B a t 0 c a n 0 and d n 0 e also κ 0 test2mac2313

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(b) a T (0) = , (c) a N (0) = , and (d) N (0) = . (e) Also, κ (0)
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TEST2/MAC2313 Page 3 of 5 ______________________________________________________________________ 5. (10 pts.) (a) Find the limit. lim t 1 3 t 2 , ln( t ) t 2 1 ,sin( π 2 t ) (b) Find parametric equations for the line tangent to the graph of r ( t ) = (2 - ln( t )) i + t 2 j at the point where t 0 = 1. ______________________________________________________________________ 6. (10 pts.) Let . f ( x , y ) y x 2 (a) Obtain an equation for the level curve for f that passes through the point (-1,2). (b) Compute the value of the directional derivative D u f ( 1,2) when u is the unit vector in the plane that is in the direction of the gradient of f at (-1,2).
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TEST2/MAC2313 Page 4 of 5 ______________________________________________________________________ 7. (10 pts.) (a) Use an appropriate form of chain rule to find z / v when z = sin( x )sin( y ) when x = u + v and y = u 2 - v 2 . z v (b) Assume that F ( x , y , z ) = 0 defines z implicitly as a function of x and y . Show that if F / z 0, then z x F / x F / z . ______________________________________________________________________ 8. (10 pts.). (a) Use limit laws and continuity properties to evaluate the following limit. lim ( x , y ) ( 1/4, π ) ( xy 2 sec 2 ( xy )) (b) Evaluate the limit, if it exists, by converting to polar coordinates.
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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 . (b) Find all second-order partial derivatives for the function f ( x , y ) = x 3 y 4 . Label correctly. ______________________________________________________________________ 10. (10 pts.) (a) Compute the total differential dz when z = tan -1 ( xy ). dz (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). ______________________________________________________________________ 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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