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Math Help - Simple Partial Derivatives Question

  1. #1
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    Simple Partial Derivatives Question

    1. The problem statement, all variables and given/known data
    Wheat production in a given year, W, depends on the average temperature T and the annual rainfall R. Scientists estimate that the average temperature is rising at a rate of 0.15 degrees celsius per year and rainfall is decreasing at a rate of 0.1 cm per year. The also estimate that, at current production levels, ∂W/∂T = -2 and ∂W/∂R = 8.

    a) what is the significance of the signs of the partials?
    b) estimate the current rate of change of wheat production dW/dt.

    3. The attempt at a solution

    So the function is W(T,R). taking the partial with respect to temperature and getting a negative value means that as the temperature is rising, Wheat production is decreasing, and the positive value for rainfall decreasing means that wheat production is rising as rainfall decreases? this doesn't make sense to me, why would the wheat production increase as rainfall decreased...am i thinking about it wrong?

    i'm not sure what they want us to do for part b though.., haven't tried that part yet.

    thanks
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  2. #2
    Senior Member
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    Dec 2007
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    Melbourne
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    a) what is the significance of the signs of the partials?
    The negative partial derivative for \frac {\delta W}{\delta T} means that increasing the temperature decreases wheat production and decreasing the temperature increases wheat production. The positive partial derivative for \frac {\delta W}{\delta R} means that as rainfall increases, wheat production will increase and that as rainfall decreases, wheat production will decrease.

    I think they just want you to apply the chain rule for part b: you have \frac {dT}{dt}, \frac {dR}{dt},  \frac {\delta W}{\delta R} and \frac {\delta W} {\delta T}. You will also need to add the impact of each of the factors on wheat production.
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