ASSIGNMENT 8
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 justifications. You will be graded on the mathematical, logical and grammatical coherence and
elegance of your solutions. Your solutions must be typed, with your name and student number at the top of
the first page. If your solutions are on multiple pages, the pages must be stapled together.
Your written assignment must be handed in before the start of your recitation on Friday, November
7. The online assignment will close at 9:00 a.m. on Friday, November 7.
1. Let f (x) = xe2/x − x. Find the vertical and horizontal asymptotes, if there are any, of f (x).
2. Let L be a line. L is a slant asymptote of a curve C if the limit of the minimum vertical distance between
C and L at x is equal to 0 as x approaches ∞ or as x approaches −∞. For example, the curve y = x + x1
has a slant asymptote y = x.
2
2
Let a, b > 0. Find the slant asymptotes, if there are any, of the hyperbola xa2 − yb2 = 1. (If you propose
any lines to be slant asymptotes, you must explain how you came up with the lines.)
3. This question refers to the 2010 paper “A Coarse-Grained Model Based on Morse Potential for Water
and n-Alkanes” [1]. A copy of the paper is available on the course webpage.
In the paper, the authors describe the benefits of using the Morse potential instead of the Lennard-Jones
potential in their force field. Recall that the Lennard-Jones potential is
12
6 !
R
R
V (r) = ε
−2
,
r
r
where r is the distance between the atoms, R is the distance at which the potential is minimal, and ε is
the depth of the potential well. The Morse potential is given in equation (1) of the paper. (Note that the
authors use R0 to describe the constant we call R.)
(a) Find the horizontal asymptotes, if there are any, of the Lennard-Jones potential and the Morse
potential.
(b) Find the vertical asymptotes, if there are any, of the Lennard-Jones potential and the Morse potential.
(c) Based on your answers above, does one model provide a better description of the potential energy of
a diatomic molecule? Explain in one or two sentences.
[1] S-W. Chiu, H.L. Scott and E. Jakobsson. “A Coarse-Grained Model Based on Morse Potential for Water and n-Alkanes.”
Journal of Chemical Theory and Computation 6 (2010), 851-863.