The size of an undisturbed fish population has been modeled by the formula p n + 1 = b p n a + p n where p n is 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 p n is 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 the 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 pn is 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 P0 > 0.
(a) Show that if {pn} is convergent, then the only possible values for its Limit are 0 and b − a.
(b) Show that Pn+1 < (b/a)pn.
(c) Use part (b) to show that if a > b, then limn→∞pn = 0; in other words, the population dies out.
(d) Now assume that a < b. Show that if P0 < b − a, then {pn} is increasing and 0 < pn < b − a. Show also that if P0> b − a, then {pn} is decreasing and pn > b − a. Deduce that if a < b, then limn→∞pn = b − a.
Find the local minima/maxima of -x + ln x if it exist
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:
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