What is the significance of the partials in wheat production changes?

In summary, wheat production in a given year is affected by the average temperature and annual rainfall, with a current rate of change of 0.15 degrees celsius per year and 0.1 cm per year respectively. Scientists estimate that at current production levels, an increase in temperature will lead to a decrease in wheat production (∂W/∂T = -2), while a decrease in rainfall will lead to an increase in wheat production (∂W/∂R = 8). The signs of the partial derivatives indicate the direction of change in wheat production with respect to each variable. To estimate the current rate of change of wheat production (dW/dt), the multivariable chain rule must be used, taking into account
  • #1
skipo
1
0
1. Homework Statement
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.
 
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  • #2
Firstly, what is your attempt at this question? To start you off, for a), note that a partial derivative of a multi-variable function with respect to a paricular variable tells you the increase or decrease (depending on whether it is positive or negative)of the function with respect to that variable alone.

For b), you have to use the multivariable chain rule here. dW/dt can be expressed in terms of partial derivatives of T and R, along with the derivatives of T and R with respect to t.
 

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