## Question

###### The resistivity $\rho$ of a conducting wire is the reciprocal of the conductivity and is measured in units of ohm-meters $(\Omega-m)$ . The resistivity of a given metal depends on the temperature according to the equation $$ \rho(t)=\rho_{20} e^{\alpha i t-20 )} $$ where $t$ is the temperature in $^{\circ} \mathrm{C}$ . There are tables that list the values of $\alpha$ (called the temperature coefficient) and $\rho_{20}$ (the resistivity at $20^{\circ} \mathrm{C} )$ for various metals.

The resistivity $\rho$ of a conducting wire is the reciprocal of the conductivity and is measured in units of ohm-meters $(\Omega-m)$ . The resistivity of a given metal depends on the temperature according to the equation $$ \rho(t)=\rho_{20} e^{\alpha i t-20 )} $$ where $t$ is the temperature in $^{\circ} \mathrm{C}$ . There are tables that list the values of $\alpha$ (called the temperature coefficient) and $\rho_{20}$ (the resistivity at $20^{\circ} \mathrm{C} )$ for various metals. Except at very low temperatures, the resistivity varies almost linearly with temperature and so it is common to approximate the expression for $\rho(t)$ by its first- or second-degree Taylor polynomial at $t=20$ . (a) Find expressions for these linear and quadratic approximations. (b) For copper, the tables give $\alpha=0.0039 /^{\circ} \mathrm{C}$ and $\rho_{20}=1.7 \times 10^{-8} \Omega-\mathrm{m} .$ Graph the resistivity of copper and the linear and quadratic approximations for $-250^{\circ} \mathrm{C} \leqslant t \leqslant 1000^{\circ} \mathrm{C}$ (c) For what values of $t$ does the linear approximation agree with the exponential expression to within one percent?