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m2bsol-20C-fa2005

m2bsol-20C-fa2005 - Print Name Math 20C Midterm Exam 2...

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Print Name: Student Number: Section Time: Math 20C. Midterm Exam 2 November 21, 2005 Read each question carefully, and answer each question completely. Show all of your work. No credit will be given for unsupported answers. Write your solutions clearly and legibly. No credit will be given for illegible solutions. 1. (6 points) (a) Find the tangent plane approximation L ( x, y ) of the function f ( x, y ) = sin(3 x + 2 y ) + 2 at the point (2 , - 3). (b) Use the approximation above to estimate the value of f (2 . 2 , - 2 . 9). (a) f x ( x, y ) = 3 cos(3 x + 2 y ) , f y ( x, y ) = 2 cos(3 x + 2 y ) , then f x (2 , - 3) = 3 cos(6 - 6) = 3 , f y (2 , - 3) = 2 cos(6 - 6) = 2 , f (2 , - 3) = sin(6 - 6) + 2 = 2 . Then, the linear approximation L ( x, y ) = f x (2 , - 3) ( x - 2) + f y (2 , - 3) ( y + 3) + f (2 , - 3) is given by, L ( x, y ) = 3( x - 2) + 2( y + 3) + 2 . (b) The linear approximation of f (2 . 2 , - 2 . 9) is L (2 . 2 , - 2 . 9), and L (2 . 2 , - 2 . 9) = 3(2 . 2 - 2) + 2( - 2 . 9 + 3) + 2 = 3(0 . 2) + 2(0 . 1) + 2 = 2 . 8 , then, the result is L (2 . 2 , - 2 . 9) = 2 . 8. # Score 1 2 3 4 Σ

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2. (6 points) Find the absolute maximum and absolute minimum of f ( x, y ) = 1 + xy - 2 y - 1 4 x 2 in the closed triangular region with vertices given by (0 , 0), (0 , 1), and (2 , 0). Justify your answer.
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