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## Differentiability for Functions of Two Variables

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**Differentiability for Functions of Two Variables**Local Linearity**Recall that when we zoom in on a “sufficiently nice”**function of two variables, we see a plane.**What is meant by “sufficiently nice”?**Suppose we zoom in on the function z=f(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**In particular. . .The Partial Derivatives Must Exist**If the partial derivatives don’t 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.**Not enough: A Puny Condition**Whoa! The existence of the partial derivatives doesn’t even guarantee continuity at the point! Suppose we have afunction • Notice several things: • Both partial derivatives exist at x=0. • The function is not locally planar at x=0. • The function is not continuous at x=0.**Directional Derivatives?**It’s 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.)**Directional Derivatives?**It’s not even good enough for all of the directional derivatives to exist! Locally Planar at the origin? What do you think?**Directional Derivatives?**If you don’t believe this is a function, just look at it from “above”. There’s one output (z value) for each input (point (x,y)).**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).**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**E(x) for One-Variable Functions**E(x) measures the vertical distance between f (x) and Lp(x) But E(x)→0 is not enough, even for functions of one variable! What happens to E(x) as x approaches p?