Since for a polar squeeze test2mac2313 page 5 of 5 9

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TEST2/MAC2313 Page 5 of 5 ______________________________________________________________________ 9. (10 pts.) (a) Calculate z / x using implicit differentiation when 3 x 2 + 4 y 2 + tan( z ) = 12. Leave your answer in terms of x , y , and z . By pretending z is a function of the two independent variables x and y and performing partial differentiations on both sides of the equation above, we have x 3 x 2 4 y 2 tan( z ) [12] x 6 x sec 2 ( z ) z x 0 z x 6 x sec 2 ( z ) provided z is not an odd integer multiple of π /2. (b) Find all second-order partial derivatives for the function f ( x , y ) = x 3 y 4 . Label correctly. Since f x ( x , y ) = 3 x 2 y 4 and f y ( x , y ) = 4 x 3 y 3 , we have f xx ( x , y ) = 6 xy 4 , f yy ( x , y ) = 12 x 3 y 2 , and f xy ( x , y ) = f yx ( x , y ) = 12 x 2 y 3 . ______________________________________________________________________ 10. (10 pts.) (a) Compute the total differential dz when z = tan -1 ( xy ). dz f x ( x , y ) dx f y ( x , y ) dy y 1 ( xy ) 2 dx x 1 ( xy ) 2 dy . (b) Assume f (1,-2) = 4 and f ( x , y ) is differentiable at (1,-2) with f x (1,-2) = 2 and f y (1,-2) = -3. Obtain an equation for the plane tangent to the graph of f at P (1,-2,4). z f (1, 2) f x (1, 2)( x 1) f y (1, 2)( y ( 2)) 4 2( x 1) 3( y 2) ______________________________________________________________________ Silly 10 Point Bonus: (a) State the definition of differentiability for a function of two variables. [You may either state the definition found in the text or the one given by the instructor in class.] (b) Then using only the definition you stated, show the function f ( x , y ) x 2 y 2 is differentiable at any point (x,y) in the plane. Say where your work is, for it won’t fit here.
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