Chapter 11.11, Problem 32E

The resistivity $\rho$ of a conducting wire is the reciprocal of the conductivity and is measured in units of ohm-meters $\left(\Omega \text{-}m\right)$. The resistivity of a given metal depends on the temperature according to the equation

$\rho \left(t\right)={\rho }_{20}{e}^{\alpha \left(t-20\right)}$

where t is the temperature in $C$. There are tables that list the values of $\alpha$ (called the temperature coefficient) and p ${\rho }_{20}$ (the resistivity at ${20}^{\circ }C$) 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 $\rho \left(t\right)$ 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 $\alpha =0.0039/\text{C}$ and ${\rho }_{20}=1.7\times {10}^{-8}\Omega \text{-}m$. Graph the resistivity of copper and the linear and quadratic approximations for $-{250}^{\circ }\text{C}\le \text{t}\le {\text{1000}}^{\circ }\text{C}$.

(c) For what values of t does the linear approximation agree with the exponential expression to within one percent?