MATH 20C Lecture 16

14.4 Differentiability, Tangent Planes, Linear Approximation

Recall

Theorem

If and both exist and are continuous on a domain
then
or

Corollary

Back to 14.4

Tangent Line for $z=f(x,y)$

Slope
Slope

Normal Vector to the Tangent Plane

So the tangent plane to the graph of at is

or tangent plane equation

Linearization of $f(x,y)$

Assume is defined in a region containing and exist
Then
is called the linearization of centered at

Differentiable

is differentiable at if
i.e. the linearization is a good approximation for near
i.e. If is close to
We say is differentiable on a domain if it is differentiable at all points in

Continuity Implies Differentiability

If and exist and are continuous on then is differentiable.

Linear Approximation

$\Delta f$

Let

Differentiable Form

Let
Define
The linear approximation can be written as