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
)
=
ρ
20
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 p
ρ
20
(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
.
(a) Find expressions for these linear and quadratic approximations.
(b) For copper, the tables give
α
=
0.0039
/
C
∘
and
ρ
20
=
1.7
×
10
−
8
Ω
-
m
. Graph the resistivity of copper and the linear and quadratic approximations for
−
250
∘
C
≤
t
≤
1000
∘
C
.
(c) For what values of t does the linear approximation agree with the exponential expression to within one percent?