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 March 8th, 2018, 10:27 PM #1 Newbie   Joined: Feb 2018 From: United States Posts: 6 Thanks: 0 Math Focus: Calculus, Linear Algebra How to solve Differential Problem? The productivity of a certain country is given by the function f(x, y) = 45x^(4/5)y^(1/5) when x units of labor and y units of capital are utilized. What is the approximate change in the number of units produced if the amount expended on labor is decreased from 243 to 240 units and the amount expended on capital is increased from 30 units to 35 units? (Round your answer to the nearest whole number.) May someone please help? Thanks!
 March 9th, 2018, 06:52 AM #2 Senior Member   Joined: Sep 2016 From: USA Posts: 521 Thanks: 294 Math Focus: Dynamical systems, analytic function theory, numerics What have you tried?
 March 9th, 2018, 07:30 AM #3 Global Moderator   Joined: Dec 2006 Posts: 19,989 Thanks: 1855 Assuming x is multiplied by 240/243 and y is multiplied by 35/30, f(x, y) is multiplied by (240/243)^(4/5)(35/30)^(1/5), which is about 1.0211. What does "productivity" mean?
March 9th, 2018, 07:55 AM   #4
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Quote:
 Originally Posted by skipjack Assuming x is multiplied by 240/243 and y is multiplied by 35/30, f(x, y) is multiplied by (240/243)^(4/5)(35/30)^(1/5), which is about 1.0211. What does "productivity" mean?
Productivity is the number of units produced (i.e. $f$) and I'm pretty sure the problem is asking to estimate the change in productivity using the linearization of $f$ at $(243,30)$. I'm not sure why you divided these.

 March 9th, 2018, 08:44 AM #5 Newbie   Joined: Feb 2018 From: United States Posts: 6 Thanks: 0 Math Focus: Calculus, Linear Algebra My math instructor told me to use partial derivatives and plug in given values.
March 9th, 2018, 09:51 AM   #6
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Quote:
 Originally Posted by SDK Productivity is the number of units produced . . .
Units of what? During what period? Isn't the productivity of a country usually given in financial terms, such as annual GDP per capita?

The problem refers to the amount expended on labor, whereas the equation given uses x for the number of units of labor. Those aren't the same.

 March 15th, 2018, 11:05 PM #7 Newbie   Joined: Jan 2018 From: Seattle, WA Posts: 20 Thanks: 6 Ask for clarification on the units of measure. Generic units of labor and capital aren't conducive to thought. Hours of labor and dollars of capital result in a function of the form hours^dollars, which is somewhat mind-boggling. So find out what the unit of measure is for f(x,y)
 April 7th, 2018, 04:28 PM #8 Math Team   Joined: Jan 2015 From: Alabama Posts: 3,261 Thanks: 894 If $\displaystyle f(x, y) = 45x^{4/5}y^{1/5}$ Then $\displaystyle \frac{\partial f}{\partial x}= (4/5)(45)x^{4/5-1}y^{1/5}= 36x^{-1/5}y^{1/5}$ and $\displaystyle \frac{\partial f}{\partial y}= (1/5)(45)x^{4/5}y^{1/5- 1}= 9x^{4/5}y^{-4/5}$. Now, if x and y are functions of some variable, t, $\displaystyle \frac{df}{dt}= \frac{\partial f}{\partial x}\frac{dx}{dt}+ \frac{\partial f}{\partial y}\frac{dy}{dt}$ $\displaystyle \frac{df}{dt}= \left(36x^{-1/5}y^{1/5}\right)\frac{dx}{dt}+ \left(9x^{4/5}y^{-4/5}\right)\frac{dy}{dt}$. "the amount expended on labor is decreased from 243 to 240 units and the amount expended on capital is increased from 30 units to 35 units" So we can take x= 243, y= 30. Taking the time in which this took place is one unit of "t", $\displaystyle \frac{dx}{dt}= 240- 243= -3$ and $\displaystyle dy/dt= 35- 30= 5$. Last edited by Country Boy; April 7th, 2018 at 04:31 PM.

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