261-S12-sg2

# 261-S12-sg2 - MA 261 Spring 2011 Study Guide 2 f f 0...

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MA 261 - Spring 2011 Study Guide # 2 0. Gradient vector for f ( x,y ): f ( x,y ) = ∂f ∂x , ∂f ∂y , properties of gradients; gradient points in direction of maximum rate of increase of f ; The maximum value of the directional derivative is equal to ∥∇ f ; f ( x 0 ,y 0 ) level curve f ( x,y ) = C and, in the case of 3 variables, f ( x 0 ,y 0 ,z 0 ) level surface f ( x,y,z ) = C : 0 (x ,y ) x y f(x,y,z)=C x y (x ,y ,z ) 0 0 0 n = 0 n = z f(x ,y ) 0 0 f(x,y)=C f(x ,y ,z ) 0 0 0 1. Relative/local extrema; critical points ( f = 0 or f does not exist); 2 nd Derivatives Test: A critical points is a local min if D = f xx f yy f 2 xy > 0 and f xx > 0, local max if D > 0 and f xx < 0, saddle if D < 0; absolute extrema; Max-Min Problems; Lagrange Multipliers: Extremize f ( x ) subject to a constraint g ( x ) = C , solve the system: f = λ g and g ( x ) = C .

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261-S12-sg2 - MA 261 Spring 2011 Study Guide 2 f f 0...

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