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Unformatted text preview: Section 11.4: Tangent planes and linear approximation (concluded) Theorem 11.4.8: If f ( x , y ) is a continuous function that is differentiable w.r.t. x and w.r.t. y at ( x , y ) = ( a , b ) (i.e., if f x ( a , b ) and f y ( a , b ) are both defined), and if f 1 ( x , y ) and f 2 ( x , y ) both exist in a neighborhood of ( a , b ) and are both continuous at ( a , b ), then f is differentiable at ( a , b ), and the graph z = f ( x , y ) has a tangent plane at the point ( a , b , c ), where c = f ( a , b ), namely the plane z – c = f x ( a , b ) ( x – a ) + f y ( a , b ) ( y – b ). We say that the function L ( x , y ) = f ( a , b ) + f x ( a , b ) ( x – a ) + f y ( a , b ) ( y – b ) is the linearization of f at ( a , b ). The approximation f ( x , y ) ≈ L ( x , y ) (valid for ( x , y ) ≈ ( a , b )) is called the linear approximation to f at ( a , b ). We can phrase this in terms of the notion of differentials discussed in Honors Calc I; writing x – a as dx and y – b as dy , where dx and dy are independent variables, we can introduce a dependent variable dz defined as dz = f x ( a , b ) dx + f y ( a , b ) dy . In the notation of differentials, our remarks about linear approximation can be written compactly as ∆ z ≈ dz where ∆ z is defined as f ( x + dx , y + dy ) – f ( x , y )....
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