MAST10006 Calculus 2, Semester 1, 2025

Assignment 6

Due by: 12pm (midday) Monday 19 May 2025

❼ Answer all questions. Of these questions, one will be chosen for marking.

❼ Submit your assignment in Canvas LMS as a single PDF file before the deadline above.

❼ Marks may be awarded for:

➒ Correct use of appropriate mathematical techniques.

➒ Accuracy and validity of any calculations or algebraic manipulations.

➒ Clear justification or explanation of techniques and rules used.

➒ Clear communication of mathematical ideas through diagrams.

➒ Use of correct mathematical notation and terminology.

❼ You must use methods taught in MAST10006 Calculus 2 to solve the assignment questions.

❼ Give any numerical answers as exact values.

The Gompertz model for a population is

where k and a are positive constants, and p(t) is the population size at time t.

The Gompertz model was first studied by Benjamin Gompertz in 1825. Gompertz was an actuary who used the model to investigate human life expectancy. It has since been used by researchers from various fields to model the growth of different ‘populations’, including by market researchers to model the uptake of a new product amongst consumers, and by oncologists to model the growth of cancer tumours.

Let

Then the Gompertz model can be expressed as

Question 1. You may use the following facts about f(p) in this question:

❼ f(p) = 0, and

❼ = 0 only when p = , and this point is a local maximum of f

(where e = 2.71828 . . . is the usual exponential base).

(a) Find the equilibrium solution(s) of the Gompertz model in terms of k and a, or show that it does not have any.

(b) Draw a phase plot for the Gompertz model. Label all important features with their values, including any local maxima or minima.

(c) Sketch the family of solutions for the Gompertz model.

(d) Describe the long-term consequences for the population predicted by this model.

(e) Suppose that initially the population is p(0) = . What is:

(i) the initial rate of growth of the population?

(ii) the maximum rate of growth of the population?

Question 2. Suppose that a population is modelled by the Gompertz model with k = 2 and a = 10, and that initially the population has size p(0) = 0.1. Find p(t) in terms of t, and find how long it takes until the population reaches size p = 5.

Question 3. Prove the two facts given in question 1:

(a) Show that f(p) = 0

(b) Show that = 0 only when p = , and that p = is a local maximum of f.