ASSIGNMENT 9
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.
The online assignment will close at 9:00 a.m. on Friday, November 14. Your written assignment must
be handed in before the start of the MATH 100 lecture on Monday, November 17.
1. Find a formula for a quartic (degree 4) polynomial whose graph is symmetric about the y-axis, which has
a y-intercept of 0, and which has global maxima (1, 3) and (−1, 3).
2. An elliptic curve is a curve of the form
y 2 = x3 + ax + b,
where a and b are constants satisfying the equation −16 4a3 + 27b2 6= 0. These curves have deep
connections with the study of prime numbers. In this question you will explore the elliptic curve denoted
E and described by the equation
y 2 = x3 − x.
(a) Find the domain, intercepts, vertical√and horizontal asymptotes, intervals of increase and decrease,
and intervals of concavity of f (x) = x3 − x.
(b) A curve is nonsingular if a single tangent line may be drawn at any point on the curve. (For example,
x2 + y 2 = 1 is nonsingular, but y 2 = x3 is not — in the latter case, a single tangent line may not be
drawn at (0, 0).) Prove that E is nonsingular.
(c) Sketch the graph of E.
(d) One of the many interesting characteristics of elliptic curves is that we can define an “addition of
points” on the curve which satisfies all the usual rules of addition. Suppose a line intersects E at
three points P , Q and R. Then we define P + Q + R = 0. (If the line intersects the curve at fewer
than three points, there are alternate definitions.) Prove that any line intersects E at no more than
three points.
3. This question refers to the 2011 paper “Fluctuations of fish populations and the magnifying effects of
fishing” [1]. A copy of the paper is available on the course webpage.
In equation (3) in the “Methods” section of the paper, the authors describe the Ricker recruitment function
R(Bt ), where Bt is the biomass of mature fish in a population at time t, and R(Bt ) is the biomass of
mature fish “recruited” from that population for the following year.
(a) Find the domain, intercepts, vertical and horizontal asymptotes, intervals of increase and decrease,
and intervals of concavity of R(Bt ).
(b) Sketch the graph of R(Bt ).
(c) Assume that β > 0. What must α be for a given population of fish to have a nontrivial sustainable
biomass of mature fish?
[1] Shelton, A.O. and Mangel, M. “Fluctuations of fish populations and the magnifying effects of fishing.” Proceedings of
the National Academy of Sciences 108(17) (2011), 7075-7080.