ASSIGNMENT 4 for SECTION 001
There are three parts to this assignment. Part A is to be completed online before 7:00 a.m. on Friday,
October 15. Part B and Part C, which require full solutions, are to be handed in at the beginning of class on
the same date.
Part A [10 marks]
This part of the assignment can be found online, labelled A4, at webwork.elearning.ubc.ca — sign in using
the MATH110 001 2010W button.
Part B [5 marks]
This part of the assignment is drawn directly from the course texts. It focuses on mathematical exposition;
you will be graded primarily on the clarity and elegance of your solutions.
From the Calculus: Early Transcendentals text, complete question 54 from section 2.5 and question 86 from
section 2.6.
Part C [15 marks]
This part of the assignment consists of more challenging questions. You are expected to provide full solutions
with complete arguments and justifications.
n ∈ N\{1}
1. The sigmoid function
P (t) =
K
ert
KP0
+ P0 1 −
1
ert
models the growth of a population P as a function of time t. P0 is a positive constant denoting the initial
population. K and r are also constants, known as the population’s carrying capacity and growth rate,
respectively. It is assumed that K > P0 .
(a) Suppose r > 0. Determine what happens to the population over time; that is, as the variable t gets
very large.
(b) What if r < 0?
2. Determine where the following function is discontinuous:
x if x = 1, 12 , 13 , 14 , . . .
f (x) =
x2 otherwise
3. Let g be a function which is continuous at a, and f be a function which is continuous at g(a). Prove that
f ◦ g is continuous at a.