Math 1320 - Lab 12
uNiD:
Name:
Due: 12/05
READ BEFORE PROCEEDING: Today we will continue with some application problems in multivariate calculus (specifically partial derivatives). 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. Present your work in such a way that someone without access to
the questions can read through your solutions and understand it. To achieve this, set-up the problem in
your own words and describe NEATLY and LEGIBLY the procedures you take in solving the problem.
Disjoint or unclear explanations may result in lost points.
1. Consider a simple circuit consisting of a power source and three resistors wired in parallel. Using Ohm’s
law V = IR and knowing that the voltage across each resistor must be the same we are able to calculate
the total resistance through the circuit using the equation
1
1
1
1
=
+
+
RT
R1
R2
R3
where R1 , R2 , R3 is the resistance (in Ohms) of the three resistors. Suppose that the resistors are rated
at the following: R1 = 20 Ohms, R2 =10 Ohms, R3 = 2 Ohms. However, due to real world imprecisions
they will never be exact and the factory that produces the resistors gives a tolerance of 1, 0.5, and 0.2
percent on the three resistors respectively. How much variation in the total resistance of the circuit is
expected?
2. The wave equation is perhaps the most important partial differential equation. It has helped physicist
and engineers describe many real world problems. The propagation of sound waves, the vibration of a
string, propagation of fluids in fluid dynamics, derivation of Maxwell’s equation in electromagnetics are
all instances of the use of the wave equation. In its simplest form, the wave equation can be defined as
follows:
∂2ψ
∂2ψ
= v2 2
2
∂t
∂x
where ψ(x, t) is the wave amplitude, v the characteristic phase velocity, x is a spatial variable, and t is
a time variable.
(a) Is the following equation a solution to the above wave equation?
ψ(x, t) = A cos(kx − ωt − φ)
where A is the wave amplitude, k is the wavenumber, ω is the angular frequency, and φ is the phase
angle. Also, note that ω = kv.
(b) Is ψ(x, t) = sin(kx) sin(vkt) also a solution?
3. In a study of frost penetration it was found that the temperature T at time t (measured in days) at a
depth x (measured in feet) can be modeled by the function
T (x, t) = T0 + T1 e−λx sin(ωt − λx)
where ω = 2π/365 and λ is a positive constant.
(a) Find ∂T /∂x. What is its physical significance?
(b) Find ∂T /∂t. What is its physical significance?
(c) Show that T satisfies the heat equation Tt = kTxx for a certain constant k.
(d) What is the physical significance of the term −λx in the expression sin(ωt − λx)?