# Week3 - ECON 117 Mathematical Economics Spring 2008 Week 3...

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ECON 117 Spring 2008 Mathematical Economics Week 3 3.5 Total Derivatives and Partial Derivatives Def) For , , i.e., , (, ) zf x y = ,, xyz \ 2 : f \\ z x ( partial derivative of with respect to z x ) is the marginal rate of change of with respect to z x given is fixed, i.e., y 0 (, ) ( , ) lim ( , ) x x x x y f x y f xy xx Δ→ ∂+ Δ == ∂Δ Likewise, z y ( partial derivative of w.r.t. ) z y 0 ) lim ( , ) y y fxy y fxy y + Δ− Δ Also, dz ( total derivative of ) is the marginal change in when both z z x and change by and , i.e., y dx dy xy dz f dx f dy = ⋅+⋅ Q) . Find 22 2 zx x y =− + + 6 y (0,1) (0,1) (0,1) , , and zz dz ∂∂ Page 1 of 10

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ECON 117 Spring 2008 Mathematical Economics Page 2 of 10
ECON 117 Spring 2008 Mathematical Economics 3.6 Convexity and Concavity of a function Let , , , i.e., () yf = x n x \ y \ : n f \\ Def) is “ concave ” if = x 12 , n ∀∈ xx \ 121 1( ) ( 1 ) ff λλ λ +− ≥⋅ +− ⋅ xxx 2 ( ) f x , where 01 ≤∀≤ Ex) ( ), , , i.e., : x x y f =∈ \ x 1 y=f(x) x 2 λ 1- λ λ x 1 +(1- λ ) x 2 x y Q) Find (1 ) f and ()( 1 ) () f xf x Def) is “ strictly concave ” if = x , n \ ) ( 1 ) >⋅ 2 ( ) f x ,

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Week3 - ECON 117 Mathematical Economics Spring 2008 Week 3...

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