Math 1320 - Lab 6
uNiD:
Name:
Due: 10/10
Today we will consider a set of application problems of Taylor’s Series. As you have learned in class, the
main use of Taylor’s series is for approximating functions which help reduce intractable problems (such as
some integrals) into simpler forms than can be easily solved. The following problems explores this use as
well as other applications of Taylor’s series. Don’t forget to write your reasoning (in COMPLETE and
CONCISE sentences) behind your approach to each problem so that someone who has taken Math 1310
would understand what you are doing.
1. 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 eα(t−20)
where t is the temperature in o C. There are tables that list the values of α (called the temperature
coefficient) and ρ20 (the resistivity at 20o 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. Find expressions for these linear and
quadratic approximations.
For copper, α = 0.0039/o C and ρ20 = 1.7 × 10−8 Ω-m. Calculate the error in your linear and quadratic
approximations when the temperatures are (a) −250o C, (b) 420o C.
2. Einstein’s special relativity is used for particles that are moving at speeds which are close to the speed
of light. In special relativity, the energy of the particle is given by
E(β) = M c2 = p
mc2
1 − β2
where c is the speed of light, M is the relativistic mass, m is the rest mass, and β = v/c where v is the
velocity of the particle. This relationship between the relativistic mass and the rest mass is given by
M=p
m
1 − (v/c)2
and is based on the Lorentz transformation T = γt where γ = √
1
,which
1−(v/c)2
was known to mathe-
maticians before Einstein developed the theory of special relativity. This particular relationship tells us
that to an outside observer, the mass of a particle in motion increases with the speed of the particle,
and is infinite in the limit v → c.
What if the speed v is small compared to the speed of light? The energy equation of special relativity
should reduce to the energy equation of Newtonian (classical) mechanics E = 21 mv 2 when v/c ≈ 0 (this
is the familiar kinetic energy you see in your physics class). Expand E(β) in a Taylor series in β about
β = 0 and show that E(β) ≈ mc2 + 12 mv 2 .
Note: The quantity mc2 is called the rest energy, since we are only interested in energy difference between
particles moving at different velocities:
1
1
1
E1 − E2 = mc2 + mv12 − mc2 + mv22 = m(v12 − v22 )
2
2
2
so the classical energy is given by E = 21 mv 2 (which gives the same value for E1 − E2 ).
3. An enthusiastic math student, having discovered that
ln x = (x − 1) −
(x − 1)2
(x − 1)3
+
− ···
2
3
decides to save money by not buying the scientific feature on his new calculator, figuring that all he has
to do is to use this formula to calculate ln x. Being somewhat lazy, he decides that the formula
ln x = (x − 1) −
(x − 1)2
2
is ”probably accurate to three decimal places.” Comment on this speculation with complete and concise
sentences and give example(s) of when this speculation fails.
Your friend tells you that the formula
ln x = (x − 1) −
(x − 1)2
2
is ”definitely accurate to three decimal places provided x is close enough to 1.” How close? (Hint: Use
Alternating Series Estimation Theorem)