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Fall 2000 Math22 Section E and G Exam# 2 Murat atmaca 1. Given the function z = p 25 - x 2 - y 2 , (a) Find and Sketch the domain of the function. (b) Describe the level curves for the given function and Sketch the level curves for the given k= 0, 3, 4, 21. 2. Show that f ( x,y ) = 1 2 ( e y - e - y )sin x satisfies Laplace’s Equation 2 f ( x,y ) ∂x 2 + 2 f ( x,y ) ∂y 2 = 0. 3. Find the limit, if it exists, or show that the limit does not exist. (a) lim ( x,y ) (0 , 1) arcsin( x y ) 1 + xy (b) lim ( x,y ) (0 , 0) y x 2 + y 2 4. Di±erentiate implicitly to find ∂z ∂y of xyz = cos( x + 2 y + 3 z ). 5. Find an equation of the tangent plane to the surface z = e 2 x +2 y at the point (0,0,1). 6. Let z = e x sin y and let x and y be functions of s and t with x(0,0)=0, y(0,0)=0, ∂x ∂s = 3, ∂y ∂s = 4, at (s,t)=(0,0). Find ∂z ∂s when (s,t)=(0,0). 7. Show that the following function is di±erentiable at (2,2), using the definition of di±erentiability. To help,
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Unformatted text preview: spaces have been provided, but you must show your work to indicate why you lled them in the way you did. f ( x,y ) = xy-5 y 2 . z = x + y + x + y 1 == 2 = 8. Find the directional derivative of the function g ( x,y,z ) = xye z at the point P(2,4,0) in the direction from P(2,4,0) to Q(0,0,0). Find f (2 , 4 , 0) and the maximum value of directional value. 9. Find the local maximum and minimum values and the saddle points of the function f ( x,y ) = x 3-6 xy + 8 y 3 . 10. Use the Lagrange multipliers to nd the indicated Extrema value of f ( x,y ) = x 2-y 2 subject to the constraint x-2 y + 6 = 0....
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