Math 1B Chapter 7 Test
fall ’06 Name________________________
Show your work for credit. Write responses on separate paper. Do not abuse a calculator.
1. Suppose
a quantity P satisfies the differential equation .
a. Find
the equilibrium solutions for P and classify each as either stable or
unstable.
b. At
what value of P is P increasing most rapidly?
2. Consider
the initial value problem .
a. Solve
for x as a function of t using the method of separating variables.
Note: You may need to use integration by parts.
If you do, show the table of u,v,du,dv expressions. Do not rely on a calculator to get the
result.
b. Use
Euler’s method with a step size of 0.25 to get a numerical approximation for . Show your calculations by completing the
table below:
3. Consider
the family of functions defined by . Find the family of orthogonal trajectories.
|
year |
Population |
|
1960 |
11209160 |
|
1970 |
14598316 |
|
1980 |
17848320 |
|
1990 |
20278957 |
|
2000 |
22191087 |
|
2005 |
23070170 |
4. Consider
the table below which tabulates the population of
a. Fit
a natural growth model to the populations in years 1960 and 1970. How well does the model predict the projected
population for 2005?
b. It
appears the growth rate may have reached a maximum in the year 1970. Assuming that is true, write a logistic
differential equation for the population of
5. Solve
the initial value problem: .
6. Suppose
that a spherical raindrop is evaporating as it falls so that its mass is
decreasing at a rate proportional to its surface area. That is, . Substitute
(where
is the mass density of water and r is
the radius of the spherical raindrop) to show that
.
7. Human
and mosquito populations in a swampy equatorial environment are modeled by the
system of differential equations,
.
a. Combine
these equations to get a separable differential equation in M and H and Show that there exists a constant A such that . What is the value of A if at some time
there are 10 humans and 1000 mosquitos?
b. Is
one or the other of these population dependent upon the other for their
survival?
Why or why not?
8. If
a ball is launched vertically upwards with an initial velocity from the surface of Earth then (according to
the law of universal gravitation) the force of gravity at a height x
above the surface is
,
where m is the mass of the ball and g is the acceleration due to
gravity at the surface. Assume that the
force of air friction is jointly proportional to the distance of the ball from
the center of Earth and the speed of the ball:
. By
so that
. Suppose that
and that the units of length are such that
. Then the equation is equivalent to
.
a. Make
the substitution , separate variables, integrate and determine
the constant of integration.