251_1 Multivariable Calculus Exam

# 251_1 Multivariable Calculus Exam - f x,y = x 2-4 xy 2 y 2...

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Multivariable Calculus Practice Exam Saturday, June 22, 2002 Chris Long 1. Find all partial derivatives for (a) f ( x, y ) = 2 xy x 2 + y (b) f ( x, y, z ) = ln( x 2 + yz 3 ) 2. Given w = yx sin( z ) , x = u + v, y = u - v, z = v 2 find ∂w ∂u and ∂w ∂v . 3. Given z = arctan( xy ) , x = t 2 , y = t 3 find dz dt . 4. If f ( x, y ) = 3 x 2 - 2 y 2 , find the directional derivative at (1 , - 3) from (1 , - 3) toward (2 , - 2). 5. Find the maximum value of f ( x, y, z ) = xyz if 6 x + 4 y + 3 z = 24. 6. Find the minimum value of
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Unformatted text preview: f ( x,y ) = x 2-4 xy + 2 y 2 if x 2 + 2 y 2 ≤ 1. 7. Let z = f ( x,y ) = x 2 y 3-2 xy . Find the total di±erential dz and the tangent plane to f ( x,y ) at the point (1 , 3), and the approximate value of f (1 . 1 , 3 . 2) using either of these. 8. Determine the local extrema of f ( x,y ) = xy 2-x 2 y + x-y ....
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