Written Assignment 1

ASSIGNMENT 1
Trung Duong Nguyen - 14624761
Syifa Nadhira Nurul Izzah - 63842660
Hussein Al Aaref - 19722024



For a particular species, the model
Rx
f (x) = , M > 0, R > 1
1 + x/M

describes the size f (x) of a population given the size x of the population in the previous generation. Because
x represents a population, the domain is restricted to x ≥ 0.

1. (a) Solve f (x) = x.
Rx
→ =x
1 + x/M
Rx
→ −x=0
1 + x/M
R
→x×( − 1) = 0
1 + x/M
R
→ x = 0 or =1
1 + x/M
→ x = 0 or R = 1 + x/M
→ x = 0 or R − 1 = x/M
→ x = 0 or x = M (R − 1)
(b) Describe in a paragraph what the solution(s) found in part (a) represent, in terms of populations.

- When letting f (x) = x it also means that the population of the initial generation and the past
generation are equal. Getting x = 0 means that if the initial population is 0, then there will be no
growth and the population will remain the same. On the other hand, at x = M (R − 1), we will get
a population where there is no growth, where the number of death and newborns are equal


2. (a) What polynomial function approximates f (x) well for large values of x? Explain your reasoning.

- For large values of x, the denominator of the function can be simplified as x/M since 1 will be too
small compared to x and can be ignored. Therefore the function will become:

Rx
f (x) =
x/M
R
→ f (x) =
1/M
→ f (x) = R ∗ M