rev1 - Sample Problems for Exam One 1. Show that lim ( x,y...

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Unformatted text preview: Sample Problems for Exam One 1. Show that lim ( x,y ) (0 , 0) ( x 3 y 2 + y 6 ) / ( x 6 + y 4 ) does not exist. 2. Show that lim ( x,y ) (0 , 0) ( x 4- 2 x 3 y + y 4 ) / ( x 2 + y 2 ) = 0. 3. Show that if F ( x,y ) is differentiable, then the first partials exist. 4. Show that any differentiable function is continuous. 5. Let f ( x,y ) = xy/ ( x 2 + y 2 ) for ( x,y ) 6 = (0 , 0) and f(0,0)=0. Show that f x and f y both exist at (0,0) but that f is not continuous there. 6. Let F ( x,y ) = x 4 x 2 + y 2 for ( x,y ) 6 = (0 , 0) and F (0 , 0) = 0. Show that F is differentiable at (0,0). 7. Define the directional derivative D v F of a function F at a point P in direction v and show that if F is differentiable, then D v F = F ( P ) v . 8. Let F x y = x 2 + y 2 2 xy 2 x + y . Find F 1 3 and give the linear approximation L x y about the point. Find the directional derivative D v F at 1 3 in the direction v = 4 / 5 3 / 5 ....
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rev1 - Sample Problems for Exam One 1. Show that lim ( x,y...

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