Practice Midterm 2 #1

Practice Midterm 2 #1 - Math 114 Sec 002, Spring 2007...

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Math 114 Sec 002, Spring 2007 Midterm 2 Instructor: David Galvin Answer key 1. Figures 1 through 4 below show the contour plots of the four functions a ( x,y ) = x 2 - y, b ( x,y ) = x + y, c ( x,y ) = x 2 - 2 y 2 and d ( x,y ) = x 2 + 2 y 2 . Match the functions with their contour plots. (No need to justify your answers) 2. Using the linearization of the function f ( x,y ) = ( x - 1) y + 3 at the point (2 , 1) , estimate . 9 4 . 1 = f (1 . 9 , 1 . 1) . Solution : Linearization is L f ( x,y ) = f (2 , 1) + f x (2 , 1)( x - 2) + f y (2 , 1)( y - 1) = 2 + 2( x - 2) + . 25( y - 1) . So . 9 4 . 1 = f (1 . 9 , 1 . 1) L f (1 . 9 , 1 . 1) = 2 + 2( - . 1) + . 25( . 1) = 1 . 825 . 3. Find an equation to the tangent plane to the surface z = sin( πx 2 y ) - cos( πxy 2 ) at the point (1 , 1 , 1) . Solution : An equation to the tangent plane is z - 1 = f x (1 , 1)( x - 1) + f y (1 , 1)( y - 1) where f ( x,y ) = sin( πx 2 y ) - cos( πxy 2 ) , or z - 1 = - 2 π ( x - 1) - π ( y - 1) . 4. Consider the function f ( x,y ) = x 2 - 4 xy + 4 y 2 - 5 at the point (1 , 1) . (a) In which direction is the rate of increase of the function the greatest?
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Practice Midterm 2 #1 - Math 114 Sec 002, Spring 2007...

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