“Many pre-calculus concepts can be taught from a discrete dynamical systems viewpoint. This approach permits the teacher to cover exciting applications and enrichment topics.” Helmut Knaust, Ph.D. Associate Professor Department of Mathematical Sciences University of Texas at El Paso January 7, 2003

We use Discrete Dynamical Systems (=first order difference equations) To study some classical functions seen in an algebra course To study some mathematical models in population biology

Excel spreadsheets Are the perfect tool to visualize the solutions of difference equations Reduce the amount of algebra performed by the students

I have used the ideas and materials presented in university courses (SCI 1300, SCI 1100, UNIV 1301) aimed at Students in Science and Engineering who were concurrently taking a remedial Mathematics course (Algebra I or Algebra II)

Objectives of these Courses: Strengthen the students’ mathematical and critical thinking skills Increase the students’ motivation to study mathematics by portraying mathematics as a useful tool in science and engineering Improve students’ computer skills

What I like about this material: This is “pre-calculus” in the true sense of the word - a preview of the power of differential equations A nice illustration of how to use technology in mathematics Conceptually hard, but algebraically not too challenging

Pierre-Simon Laplace (1749-1827) “The simplicity of nature is not to be measured by that of our conceptions. Infinitely varied in its effects, nature is simple only in its causes, and its economy consists in producing a great number of phenomena, often very complicated, by means of a small number of general laws.” Pierre-Simon Laplace (1749-1827)

“Difference Equation” An Introductory Example: Linear Models Converting temperature from ºC to ºF The defining ingredients: 0 ºC corresponds to 32 ºF Every 1 degree increase in ºC corresponds to a 1.8 degree increase in ºF “Initial Data” “Difference Equation”

f(0)=32 f(c+1)-f(c) = 1.8 f(c+1) = f(c) + 1.8 (c = degrees Celsius, f = degrees Fahrenheit) Initial Data: f(0)=32 Difference Equation: f(c+1)-f(c) = 1.8 f(c+1) = f(c) + 1.8

c f 32.0 1 33.8 2 35.6 3 37.4 4 39.2 5 41.0 6 42.8 7 44.6 8 46.4 Initial Data f(5) = f(4)+1.8 = 39.2+1.8 = 41.0

Linear Models c f 32.0 1 33.8 2 35.6 3 37.4 4 39.2 5 41.0 6 42.8 7 44.6 8 46.4 Data on the left are in “arithmetic progression” (= constant differences between consecutive terms) Data on the right are in “arithmetic progression” (= constant differences between consecutive terms)

Exponential Models: Population Growth “The change in population from one year to the next is proportional to the present population” Difference Equation: P(n+1) - P(n) = k P(n) (P = population at time n, n = time (in years)) The change in population from one year to the next… …is proportional to… …to the present population

Exponential Models P(n+1) - P(n) = k P(n) P(n+1) = (1+k) P(n) Spreadsheet

Exponential Models: Population Growth P(n) 5000 1 5500 2 6050 3 6655 4 7321 5 8053 6 8858 7 9744 8 10718 Data on the left are in “arithmetic progression” (= constant differences between consecutive terms) Data on the right are in “geometric progression” (= constant ratios between consecutive terms)

Student Activity Fill in the missing data in the table on the right such that the y-data are in geometric progression: x y 0.0 1.000 0.5 1.0 2.000 1.5 2.0 4.000 2.5 3.0 8.000

First Historical Aside: The Babylonian Algorithm Compute approximations for the square root of 2. Take as a first guess x(0)=1. x(0)=1 is too small, since 12 < 2; consequently 2/x(0)=2 is too big; try their average next: x(1)=1/2 [ x(0) + 2/(x(0) ]=1.5

First Historical Aside: The Babylonian Algorithm This leads to the recurrence relation x(n+1)=1/2 [ x(n) + 2/x(n) ]

First Historical Aside: The Babylonian Algorithm x(n) x(n)2 1.000000000000 1 1.500000000000 2.250000000000 2 1.416666666667 2.006944444444 3 1.414215686275 2.000006007305 4 1.414213562375 2.000000000005 5 1.414213562373 2.000000000000

Second Historical Aside: Fibonacci Numbers Leonardo of Pisa, better known as Fibonacci, might have been the first to propose a model for population growth. In 1202 he proposed the following model for an imaginary rabbit population.

Second Historical Aside: Fibonacci Numbers We start with one pair of rabbits (one female and one male) that matures to reproductive age in a fixed period of time, say a month. At that time they produce a new pair, one female and one male. The original pair will reproduce one more time, after one more month, and again the offspring will be a pair of rabbits. In the sequel, each pair of rabbits will reproduce twice, at intervals separated by a month, and at each reproduction, the new pair will go on in a similar fashion. All of the reproduction happens at the same time, and each pair reproduces exactly twice.

Second Historical Aside: Fibonacci Numbers We can model Fibonacci’s model as follows: Let R(t) be the number of rabbit pairs that are born at the beginning of the t’th month. The first pair appears at time t = 0. R(0) = 1 This first pair bears another pair at time t = 1. R(1) = 1 It follows from the description above that for all later times R(t) = R(t-1) + R (t-2).

Second Historical Aside: Fibonacci Numbers R(n) R(n)/R(n-1) 1 11 144 1.6179775 1.0000000 12 233 1.6180556 2 2.0000000 13 377 1.6180258 3 1.5000000 14 610 1.6180371 4 5 1.6666667 15 987 1.6180328 8 1.6000000 16 1597 1.6180344 6 1.6250000 17 2584 1.6180338 7 21 1.6153846 18 4181 1.6180341 34 1.6190476 19 6765 1.6180340 9 55 1.6176471 20 10946 10 89 1.6181818 17711

Second Historical Aside: Fibonacci Numbers Cheating, by assuming that the ratio r of consecutive terms is eventually constant, we can compute r: r = R(n+2)/R(n+1) = R(n+1)/R(n) Using R(n+2)=R(n+1)+R(n), we obtain r = 1 + 1/r, i.e. r2 – r – 1 = 0 Solving for r yields the “Golden Ratio” as the positive solution:

Modeling with Difference Equations: Logistic Growth Population growth with limited resources Introducing the concept of a population ceiling N If the population is much smaller than N, growth should be “exponential” If the population is close to N, growth should be close to 0 If population exceeds N, growth should be negative

Modeling with Difference Equations: Logistic Growth Spreadsheet

Modeling with Difference Equations: Predator-Prey-Model

Modeling with Difference Equations: Predator-Prey-Model

Modeling with Difference Equations: Predator-Prey-Model Spreadsheet (a, b, c, d and N are positive constants)

All Questions Answered, All Answers Questioned* Contact: helmut@math.utep.edu Web: http://www.math.utep.edu/Faculty/helmut *Borrowed from D. Knuth