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math10c fall 09 real midterm2

# math10c fall 09 real midterm2 - MIDTERM II SOLUTIONS 1...

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MIDTERM II SOLUTIONS 1. Consider the points P , Q , R with coordinates (2 , 1 , 0) , (0 , 1 , 3) and (1 , 0 , 1) respectively. (i) Find the area of the parallelogram spanned by the vectors ~ PQ and ~ PR . (ii) Find the equation of the plane through P , Q , R . Answer: (i) We compute ~ PQ = ( - 2 , 0 , 3) , ~ PR = ( - 1 , - 1 , 1) . The cross product equals ~ PQ × ~ PR = (3 , - 1 , 2) . The area of the parallelogram is || ~ PQ × ~ PR || = p 3 2 + ( - 1) 2 + 2 2 = 14 . (ii) The plane has equation 3( x - 2) - ( y - 1) + 2 z = 0 = 3 x - y + 2 z = 5 . 2. Find the second order Taylor polynomial near (1 , - 1) for the function f ( x, y ) = x 3 y. Answer: We compute f (1 , - 1) = - 1 , f x = 3 x 2 y = f x (1 , - 1) = - 3 f y = x 3 = f y (1 , - 1) = 1 , f xx = 6 xy = f xx (1 , - 1) = - 6 , f xy = 3 x 2 = f xy (1 , - 1) = 3 , f yy = 0 . The Taylor polynomial is P 2 = - 1 - 3( x - 1) + ( y + 1) - 3( x - 1) 2 + 3( x - 1)( y + 1) . 3. Consider the function f ( x, y ) = x 4 y 3 . (i) Write down the equation of the tangent plane at the graph of the function at the point (1 , 1 , 1) . (ii) Write down an expression for the change, Δ z , in z = f ( x, y ) depending on Δ x and Δ y , the change in x and y, respectively, near the point x = y = 1 . Is the function f ( x, y )

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math10c fall 09 real midterm2 - MIDTERM II SOLUTIONS 1...

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