Chapter 8 Application Exercises
Math 152 Spring 2008
1.
A model for the ability M of a child to memorize, measured on a scale from 0 to 10, is given by
1 1.6 ln,0 4
M t t t
where t is the child’s age in years. Find the average value of this
model a) between the child’s first and second birthdays, b) between the child’s third and fourth
birthdays.
2.
A string stretched between the two points (0,0) and (2,0) is plucked by dis
placing the string h
units at its midpoint. The motion of the string is modeled by a Fourier Sine Series whose
coefficients are given by
1 2
0 1
sin 2 sin
2 2
n
n x n x
b h x dx h x dx
. Find b
n
.
3.
The field strength H
of a magnet of length 2L on a particle r units from the c
enter of the magnet
is
3
2 2
2
2
ml
H
r L
where
±
m are the poles of the magnet. Find the average strength as the
particles moves from 0 to R units from the center by evaluating the integral
3
2 2
2
0
1 2
R
ml
dr
R
r L
.
4.
A single infected individua
l enters a community of n susceptible individuals. Let x be the
number of newly infected individuals at time t. The common epidemic model assumes that the
disease spreads at a rate proportional to the product of the total number of infected and the
numbe
r not yet infected. So
1
dx
k x n x
dt
and you obtain
1
1
dx kdt
x n x
Solve for x as a function of t.
5.
The cross section of a precast concrete beam for a building is bounded by the graphs of the
equations
2 2
2 2
,,0,3
1 1
x x y y
y y
, where x and y are measured in feet. The
length of the beam is 20 feet. A) Find the volume V and the weight W of the beam. Assume the
concrete weight 148 lbs. per cubic foot. B) Find the centroid of a cross section of the beam. (See
figure pg. 566 und
er #85)
6.
The velocity v of an object falling through a resisting medium such as air or water is given by
0
32
1
32
kt
kt
v ke
v e
k
where v
0
is the initial velocity, t is the time in seconds, and k is the
resistance constant of the medium. Use L’Hôpital’
s Rule to find the formula for the velocity of a
falling body in a vacuum by fixing v
0
and t and letting k approach zero. (Assume that the
downward motion is positive.)
7.
The magnetic potential P at a point on the axis of a circular coil is given by
3
2 2
2
2 1
c
NIr
P dx
k
r x
, where N, I, r and k are constants. Find P.
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