ASSIGNMENT 2
There are two parts to this assignment. The first part is on WeBWorK — the link is available on the course
webpage. WeBWorK questions can be submitted directly online. They should be completed before you attempt
the second part of the assignment, which consists of the questions on this page. For these questions, you
are expected to provide full solutions with complete arguments and justifications. You will be graded on the
correctness, clarity and elegance of your solutions. Your answers must be typeset or very neatly written.
They must be stapled, with your name and student number at the top of each page.
1. Let f (x) = x−n where n is a positive integer. Prove, using the limit definition of derivative, that
f 0 (x) = −nx−n−1 .
2. Let
(
f (x) =
1
q
if x is a rational number
0
if x is irrational
p
q
in lowest terms with q > 0
.
In Assignment 1, you proved that f is discontinuous, hence nondifferentiable, at all rational numbers. f
is continuous at all irrational numbers (the proof of this was a bonus problem). Prove, using the limit
definition of derivative, that f is nondifferentiable at all irrational numbers.
Hint: let c be irrational. Consider
f (x) − f (c)
.
x−c
What happens if you restrict x to be irrational? What if you restrict x to be rational?
lim
x→c
3. Recall the function f that we defined in class to describe the spawning salmon population in a branch of
the Fraser River:
f : number of salmon in a given year → number of salmon in the following year.
One canonical model describes f as follows:
f (x) = xer(1− k ) , where r, k > 0 are constants.
x
Determine whether, according to this model, there is a sustainable number of salmon; and if so, under
what conditions that number is stable.