Math 1320 Lab 11
Name:
1. If a sound with frequency fs is produced by a source traveling along a line with speed vs
and an observer is traveling with speed vo along the same line from the opposite direction
toward the source, then the frequency of the sound heard by the observer is
c + vo
fo =
fs
c − vs
where c is the speed of sound, about 332 m/s. (This is the Doppler effect). Suppose
that, at a particular moment, you are in a train traveling at 34 m/s and accelerating
at 1.2 m/s2 . A train is approaching you from the opposite direction on the other track
at 40 m/s, accelerating at 1.4 m/s2 , and sounds its whistle, which has a frequency of
460Hz. At that instant, what is the perceived frequency that you hear and how fast is
it changing?
2. Consider the function
xy
f (x, y) = x2 + y 2
0


if (x, y) 6= (0, 0)
if (x, y) = (0, 0)
(a) Show that f is not continuous (and hence not differentiable) at (0, 0).
(b) However, show that fx (0, 0) and fy (x, y) exist using the definition of partial derivative.
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(c) Find fx (x, y) for (x, y) 6= (0, 0).
(d) Show that fx (x, y) is not continuous at (0, 0).
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3. If R is the total resistance of three resistors, connected in parallel, with resistances R1 ,
R2 , R3 , then
1
1
1
1
=
+
+
.
R
R1 R2 R3
If the resistances are measured in ohms as R1 = 25 Ω, R2 = 40 Ω, and R3 = 50 Ω, with a
possible error of 0.5% in each case, estimate the maximum error in the calculated value
of R.
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4. A Physical Interpretation of the Gradient
Consider the following function.
U (x, y, z) = −
(x2
GM m
+ y 2 + z 2 )1/2
Here G is the gravitational constant, M is a large point mass fixed at the origin, and
m is a point mass which is free to move. The function U represents the gravitational
potential energy of m. This question is about finding a physical interpretation for the
gradient of U .
(a) Calculate the partial derivative Ux .
(b) Use part (a) to find formulas for the partial derivatives Uy and Uz .
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(c) Use your previous work to show that the gradient of the potential energy U (x, y, z)
is equal to the following expression.
∇U =
(x2
GM m
(xi + yj + zk)
+ y 2 + z 2 )3/2
(d) (No work needed for this part). Here is the physical interpretation: Let F(x, y, z)
be the gravitational force acting on m. Let r = xi + yj + zk be the position vector
of m and let r = |r| = (x2 + y 2 + z 2 )1/2 . In this notation, we would have:
∇U =
GM m
r
r3
From physics, we know that:
F(x, y, z) = −
GM m
r
r3
So your computation shows that F = −∇U . This is the generalization of the
one-dimensional relationship between force and energy to three dimensions.
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