03.13.08_Page_4

03.13.08_Page_4 - Z f x07y0)=0 Z=f(x0_y0 5_f 5 5f 5—x...

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Unformatted text preview: +Z- f( x07y0)=0 Z=f(x0 _y0)+ +5_f +5)! 5f 5—x( x0,y0)(x—x0) —(x0,y0)(y- yo) Differentiability A 2-variable function f(x, y) is differentiable at (x0, yo) if, when we zoom in on the graph athe point (x0, yo, f(x0, yo )), the graph eventually “looks like” a plane. 51% Dam—xv j—f More precisely. the function (x, y) —> f(x0,y0) —>— 5 x —(x0,yo )(y- yo) Should be a “good” approximation to f Still more precisely f is differentiable at (x0, yo) if 5 error<x,y)=f(x,y)—[f(x xo,y0)+—f(x x,y0)(x— x >+5i<xmyo)(y—yo) \$90 as (x,y)—>(x ,yo) V‘(x_x0)2+(y_y0)2 Diagram3 Useful criteria —f and— 5f 6x 5y e. g. f (x, y) = sin(xy) is differentiable everywhere f 1s differentiable at (a 19) provided that ...
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