“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 ProfessorDepartment of Mathematical

SciencesUniversity of Texas at El Paso

January 7, 2003

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• 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

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• Excel spreadsheets – Are the perfect tool to visualize the solutions of

difference equations– Reduce the amount of algebra performed by

the students

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• 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)

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• 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

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• 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

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“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)

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• 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”

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(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

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c f

0 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

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c f

0 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

• Linear Models

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)

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• 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

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• Exponential Models

P(n+1) - P(n) = k P(n)

P(n+1) = (1+k) P(n)

Spreadsheet

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• Exponential Models: Population Growth

n P(n)

0 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)

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• 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

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• 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

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• First Historical Aside: The Babylonian Algorithm

– This leads to the recurrence relationx(n+1)=1/2 [ x(n) + 2/x(n) ]

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• First Historical Aside: The Babylonian Algorithm

n x(n) x(n)2

01.0000000000

001.0000000000

00

11.5000000000

002.2500000000

00

21.4166666666

672.0069444444

44

31.4142156862

752.0000060073

05

41.4142135623

752.0000000000

05

51.4142135623

732.0000000000

00

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• 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.

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• Second Historical Aside: Fibonacci NumbersWe 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.

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• 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) = 1This first pair bears another pair at time t = 1.

R(1) = 1It follows from the description above that for all

later timesR(t) = R(t-1) + R (t-2).

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• Second Historical Aside: Fibonacci Numbers

n R(n) R(n)/R(n-1) n R(n) R(n)/R(n-1)

0 1 11 144 1.6179775

1 1 1.0000000 12 233 1.6180556

2 2 2.0000000 13 377 1.6180258

3 3 1.5000000 14 610 1.6180371

4 5 1.6666667 15 987 1.6180328

5 8 1.6000000 16 1597 1.6180344

6 13 1.6250000 17 2584 1.6180338

7 21 1.6153846 18 4181 1.6180341

8 34 1.6190476 19 6765 1.6180340

9 55 1.6176471 20 10946 1.6180340

10 89 1.6181818 21 17711 1.6180340

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• 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:

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• 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

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• Modeling with Difference Equations: Logistic Growth

Spreadsheet

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• Modeling with Difference Equations: Predator-Prey-Model

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• Modeling with Difference Equations: Predator-Prey-Model

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• Modeling with Difference Equations: Predator-Prey-Model

Spreadsheet

(a, b, c, d and N are positive constants)

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All Questions Answered,All Answers Questioned*

Contact: helmut@math.utep.edu

Web: http://www.math.utep.edu/Faculty/helmut*Borrowed from D. Knuth