable at those points.

So a good example of that,

let me actually get rid of some of these

just so that we can really zoom in.

Let's say y is equal to x

to the, and I'm gonna make
a very large exponent here,

so x to the 10th power.

It's starting to look at
little bit like a corner there.

Let's make it to the 100th power.

Well now it's looking even
more like a corner there.

Let me go to the 1,000th
power just for good measure.

So at this scale,

it looks like we have
a corner at the point

one comma zero.

Now this curve actually
does not go to the point

one comma zero.

If x is one,

then y is going to be one,

and we'll see that as we zoom in,

this what looks like a hard corner

is going to soften.

And that's good because this function

is actually differentiable
at every value of x.

It's a little bit more exotic

that what we typically see,

but as we zoom in,
we'll actually see that.

Let's just zoom in on what looks like

a fairly hard corner,

but if we zoom sufficiently enough,

even at the part that
looks like the hardest part

of the corner,

the real corner, we'll see
that it starts to soften

and it curves.

And if we zoom in sufficiently,

it will actually look like a line.

It's hard to believe when
you're really zoomed out,

and I'm going at the point that
really looked like a corner

from a distance.

But as we zoom on in,

we see once again this local linearity

that's a non-vertical line.

And so once again, this is true
at any point on this curve,

that we are going to be differentiable.

So the whole point here is,

sometimes you might have to zoom in a lot,

a tool like Desmos which
I'm using right now

is very helpful for doing that.

And this isn't rigorous mathematics,

but it's to give you an intuitive sense

that if you zoom in sufficiently,

and you start to see a
curve looking more and more

like a line,

good indication that
you are differentiable.

If you keep zooming in and it still looks

like a hard corner,

of if you zoom in and it looks like

the tangent might be vertical,

well then some questions
should arise in your brain.