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MATH_200_200703_exam_3_with_solutions

# MATH_200_200703_exam_3_with_solutions - Exam Three MATH 200...

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Page 1 of 6 Exam Three MATH 200 Spring, 2008 Name_______________________________ Section____________ Show all your work on the exam paper, legibly and in detail, to receive full credit. No Calculators. 1. Consider the ellipsoid 4 21 3 2 2 2 2 z y x a)(10 pts) For what value of b is the point (1/2, b , 1/2) on the given ellipsoid? b)(10 pts) What is the direction of the normal vector at this point? Pg 1 (20 pts) Pg 2 (25 pts) Pg 3 (25 pts) Pg 4 (20 pts) Pg 5 (10 pts) Pg 6 (4 pts) Total

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Page 2 of 6 2a)(10 pts) Find an equation for the tangent plane to the surface given by the equation ) 1 ( ) ( 3 2 2 y x x z , at the point 2 , 1 , 1 P . 2b)(10 pts) Find the parametric equation for the normal line to the surface of part a), again at the point 2 , 1 , 1 P . 2c)(5 pts) Find a vector which is both perpendicular to the normal vector found in part b), and the unit vector (1,0,0).
Page 3 of 6 3(a)(15 points) The following function has exactly one critical point. What are the coordinates of that critical point? 4 3 3 , 2 2 y x y xy x y x f 3(b)(10 points) Using the second derivative test, show that the critical point that

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