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23 Sample Exam 2

23 Sample Exam 2 - MATH 23 Sample Second Exam was April...

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MATH 23 Sample Second Exam was: April, 2003 NAME : Section (Last, First) 1. (10 points ) Sketch the level curves corresponding to the values -1, 0 and 2 for the function f ( x, y ) = y - 4 x 2 . 2. (10 points ) Let f ( x, y ) = ye xy . Find each of the following. (a) ∂f ∂x (b) ∂f ∂y (c) ∂f ∂y (0 , 2) 3. (10 points ) Find an equation of the tangent plane to the surface z = x 2 - 3 y 2 at the point ( - 3 , 2 , - 3) . 4. (15 points ) Let f be the function f ( x, y ) = cos(2 xy ) . (a) Find the directional derivative of f at the point (2 , π 8 ) in the direction of the vector a = < - 1 , 3 > = - i + 3 j. (b) Find the maximal rate of change of f at the point (2 , π 8 ) and the direction in which it occurs. 5. (15 points ) The function h ( x, y ) = xy - x 2 - y 2 + 3 y + 1 has only one critical point. Find the critical point and determine whether h has a local maximum, local minimum or saddle point. Give the value. 6. (10 points ) Use six squares with width 1 to find the estimate for the volume of the solid over the rectangle 0 x 3 , 0 y 2 that has height x 2 + y 2 at the point P ( x, y ) ,

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