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math-2011-note8

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

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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 0 = ( x 0 1 , ··· , x 0 k ) is defined as F x i ( x 0 ) = lim h 0 F ( x 0 1 , ··· , x 0 i + h , ··· , x 0 k ) - F ( x 0 1 , ··· , x 0 i , ··· , x 0 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 0 , L 0 ) is the rate at which output changes with respect to labor L , keeping K fixed at K 0 . If labor increases by Δ L , then output will increase by Δ Q F L ( K 0 , L 0 ) Δ L . Setting Δ L = 1, we see that F L ( K 0 , L 0 ) estimates the change in output due to one unit increase in labor. Hence, F L ( K 0 , L 0 ) 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

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12.2 Approximation by Differentials 1 Since f ( x 0 + h ) - f ( x 0 ) h f prime ( x 0 ) for small h , one can write f ( x 0 + h ) f ( x 0 )+ f prime ( x 0 ) h . Write x = x 0 +( x - x 0 ) . When x - x 0 is small we can write f ( x ) = f ( x 0 +( x - x 0 )) f ( x 0 )+ f prime ( x 0 )( x - x 0 ) . Hence, f ( x ) can be approximated by a translationof a linear function f ( x 0 )+ f prime ( x 0 )( x - x 0 ) around x 0 . Figure 22: Approximation by a (translated) linear function 12.3 Approximation by Differentials 2 F ( x 0 + Δ x , y 0 ) - F ( x 0 , y 0 ) F x ( x 0 , y 0 ) Δ x F ( x 0 , y 0 + Δ y ) - F ( x 0 , y 0 ) F y ( x 0 , y 0 ) Δ y F ( x 0 + Δ x , y 0 + Δ y ) - F ( x 0 , y 0 ) F x ( x 0 , y 0 ) Δ x + F y ( x 0 , y 0 ) Δ y That is, a function F ( x , y )
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