Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 14 - Partial Derivatives - 14.5 The Chain Rule - 14.5 Exercises - Page 984: 36

Answer

a) $W$ decreases as the $T$ increases and $W$ increases as the $R$ increases. b) $$-1.1$$

Work Step by Step

a) (1) It has been notice that the rate of change of wheat production (W) with respect to the average temperature (T) is negative. So, when $W_T \lt 0$, then we have $W$ decreases as the $T$ increases. 2) It has been notice that the rate of change of wheat production (W) w.r.t. the annual rainfall (R) is positive. So, when $W_R \gt 0$. then , we have $W$ increases as the $R$ increases. Hence, $W$ decreases as the $T$ increases and $W$ increases as the $R$ increases. (b) We need to apply chain rule. $$\dfrac{dW}{dt}=(\dfrac{\partial W}{\partial T})(\dfrac{dT}{ dt})+(\dfrac{\partial W}{\partial R})(\dfrac{dR}{ dt}) \\=(-2)(0.15)+(8) (-0.1)\\=-0.3-0.8 \\=-1.1$$
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