Wheat production W in a given year 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 °C/year and rainfall is decreasing at a rate of 0.1 cm/year. They also estimate that, at current production levels, ∂W/∂T = −2 and ∂W/∂R = 8.
(a) What is the significance of the signs of these partial derivatives?
(b) Estimate the current rate of change of wheat production, ∂W/∂t.

a) Check Explanation.

b) -1.1.

The wheat production is decreasing at a rate of 1.1 units/year.

Step-by-step explanation:

a) The signs indicate whether the rate of change is increasing or decreasing.

Plus sign in front of the derivative indicates an increasing rate of change while the minus sign indicates a decreasing rate of change.

(∂W/∂T) = −2 indicate that the wheat production reduces with increase in temperature

And (∂W/∂R) = 8 indicate that wheat production increases with increase in temperature.

b) Since wheat production depends on the average temperature, T, and annual rainfall, R, the total rate of change of wheat production with time will be given mathematically as

(dW/dt) = [(∂W/∂T)(dT/dt)] + [(∂W/∂R)(dR/dt)]

(∂W/∂T) = - 2

(dT/dt) = 0.15 °C/year

(∂W/∂R) = 8

(dR/dt) = -0.1 cm/year (annual rainfall is decreasing with time)

(dW/dt) = (-2×0.15) + (8×-0.1) = -0.30 - 0.8 = -1.1

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