MATH 241 -- Multivariable Calculus
Differentiability for 
September 21, 2007
We consider the function 
for 
(and 
This function has partial derivatives 
and 
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(1) |
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The graph together with its ``tangent plane''  |
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![plot3d([0, f(x, y)], x = -3 .. 3, y = -3 .. 3, grid = [60, 60]); 1](images/Diff_10.gif) |
The tangent plane does not seem to have anything to do with the shape of the graph
in this example. Indeed, it it is not even the case that 
as
Next, consider the function 
given by:
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(2) |
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![plot3d(g(x, y), x = -3 .. 3, y = -3 .. 3, grid = [60, 60]); 1](images/Diff_17.gif) |
Here again, 
and 
and unlike the first example 
But from
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![plot3d(`/`(`*`(`+`(g(x, y), 0)), `*`(sqrt(`+`(`*`(`^`(x, 2)), `*`(`^`(y, 2)))))), x = -3 .. 3, y = -3 .. 3, grid = [50, 50]); 1](images/Diff_22.gif) |
we suspect that the approximation is not ``especially good.''
Let's see what is true about the partial derivative functions for 
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)](images/Diff_25.gif) |
 |
(3) |
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)](images/Diff_27.gif) |
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(4) |
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![plot3d(gx(x, y), x = -3 .. 3, y = -3 .. 3, grid = [50, 50]); 1](images/Diff_29.gif) |
From this, we can see that 
does not exist, so 
is not
continuous at (0,0).
