ASSIGNMENT 5
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 typed or very neatly written. They must be stapled, with your name
and student number at the top of each page.
This assignment is due at 10:00 a.m. on Friday, November 8. The written assignment is to be handed
in by that time. The online assignment will also close at that time.
1. Determine whether each of the following statements is true or false. If it is true, provide a proof; if it is
false, provide a counterexample.
(a) If f is increasing and concave up on the interval (−∞, ∞), there is a number M such that f (x) > M
for all x.
(b) If f 00 (x) > 0 everywhere on the interval (−∞, ∞), then f (x) > 0 somewhere on the interval (−∞, ∞).
2. Determine the domain, intercepts, extrema, intervals of increase and decrease, inflection points and intervals of concavity for the function
x2 − 2
.
f (x) = 2
x −1
Then sketch the graph of the function, indicating all of this information in the sketch.
3. The Morse potential function
2
V (r) = De 1 − e−a(r−re )
(here r is internuclear distance; De is well depth; re is bond length; and a is a positive constant depending
on De , the vibrational constant, and the molecule’s reduced mass) and Lennard-Jones potential function
σ 12 σ 6
V (r) = 4ε
−
r
r
(here ε is well depth, r is internuclear distance, and σ is a positive constant describing the equilibrium
distance) both model the potential energy of a diatomic molecule. Describe one aspect in which the
graphs of the models differ. Then compare the accuracy of the two models with respect to that aspect.
For example, you might show that the Morse potential has a horizontal asymptote but the Lennard-Jones
potential increases without bound, and then argue for the superiority of the former based on the fact
that a molecule’s potential energy is bounded by physical constraints. (Please note that this example is
incorrect, and intended only as an illustration.)