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Study guide 2 - MA 261 Fall 2010 Study Guide 2 1...

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Unformatted text preview: MA 261 - Fall 2010 Study Guide # 2 1. Relative/local extrema; critical points ( ∇ f = vector 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 ( vector x ) subject to a constraint g ( vector x ) = C , solve the system: ∇ f = λ ∇ g and g ( vector x ) = C . 2. Double integrals; Midpoint Rule for rectangle : integraldisplayintegraldisplay R f ( x,y ) dA ≈ m summationdisplay i =1 n summationdisplay j =1 f ( x i , y j ) Δ A ; 3. Type I region D : braceleftbigg g 1 ( x ) ≤ y ≤ g 2 ( x ) a ≤ x ≤ b ; Type II region D : braceleftbigg h 1 ( y ) ≤ x ≤ h 2 ( y ) c ≤ y ≤ d ; iterated integrals over Type I and II regions: integraldisplayintegraldisplay D f ( x,y ) dA = integraldisplay b a integraldisplay g 2 ( x ) g 1 ( x ) f ( x,y ) dy dx and integraldisplayintegraldisplay D f ( x,y ) dA = integraldisplay d c integraldisplay h 2 ( y ) h 1 ( y ) f ( x,y ) dxdy , respectively; Reversing Order of Integration (regions that are both Type I and Type II); properties of double integrals. 4. Integral inequalities: mA ≤ integraldisplayintegraldisplay D f ( x,y ) dA ≤ MA , where A = area of D and m ≤ f ( x,y ) ≤ M on D ....
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