math-2011-note8 - 12 Calculus of Several Variables (ch.14)...

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Unformatted text preview: 12 Calculus of Several Variables (ch.14) 12.1 Partial Derivative Definition 25 (p.300) . Let F : R k → R . The partial derivative of f with respect to x i at x = ( x 1 , ··· , x k ) is defined as ∂ F ∂ x i ( x ) = lim h → F ( x 1 , ··· , x i + h , ··· , x k )- F ( x 1 , ··· , x i , ··· , x k ) h , if the limit exists. Example 3. F ( x 1 , x 2 ) = 3 x 2 1 x 2 2 + 4 x 1 x 3 2 + 7 x 2 . ∂ F ∂ x 1 ( x ) = ? , ∂ F ∂ x 2 ( x ) = ? Economic Example: Marginal Product Let Q = F ( K , L ) be a production function. Then the partial derivative ∂ F ∂ L ( K , L ) is the rate at which output changes with respect to labor L , keeping K fixed at K . If labor increases by Δ L , then output will increase by Δ Q ≈ ∂ F ∂ L ( K , L ) Δ L . Setting Δ L = 1, we see that ∂ F ∂ L ( K , L ) estimates the change in output due to one unit increase in labor. Hence, ∂ F ∂ L ( K , L ) is called the marginal product of labor. Example 4 (Cobb-Douglas production function) . Q = 4 K 3 4 L 1 4 , ∂ F ∂ L ( 10 , 60 ) = ? Figure 21: Graph of a Cobb-Douglas function with or without K fixed 33 12.2 Approximation by Differentials 1 Since f ( x + h )- f ( x ) h ≈ f prime ( x ) for small h , one can write f ( x + h ) ≈ f ( x )+ f prime ( x ) h . Write x = x +( x- x ) . When x- x is small we can write f ( x ) = f ( x +( x- x )) ≈ f ( x )+ f prime ( x )( x- x ) . Hence, f ( x ) can be approximated by a translationof a linear function f ( x )+ f prime ( x )( x- x ) around x . Figure 22: Approximation by a (translated) linear function 12.3 Approximation by Differentials 2 F ( x + Δ x , y )- F ( x , y ) ≈ ∂ F ∂ x ( x , y ) Δ x F ( x , y + Δ y )- F ( x , y ) ≈ ∂ F ∂ y...
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This note was uploaded on 03/26/2012 for the course ECON 205 taught by Professor Mr.lee during the Spring '11 term at Korea University.

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math-2011-note8 - 12 Calculus of Several Variables (ch.14)...

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