James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

610 Chapter 9 Differential Equations
;
3. Let t 1 be the time that the ball takes to reach its maximum height. Show that
t 1 − S m p
ln mt 1 pv0D
mt
Find this time for a ball with mass 1 kg and initial velocity 20 mys. Assume the air resistance
is 1 10 of the speed.
4. Let t 2 be the time at which the ball falls back to earth. For the particular ball in Prob lem 3,
estimate t 2 by using a graph of the height function ystd. Which is faster, going up or coming
down?
5. In general, it’s not easy to find t 2 because it’s impossible to solve the equation ystd − 0
explicitly. We can, however, use an indirect method to determine whether ascent or descent is
faster: we determine whether ys2t 1d is positive or negative. Show that
ys2t 1d − m 2 t
Sx 2 D
1 p 2 x 2 2 ln x
where x − e pt 1ym
. Then show that x . 1 and the function
f sxd − x 2 1 x 2 2 ln x
is increasing for x . 1. Use this result to decide whether ys2t 1d is positive or negative.
What can you conclude? Is ascent or descent faster?
In this section we investigate differential equations that are used to model population
growth: the law of natural growth, the logistic equation, and several others.
The Law of Natural Growth
One of the models for population growth that we considered in Section 9.1 was based
on the assumption that the population grows at a rate proportional to the size of the
population:
dP
− kP
dt
Is that a reasonable assumption? Suppose we have a population (of bacteria, for instance)
with size P − 1000 and at a certain time it is growing at a rate of P9 − 300 bacteria per
hour. Now let’s take another 1000 bacteria of the same type and put them with the first
pop ulation. Each half of the combined population was previously growing at a rate of
300 bac teria per hour. We would expect the total population of 2000 to increase at a rate
of 600 bacteria per hour initially (provided there’s enough room and nutrition). So if
we double the size, we double the growth rate. It seems reasonable that the growth rate
should be proportional to the size.
In general, if Pstd is the value of a quantity y at time t and if the rate of change of P
with respect to t is proportional to its size Pstd at any time, then
1
dP
dt
− kP
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