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Unformatted text preview: SOLUTIONS TO 18.02 MIDTERM #1 BJORN POONEN October 1, 2009 1) Find the equation of the tangent plane to the graph of f ( x,y ) := x 2 y 3 y 2 at the point where x = 2 and y = 1. Solution: We have f x = 2 xy and f y = x 2 6 y . At (2 , 1), we have f = 1, f x = 4, f y = 2. So the tangent plane at (2 , 1 , 1) is z 1 = 4( x 2) + ( 2)( y 1) . Alternative forms of the answer include z = 4 x 2 y 5 and 4 x 2 y z = 5 . 2) Given that x 1 ,x 2 ,x 3 ,y 1 ,y 2 ,y 3 are real numbers satisfying the conditions x 2 1 + x 2 2 + x 2 3 = 4 y 2 1 + y 2 2 + y 2 3 = 9 , what is the range of possible values of x 1 y 1 + x 2 y 2 + x 3 y 3 ? Solution: Equivalently, given that ~x is a vector of length 2, and ~ y is a vector of length 3, what is the range of possible values of ~x · ~ y ? The angle θ between ~x and ~ y can lie anywhere in the range [0 ,π ], so cos θ can lie anywhere in [ 1 , 1], so ~x · ~ y =  ~x  ~ y  cos θ = 6 cos θ can lie anywhere in the range [ 6 , 6]. 3) Let f ( x,y ) = x 3 + 3 xy y 3 . (a) (15 pts.) Find all points (if any) at which f ( x,y ) has a local minimum. Solution to (a): We have f x = 3 x 2 + 3 y f y = 3 x 3 y 2 . These exist everywhere, so at a local minimum, both f x and f y must be 0. This leads to 3 x 2 + 3 y = 0 3 x 3 y 2 = 0 1 or equivalently y = x 2 x = y 2 , so x = ( x 2 ) 2 = x 4 , so x = 0 or 1 = x 3 . The first case....
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This note was uploaded on 03/08/2010 for the course MATH 18.02 taught by Professor Auroux during the Fall '08 term at MIT.
 Fall '08
 Auroux
 Multivariable Calculus

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