Department of Mathematical and Statistical Sciences
Wim Ruitenburg's Fall 2024 MATH 1300-101
Last updated: 24 November 2024
Comments and suggestions: Email wim.ruitenburg@marquette.edu
Growth from chapter 9
From Section 9.1 we can learn:
- There are many ways to describe a sequence.
Sometimes very precise but cumbersome.
Sometimes informal and not always clear.
The sequence 1, 11, 21, 1211, 111221, ..... is given by a `weird' rule.
- An example of a common informal unclear sequence: 2, 3, 5, 8, .....
What is the next number?
Students suggest 12 or 13.
There is no reasonable and unique correct answer.
Why not 696895?
- Shortly after 1200, Fibonacci (Leonardo Pisano) popularized the
decimal `Arabic' number system in Italy.
From Section 9.2 we can learn:
- A linear growth sequence is also called an arithmetic sequence.
- Given a linear sequence that begins as 7, 13, ..., what is the next
number?
- Quick and easy ways to add a long list of numbers of a linear sequence,
in particular if one knows the length of the sequence.
From Section 9.3 we can learn:
- An exponential growth sequence is also called a geometric sequence.
- Exponential growth talks about increasing geometric sequences as well as
about decreasing geometric sequences.
- Given a geometric sequence that begins as 4, 12, ..., what is the next
number?
- Quick and easy ways to add a long list of numbers of a geometric
sequence.
From Section 9.4 we can learn:
- Logistic growth differs from exponential geometric growth by taking into
account that we deal with limited resources.
- Normally generations p(0), p(1), p(2), (p(3), ... should have values in
the range from 0 to 1 (population densities relative to a maximum population).
- Instead of a fixed geometric growth rate r, we have a varying growth
rate r * (1 - p(n)) from generation density p(n) to generation density p(n+1).
Hence the formula p(n+1) = r * (1 - p(n)) * p(n).
Normally the value of r is in the range from 1 to 4, although mathematically
all values in the range from 0 to 4 are workable.
- A logistic growth sequence at first looks like a geometric sequence
when we start with numbers p(n) that are very small.
After a while a logistic growth sequence may look like a constant sequence.
Some logistic sequences may turn chaotic rather than constant.
Example Problems
-
(This exercise is about why approximate mathematical models are a good thing.)
We have 2 clocks.
The 1st clock stands still, while the 2nd clock runs but is a little bit off.
So the 1st clock is perfectly accurate twice a day, while the 2nd clock is
never accurate.
Give sound reasons why the 2nd clock is better than the 1st.
- Recommended problems from Chapter 9 of the book:
1, 5, 10a
- Recommended problems from Chapter 9 of the book:
19ab, 29, 30
- Recommended problems from Chapter 9 of the book:
37ac, 39, 47, 50
- Recommended problems from Chapter 9 of the book:
53a, 54a
- Consider the logistic equation p(n+1) = 4 * p(n) * (1 - p(n)).
So r = 4.
- If p(1) = 0.001 (which is small), give a decent approximation of p(2)
`without the calculator'.
- If p(1) = 0.5, compute p(2) and p(3).
- Suppose x = p(1) = p(2) is positive, so the population is stable.
Compute x.