Chapter 11.11, Problem 32E

The resistivity ρ of a conducting wire is the reciprocal of the conductivity and is measured in units of ohm-meters (Ω-m). The resistivity of a given metal depends on the temperature according to the equation

ρ(t) = ρ₂₀ e^α(t−20)

where t is the temperature in °C. There are tables that list the values of α (called the temperature coefficient) and ρ₂₀ (the resistivity at 20°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 ρ(t) by its first- or second-degree Taylor polynomial at t = 20.

1. (a) Find expressions for these linear and quadratic approximations.
2. (b) For copper, the tables give α = 0.0039/°C and ρ₂₀ = 1.7 × 10⁻⁸ Ω-m. Graph the resistivity of copper and the linear and quadratic approximations for −250°C ≤ t ≤ 1000°C.
3. (c) For what values of t does the linear approximation agree with the exponential expression to within one percent?