SCIE 001 MATHEMATICS ASSIGNMENT 5 (Due Nov. 5, 2012)
There are two parts to this assignment. The first part is on WeBWorK and is due at 10:00 pm on Wed.
Oct. 31. The second part consists of the questions on this page, and is due before 10:00 am on Mon. Nov.
5. For these questions, you are expected to provide full solutions with complete arguments and justifications.
You will be graded primarily 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. For each of the following functions, determine: i) the domain, ii) all critical numbers, iii) intervals where
the function is increasing or decreasing, iv) all local and absolute extreme values, v) intervals where the
graph is concave upward or downward, vi) all inflection points. Then sketch the graph of the function,
showing the information determined above, and showing careful detail near any critical numbers.
(a) f (x) = x1/3 − x2/3 .
(b) g(t) = t2/3 (t − 2)2 .
2. The Lennard-Jones potential (an alternative model to the Morse potential) is
r 6 req 12
eq
V (r) = ε
−2
, 0 < r < ∞,
r
r
where ε and req are both positive constants. The first term models the potential for a short-range
repulsive force, and the second term corresponds to a longer-range attractive force. A bond forms due to
the balance of these forces.
(a) Determine: i) all horizontal or vertical asymptotes (definitions p. 311) ii) all critical numbers, iii)
intervals where the function is increasing or decreasing, iv) all local and absolute extreme values, v)
intervals where the graph is concave upward or downward, vi) all inflection points. Then sketch the
graph of the potential, showing the information determined above.
(b) Interpret physically each of the mathematical features determined in part (a), if V (r) is the potential
energy corresponding to a conservative force between two particles a distance r apart.
(c) Describe the similarities and differences between the graphs of the Morse potential and Lennard-Jones
potential.