Differentiability 1. Example 1: Given two functions in R2 : g(x, y) = 8 − 2x2 − 3y 2 and f (x, y) = 4 − x2 − y 2 . (a) Find the tangent plane P to the graph of g at the point (1, 2, −6); (b) Find the point on the graph of f which has tangent plane parallel to P ; 2. Example 2: Given a function f in R2 defined as f (x, y) = 1 if xy = 0, f (x, y) = 0 otherwise. (a) Sketch the graph of f ; (b) Find lim(x,y)→(0,0) f (x, y); (c) Is f continuous at (0, 0)? (d) Find the linear approximation of f at (0, 0); (e) Is f differentiable at (0, 0)? Properties of derivatives (Theorem 10 of section 2.5) 1. Given f : U ⊂ Rn → Rm is differentiable at X0 and c is a scalar. Then, h(x) = cf (x) is differentiable at X0 and Dh(X0 ) = cDf (X0 ); 2. Given f : U ⊂ Rn → Rm , g : U ⊂ Rn → Rm are differentiable at X0 and let h(x) = f (x) + g(x). Then, h is differentiable at X0 and Dh(X0 ) = Df (X0 ) + Dg(X0 ); 3. Given f : U ⊂ Rn → R, g : U ⊂ Rn → R are differentiable at X0 and let h(x) = f (x)g(x). Then, h is differentiable at X0 and Dh(X0 ) = g(X0 )Df (X0 )+f (X0 )Dg(X0 ); f (x) g(x) g(X0 )Df (X0 )−f (X0 )Dg(X0 ) ; g(X0 )2 4. Given f : U ⊂ Rn → R, g : U ⊂ Rn → R are differentiable at X0 and let h(x) = (g(x) 6= 0 in U ). Then h is differentiable at X0 and Dh(X0 ) =

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