The size of an undisturbed fish population has been modeled by the formula p n + 1 = b p n a + p n where is p n the fish population after n years and a and b are positive constants that depend on the species and its environment. Suppose that the population in year 0 is p 0 > 0 . (a) Show that if { p n } is convergent, then the only possible values for its limit are 0 and b − a . (b) Show that p n + 1 < ( b / a ) p n . (c) Use part (b) to show that if a > b , then lim n → ∞ p n = 0 ; in other words, the population dies out. (d) Now assume that a < b Show that if p 0 < b − a , then { p n } is increasing and 0 < p n < b − a . Show also that if p 0 > b − a , then { p n } is decreasing and p n > b − a . Deduce that if a < b then lim n → ∞ p n = b − a .
The size of an undisturbed fish population has been modeled by the formula p n + 1 = b p n a + p n where is p n the fish population after n years and a and b are positive constants that depend on the species and its environment. Suppose that the population in year 0 is p 0 > 0 . (a) Show that if { p n } is convergent, then the only possible values for its limit are 0 and b − a . (b) Show that p n + 1 < ( b / a ) p n . (c) Use part (b) to show that if a > b , then lim n → ∞ p n = 0 ; in other words, the population dies out. (d) Now assume that a < b Show that if p 0 < b − a , then { p n } is increasing and 0 < p n < b − a . Show also that if p 0 > b − a , then { p n } is decreasing and p n > b − a . Deduce that if a < b then lim n → ∞ p n = b − a .
Solution Summary: The author explains that if leftp_nright is convergent, the only possible values for its limit are 0 and b-a.
The size of an undisturbed fish population has been modeled by the formula
p
n
+
1
=
b
p
n
a
+
p
n
where is
p
n
the fish population after n years and a and b are positive constants that depend on the species and its environment. Suppose that the population in year 0 is
p
0
>
0
.
(a) Show that if
{
p
n
}
is convergent, then the only possible values for its limit are 0 and
b
−
a
.
(b) Show that
p
n
+
1
<
(
b
/
a
)
p
n
.
(c) Use part (b) to show that if
a
>
b
, then
lim
n
→
∞
p
n
=
0
; in other words, the population dies out.
(d) Now assume that
a
<
b
Show that if
p
0
<
b
−
a
, then
{
p
n
}
is increasing and
0
<
p
n
<
b
−
a
. Show also that if
p
0
>
b
−
a
, then
{
p
n
}
is decreasing and
p
n
>
b
−
a
. Deduce that if
a
<
b
then
lim
n
→
∞
p
n
=
b
−
a
.
A population of a town grows according to the law of uninhibited growth modeled
by the function
P(t) = Poekt
!!
where P is measured in thousands and t is measured in years.
If the initial population of this town was 12,000 and it doubles every 8 years, find the
growth rate, k. Provide EXACT value for k. Then approximate it to FOUR decimal places.
Show all the steps and write the final answer in the space provided.
Exact
k=
Approximate:
Suppose that 40 deer are introduced in a protected area. The population of the herd P can be modeled by P = 40 + 20x / 1 + .05x where x is the time in years since introducing the deer. Determine the time required for the deer population to reach 200.
On the first day of training a new employee, a company at first loses money at the start of the day. A statistician has developed
a function to model the rate at which money is being gained or lost as time goes on, which depends on how long the employee will work
before her first break at time T after she starts her day. If the employee will work continuously 0 < t = T < 2 hours before her first break,
πC²
2
the RATE at which money is gained or lost is modeled by f(t, T) = 3-
- 3 ln(T + t)ln(T − t).
If the employee begins work at 8:00am that first day, and works until she is profitable, at what time, to the nearest minute,
does the she finally become profitable?
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