Differentiability for Functions of Two Variables

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Differentiability for Functions of Two Variables

• Local Linearity

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Recall that when we zoom in on a sufficiently
nice function of two variables, we see a plane.
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What is meant by sufficiently nice?

• Suppose we zoom in on the function zf(x,y)
  centering our zoom on the point (a,b) and we see
  a plane. What can we say about the plane?
• The partial derivatives for the plane at the
  point must be the same as the partial derivatives
  for the function.
• Therefore, the equation for the tangent plane is

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In particular. . .The Partial Derivatives Must
Exist

• If the partial derivatives dont exist at the
  point (a,b), the function f cannot be locally
  planar at (a,b).

Example (as given in text) A cone with vertex
at the origin cannot be locally planar there, as
it is clear that the x and y cross sections are
not differentiable there.
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Not enough A Puny Condition
Whoa! The existence of the partial derivatives
doesnt even guarantee continuity at the point!
Suppose we have a function

• Notice several things
• Both partial derivatives exist at x0.
• The function is not locally planar at x0.
• The function is not continuous at x0.

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Directional Derivatives?

• Its not even good enough for all of the
  directional derivatives to exist!
• Just take a function that is a bunch of straight
  lines through the origin with random slopes. (One
  for each direction in the plane.)

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Directional Derivatives?

• Its not even good enough for all of the
  directional derivatives to exist!

Locally Planar at the origin? What do you think?
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Directional Derivatives?

• If you dont believe this is a function, just
  look at it from above.
• Theres one output (z value) for each input
  (point (x,y)).

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Differentiability
The function z f(x,y) is differentiable
(locally planar) at the point (a,b) if and
only if the partial derivatives of f exist and
are continuous in a small disk centered at (a,b).

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Differentiability A precise definition

• A function f(x,y) is said to be differentiable at
  the point (a,b) provided that there exist real
  numbers m and n and a function E(x,y) such that
  for all x and y
• and

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E(x) for One-Variable Functions
But E(x)?0 is not enough, even for functions of
one variable!
E(x) measures the vertical distance between f (x)
and Lp(x)
What happens to E(x) as x approaches p?