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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