Chapter 13 - Functions of Several Variables - 13.4 Exercises - Page 906: 32

$$\Delta R \approx - 0.14$$

$$\eqalign{ & \frac{1}{R} = \frac{1}{{{R_1}}} + \frac{1}{{{R_2}}} \cr & {\text{Solve for }}R \cr & \frac{1}{R} = \frac{{{R_2} + {R_1}}}{{{R_1}{R_2}}} \cr & R = \frac{{{R_1}{R_2}}}{{{R_2} + {R_1}}} \cr & {\text{Find the differential total differential }}dR \cr & dR = \frac{{\partial R}}{{\partial {R_1}}}d{R_1} + \frac{{\partial R}}{{\partial {R_2}}}d{R_2} \cr & dR = \frac{\partial }{{\partial {R_1}}}\left[ {\frac{{{R_1}{R_2}}}{{{R_2} + {R_1}}}} \right]d{R_1} + \frac{\partial }{{\partial {R_2}}}\left[ {\frac{{{R_1}{R_2}}}{{{R_2} + {R_1}}}} \right]d{R_2} \cr & {\text{*By the quotient rule}} \cr & \frac{\partial }{{\partial {R_1}}}\left[ {\frac{{{R_1}{R_2}}}{{{R_2} + {R_1}}}} \right] = \frac{{\left( {{R_2} + {R_1}} \right){R_2} - {R_1}{R_2}}}{{{{\left( {{R_2} + {R_1}} \right)}^2}}} \cr & \frac{\partial }{{\partial {R_1}}}\left[ {\frac{{{R_1}{R_2}}}{{{R_2} + {R_1}}}} \right] = \frac{{{{\left( {{R_2}} \right)}^2}}}{{{{\left( {{R_2} + {R_1}} \right)}^2}}},{\text{ }} \cr & {\text{similarly }}\frac{\partial }{{\partial {R_2}}}\left[ {\frac{{{R_1}{R_2}}}{{{R_2} + {R_1}}}} \right] = \frac{{{{\left( {{R_1}} \right)}^2}}}{{{{\left( {{R_2} + {R_1}} \right)}^2}}} \cr & {\text{Therefore,}} \cr & dR = \frac{{{{\left( {{R_2}} \right)}^2}}}{{{{\left( {{R_2} + {R_1}} \right)}^2}}}d{R_1} + \frac{{{{\left( {{R_1}} \right)}^2}}}{{{{\left( {{R_2} + {R_1}} \right)}^2}}}d{R_2} \cr & dR = \frac{1}{{{{\left( {{R_2} + {R_1}} \right)}^2}}}\left[ {{{\left( {{R_2}} \right)}^2}d{R_1} + {{\left( {{R_1}} \right)}^2}d{R_2}} \right] \cr & {\text{With: }} \cr & \Delta {R_1} \approx d{R_1} = 10.5 - 10 = 0.5 \cr & \Delta {R_2} \approx d{R_2} = 13 - 15 = - 2 \cr & {\text{So}},{\text{ the change can be approximated by}} \cr & \Delta R \approx dR = \frac{{\partial R}}{{\partial {R_1}}}d{R_1} + \frac{{\partial R}}{{\partial {R_2}}}d{R_2} \cr & \Delta R \approx \frac{1}{{{{\left( {{R_2} + {R_1}} \right)}^2}}}\left[ {{{\left( {{R_2}} \right)}^2}d{R_1} + {{\left( {{R_1}} \right)}^2}d{R_2}} \right] \cr & \Delta R \approx \frac{1}{{{{\left( {10 + 15} \right)}^2}}}\left[ {{{\left( {15} \right)}^2}\left( {0.5} \right) + {{\left( {10} \right)}^2}\left( { - 2} \right)} \right] \cr & \Delta R \approx - 0.14 \cr} $$