9th - f x,y,z = x 2 yz-5 on the sphere x 2 y 2 z 2 = 1 7 Find an equation of the tangent line at P ◦ =(1-1 1 to the curve of intersection of the

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MATH 120 RECITATION QUESTIONS (WEEK 9) 1. Let f ( x,y ) = ( 2 xy x 2 + y 2 if ( x,y ) 6 = (0 , 0) 0 if ( x,y ) = (0 , 0) Show that D ~u f (0 , 0) exists in every direction ~u = h cosθ,sinθ i , 0 θ 2 π , but the formula D ~u f (0 , 0) = f (0 , 0) · ~u is true only for ~u = ± ~ i or ~u = ± ~ j . 2. Determine the points and the directions for which the directional derivative of f ( x,y ) = 3 x 2 + y 2 has its largest value if ( x,y ) is restricted to the points on the circle x 2 + y 2 = 1 . 3. Let f ( x,y ) be a function having a directional derivative in every direction at P = ( a,b ) . It is also given that for ~u = 1 2 ( ~ i - ~ j ) and ~v = 1 2 ( ~ i + ~ j ) , the directional derivatives are D ~u f ( a,b ) = 1 2 , and D ~v f ( a,b ) = 2. What is the maximum rate of increase of f at ( a,b ) ? 4. Find and classify all critical points of f ( x,y ) = x 2 y + xy 2 + 3 xy . 5. Find the minumum and the maximum values of f ( x,y ) = 3 x 2 + 3 y 2 + 2 xy + 1 on the closed disk x 2 + y 2 1. 6. Find the maximum and the minumum values of
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Unformatted text preview: f ( x,y,z ) = x 2 + yz-5 on the sphere x 2 + y 2 + z 2 = 1. 7. Find an equation of the tangent line at P ◦ = (1 ,-1 , 1) to the curve of intersection of the surfaces x 2 + y 2 = 2 and y 2 + z 2 = 2. 8. Use Lagrange multipliers to determine the point P ◦ on the plane 2 x-y + 3 z = 5 at which f ( x,y,z ) = 4 x 2 + y 2 + 3 z 2 has a local extremum. Is the local extremum you have found an absolute extremum? 9. Find the absolute extrema of the function f ( x,y ) = xye-x-y on the triangular region with vertices (0 , 0) , (0 , 4) , and (4 , 0). 10. Express the number 12 as the sum of three positive integers x , y , and z such that xy 2 z 3 is a maximum. 1...
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This note was uploaded on 04/30/2010 for the course MATHEMATIC MATH 119 taught by Professor Tor during the Spring '10 term at Middle East Technical University.

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