Question 5 27 points consider the function f x y e 2

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Question 5 (27 points) . Consider the function f ( x, y ) = e 2 x + y . (a) (8 points) Compute the gradient vector of f at the point (0 , 0) .
(b) (4 points) Find the directions in which the directional derivative of f at (0 , 0) has the value 1 .
(c) (12 points) Find a quadratic function g ( x, y ) that best approximates f ( x, y ) = e 2 x + y near (0 , 0) .
(d) (3 points) Use g to estimate f (0 . 2 , - 0 . 1) .
Math 230 Final Exam Solutions Winter Quarter 2014 Page 9 of 16 Question 6 (15 points) . Let f ( x, y ) = x 2 + 2 y 2 - 2 x - 4 y . Find the absolute maxi- mum and minimum values of f over the closed triangular region with vertices (0 , 0) , (0 , 2) , and (4 , 2) .
Math 230 Final Exam Solutions Winter Quarter 2014 Page 10 of 16 Question 7 (18 points) . Consider the surface z ( x 2 + y 2 ) = y 3 . (a) (15 points) Find an equation for the plane tangent to the surface at ( - 2 , 2 , 1) .
Math 230 Final Exam Solutions Winter Quarter 2014 Page 11 of 16
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