20141119131112_L76_W11 - Partial Differentiation - LILY

20141119131112_L76_W11 - Partial Differentiation - LILY - &...

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Functions of several variables & Partial differentiation RA Mislihah, MM 1
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Determine maximum Revenue Example : Given: TR = 50 Q - 2 Q 2 Use differentiation to calculate maximum TR! Only 1 type of good sold!! 
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3 A monopolist produces two goods, X and Y. The demand function for each good is given as PX=36-3x and PY=56-4y (a) Find the quantities of each good which must be sold in order to maximize the total revenue. (b) What is the maximum total revenue? Determine maximum Revenue
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Functions of two variables 4 General form: z= f (x,y) z = x + 2y +4 z = x2(1+2y) z = ln(x)+3ln(y) Production function: Output= f (Labor, Kapital) Q=5LK Q=80L0.5K0.2 Total revenue for 2 goods PX=36-3x and PY=56-4y TR = (36-3x)x + (56-4y)y
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Partial differentiation: first-order partial derivatives 5    :the first partial derivative of  z  w.r.t.  X (treat other variables as constant) x z x z or    :the first partial derivative of  z  w.r.t.  Y (treat other variables as constant) y z y z or
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6 Exercises: Find the first-order partial derivatives for each of the following functions. (a) (b) (c) 5 3 2 2 y x x z 3 . 0 7 . 0 10 K L Q U x y 2 5 y x z x 3 4 x z y 3 3 . 0 3 . 0 7 K L Q L 7 . 0 7 . 0 3 K L Q K Partial differentiation: first-order partial derivatives
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7 xx z x z x z x or 2 2 yy z y z y z y or 2 2 yx z y x z y z x or 2 The straight second-order partial derivatives The mixed second-order partial derivatives Partial differentiation: second-order partial derivatives
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8 Find the second order partial derivatives for each of the following functions. (a) (b) 5 3 2 2 y x x z 3 . 0 7 . 0 10 K L Q U x y 2 5 4 xx z y x z x 3 4 3 xy z x z y 3 0 yy z 3 yx z Partial differentiation: second-order partial derivatives
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Differential and small changes  (incremental changes) 9
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Differential and small changes (incremental changes) 10 Concern: What’s the approximate change in y if x is increased/decreased by a small change? Example: Given P = Q2 If Q is increased by 4%, use differentials to find the approximate change in P!
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