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Application Examples
| If it is not already on your hard drive, you will need to download
the free DPGraph Viewer to view some of
the pictures linked to on this web site. |
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QuickTime free download. |
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Video for
Section 5.6, Example 7, Maximizing an Angle
(5 point bonus if you can find the error I made--I changed the name of the
measure of an angle and forgot to change it one place in the presentation.)
PowerPoint Only
Quicktime Movie for 5.6#'s 89, 90 (97, 98
in the 9th edition)
Maple Worksheet for Section 5.7 #71 (79 in
the 9th edition)
| Solving a
Separable Differential Equation
First
Order DE Solution Grapher
Another One Using Euler's Method |
 The graph of the solution is shown above. Click on the
left graph to see an animation of the direction field
vectors moving across the screen for increasing values of x along with an
animated solution point. Solutions corresponding to initial
conditions y(0) = -1, -0.4, 0.4, 1 are shown below along with the
direction field. Click on the picture to see an enlargement. 
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Here are more
examples of solutions to first order
separable differential equations from my Differential Equations course web
site.
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Exponential Population Growth
The population of a country is growing at a rate
that is proportional to the population of the country. The
population in 1990 was 20 million and in 2000 the population was 22
million. Estimate the population in 2020. |
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| Solution
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Extra Credit
A population of mice has been
accidentally introduced onto a fairly large island in the Pacific.
Researchers have estimated that the island is capable of supporting a population
of up to 100,000 mice and that the growth of the mouse population would be
accurately represented by a logistic growth model. The researchers have
also estimated that the growth factor (b) in the logistic growth model will be 1
if time (t) is measured in years. Thus they are predicting that the rate
of change (dP/dt) of the mouse population on the island will be
(1)
where b is the growth factor, C is the
largest sustainable mouse population, and P is the mouse population (as a
function of time). In this example
One hundred mice (about equal numbers of
male and female mice) were initially introduced onto the island. Assign t
= 0 as the time the 100 mice were initially introduced onto the island. In
our logistic population growth model
and thus we have
Solve differential equation (1) above to
show that
Hint
Substitute in the appropriate values for a,
b, and C to construct the logistic growth function P(t) for this example.
From your P(t) function estimate the mouse population on the island (to the
nearest whole mouse) one year after the 100 mice were first first introduced
onto the island (i.e., find P(1)). Also estimate the mouse population on
the island 2 years, 5 years, 10 years, and 20 years after the original 100 mice
were introduced onto the island.
Draw a graph of the mouse population (P) as
a function of time (t in years) for the first 20 years after the mouse
population is introduced onto the island.
How long (to the nearest tenth of a year)
does it take for the mouse population on the island to grow from 100 to 50,000?
| NEWTON'S LAW OF COOLING PROBLEM: A pot of
liquid is put on the stove to boil. The temperature of the liquid reaches
170oF and then the pot is taken off the burner and placed on a
counter in the kitchen. The temperature of the air in the kitchen is 76oF.
After two minutes the temperature of the liquid in the pot is 123oF.
How long before the temperature of the liquid in the pot will be 84oF?
Click
here for more on Newton's Law of Cooling.
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Solution
Function Graph
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Differential Equations
Sailing Application Example (Same
type DE as in Newton's Law of Cooling)
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Work Pumping
Water
The
water in a large horse watering trough weighs 62.4 pounds per cubic
foot. The ends of the trough are isosceles triangles with a base
of length 10 feet, equal length sides 13 feet, height 12 feet, with the
base up as shown in the picture. The trough is 30 feet long and
held in an upright position by supports on the sides. The trough
is completely filled with water. How much work is done in pumping
the water over the edge of the trough to completely empty it?
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Work Lifting
a Chain and Weight
| How much work is done against gravity in
lifting the chain and the weight attached to the end of it up to the
ceiling? animation
(see the picture on the right) The chain weighs one pound per foot. After the "oops" the weight comes loose and falls
back to the floor. Extra Credit:
How fast is the weight traveling (neglecting air
resistance) when it hits the floor?
Solution Below |
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| Spring Stretching Example
A force of 100 pounds will stretch a spring 2
feet beyond its equilibrium length of 5 feet. Find the work done in
stretching the spring from a length of 5 feet to a length of 8 feet.
Click here to see an animation
and click here to see an
animation with scales.
100 = 2k so the spring constant is 50. The work
done would be
How much additional work would be done in
stretching the spring two more feet (assuming we are still within the
elastic limits of the spring and Hooke's Law holds)?
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Center of Mass
Find
the center of mass of a planar lamina whose density is 3 units/square
unit and whose boundaries are formed by the graphs of the functions
given below.
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Center of Mass
Find
the center of mass of a planar lamina whose density is 2 units/square
unit and whose boundaries are formed by the graphs of the functions
given below.
Extra Credit: Find the mass of a
plate with the same boundaries but whose density is the function of x
given below. The plate is pictured at the right, colored to
reflect the density function. Here is a
DPGraph picture of the plate colored
based on the density function.
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Maple Worksheet
for the two center of mass examples
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| Fluid Force
Find the total fluid force on a vertical
circular porthole of radius 2 feet whose center is 14 feet below the
surface of water whose weight is 62.4 pounds per cubic foot.
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| Fluid Force
Find the total fluid force on one side of a vertical
isosceles triangle whose base is 10 feet wide, height is 12 feet, and
the base is resting on the bottom of a tank filled with water to a depth
of 16 feet. The weight of the water is 62.4 pounds per cubic foot.
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