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March 8th, 2018, 09:27 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, 05:52 AM  #2 
Senior Member Joined: Sep 2016 From: USA Posts: 435 Thanks: 247 Math Focus: Dynamical systems, analytic function theory, numerics 
What have you tried?

March 9th, 2018, 06:30 AM  #3 
Global Moderator Joined: Dec 2006 Posts: 19,508 Thanks: 1741 
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, 06:55 AM  #4 
Senior Member Joined: Sep 2016 From: USA Posts: 435 Thanks: 247 Math Focus: Dynamical systems, analytic function theory, numerics  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, 07:44 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, 08:51 AM  #6 
Global Moderator Joined: Dec 2006 Posts: 19,508 Thanks: 1741  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, 10:05 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 mindboggling. So find out what the unit of measure is for f(x,y) 
April 7th, 2018, 03:28 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 $\displaystyle \frac{\partial f}{\partial x}= (4/5)(45)x^{4/51}y^{1/5}= 36x^{1/5}y^{1/5}$ 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 03:31 PM. 

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differential, multivariable, problem, solve 
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