a4 - x,y ) 6 = (0 , 0) , if ( x,y ) = (0 , 0) . b) g ( x,y...

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Math 237 Assignment 4 Due: Friday, Feb 4th * 1. Determine all points where f ( x,y ) = ( x 3 - y 4 x 2 + y 2 + 1 if ( x,y ) 6 = (0 , 0) 1 if ( x,y ) = (0 , 0) is differentiable. * 2. Let f ( x,y ) = ( xy 2 + y 3 x 2 + y 2 if ( x,y ) 6 = (0 , 0) 0 if ( x,y ) = (0 , 0) . a) Prove that f is continuous at (0 , 0). b) Determine all points where f is differentiable. * 3. A proctor is walking around the exam room. His position is given by ( x ( t ) ,y ( t )) = (cos t, sin t ). At position ( x,y ) his cellphone gets a signal strength of F ( x,y ) = e x y 2 . Using the Chain Rule, find the rate of change of the signal strength with respect to time at t = π/ 2. 4. Determine all points where the function is differentiable. a) f ( x,y ) = ( x 3 + y 3 x 2 + y 2 , if (
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Unformatted text preview: x,y ) 6 = (0 , 0) , if ( x,y ) = (0 , 0) . b) g ( x,y ) = | x | 1 / 2 . 5. Prove that if all of the second partial derivative of f are continuous at ( a,b ), then f is continuous at ( a,b ). 6. Consider the theorem, If f is dierentiable at ( a,b ), the f is continuous at ( a,b ). Prove that the converse is false by nding a function f which is continuous at (1 , 0), but not dierentiable at (1 , 0). Prove that your function satises the conditions. NOTE: Only * questions will be graded...
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