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Engineering Calculus Notes 249

Engineering Calculus Notes 249 - 237 3.3 DERIVATIVES A...

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3.3. DERIVATIVES 237 A third example is defined by a straightforward formula (no “cases”): f ( x,y ) = x 1 / 3 y 1 / 3 . Again, the function is constant along the coordinate axes, so both partials are zero. However, if we try to evaluate the limit in Equation ( 3.6 ) using any vector not pointing along the axes, we get d (0 , 0) f ( α −→ ı + β −→ ) = d dt vextendsingle vextendsingle vextendsingle vextendsingle t =0 bracketleftBig α 1 / 3 β 1 / 3 t 2 / 3 bracketrightBig ; since t 2 / 3 is definitely not differentiable at t = 0, the required linear map d (0 , 0) f ( α −→ ı + β −→ ) cannot exist. From all of this, we see that having the partials at a point −→ x 0 is not enough to guarantee differentiability of f ( −→ x ) at −→ x = −→ x 0 . It is not even enough to also have partials at every point near −→ x 0 —all our examples above have this property. However, a slight tweaking of this last condition does guarantee differentiability. We call f ( −→ x ) continuously differentiable at −→ x 0
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