ASSIGNMENT 2·6
There are two parts to this assignment. The first part is on WeBWorK — the link is available on the course
webpage. The second part consists of the questions on this page. 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. They must be stapled, with your name and student number
at the top of each page.
x2 − 2
. Sketch the graph of the function, indicating:
x2 − 1
(a) its domain, intercepts and asymptotes;
(b) its intervals of increase and decrease, and local extrema; and
(c) its intervals of concavity and inflection points.
1. Let f (x) =
You must justify your answers, though you do not need to show all of the steps of your calculations.
2. Let f (x) = x1/x be defined on the interval (0, ∞). Sketch the graph of the function, indicating:
(a) its asymptotes; and
(b) its intervals of increase and decrease, and local extrema.
Unlike the first question, you should show all of your calculations in your solution to this question.
3. The Morse potential is a model describing the potential energy of a two-atom molecule such as oxygen
(O2 ) as a function of the distance between the atoms. It is given by
f (r) = De 1 −
1
ea(r−re )
2
,
where f (r) is the potential energy when the atoms are a distance r apart, re is the distance yielding the
lowest potential energy (the “equilibrium distance”), and De and a are positive constants describing the
“elasticity” of the molecule.
(a) Find the horizontal asymptote of the function f (r).
(b) In molecular terms, what do you think this horizontal asymptote represents? Describe your answer
in a few sentences.