Using Technology to Uncover the Mathematics

August 3-6, 2015

Dave Brown slides available atProfessor, Ithaca College http://faculty.ithaca.edu/dabrown/geneva/dabrown@ithaca.edu

RecursionModeling Population– Fish and Wildlife Management monitors trout population

in a stream, with its research showing that predation along with pollution and fishing causes the trout population to decrease at a rate of 20% per month. The Management team proposes to add trout each month to restock the stream. The current population is 300 trout.

1. If there is no restocking, what will happen to the trout population over the next 10 months?

2. What is the long-term of effect of adding 100 trout per month?3. Investigate the result of changing the number of trout introduced each

month. What is the long-term effect on trout population?4. Investigate the impact of changing the initial population on the long-

term trout population.5. Investigate the impact of changing the rate of population decrease on

the long-term trout population.

Day 2 – Implicit Plots and Parametric Equations

Goals ~ Answer the following– What are implicit curves?– What are parametric equations?– Why are they important?– How can use them in applications?– How can we use them to explore math?

Day 2, Session 1

Implicit Curves – Technology as exploration– Intro to Desmos– Desmos.com– Play a little

Day 2, Session 1

Implicit Curves – Technology as exploration– Is this the graph of a function? Why or

why not?

This is the curve y2=x3+x2-3x+2

What does this mean?

Day 2, Session 1

Implicit CurvesPlotting and exploration using parametersOn to Activities!

Day 2, Session 2

Day 1 – Described plane curves via1. Explicitly: y as a function of x -> y=f(x)• y=3x+2

2. Implicitly: relation between x and y• x2+y2=1• xy+x3=y4

– The first is easy to plot and visualize– The second requires understanding what

points in Cartesian plane satisfy the relation.

Curve vs Parametric

We see Billy’s path, but what are we missing?

Figure Skating

Introduction

Imagine that a particle moves along the curve C shown here.

• Is it possible to model C via an equation y=f(x)?

• Why or why not?

Introduction

Imagine that a particle moves along the curve C shown here.

• We think of the x- and y-coordinates of the particle as functions of “time”.

• Like the skater in motion• We write x=f(t) and y=g(t)

Very convenient way of describing a curve!

Parametric Equations

Suppose that x and y are given as functions of a third variable t (called a parameter) by the equations

x = f(t) and y = g(t)

These are called parametric equations.

Explorations

Activity 1 – Inch worm racesDiscussionActivity 2 – Non-linear inch wormDiscussionActivity 3 – Multiple inch worms and collisionsDiscussion

Day 2, Session 3

Linear motion – Ant on the Picnic Table Activity

Day 2, Session 3

Example

Day 2, Session 3

Exploration with graphing calculatorx=A cos t, y=B sin t, with A,B any numbersTry A=3, B=2; Try A=1, B=1Explore the curve for several values of A, B What is the curve when A<B?What is the curve when A=B?What is the curve when A>B?Can you eliminate the parameter to confirm?

Day 2, Session 3

Parameter Elimination ActivityFerris Wheel Activity

Day 2, Session 3

Exploration 2 (cos(at), sin(bt))Explore for various choices of a and b.What if a,b are integers? How about a=4, b=2?How about a=1.5, b=3? Ratio?

Day 2, Session 3

Trammel of Archimedes

Day 2, Session 4

Brachistochrone Problem and the cycloidFascinating history

Jakob Bernoulli (1654-1705) and Johann Bernoulli (1667-1748)

Acta Eruditorum, June 1696

I, Johann Bernoulli, address the most brilliant mathematicians in the world.

Acta Eruditorum, June 1696

I, Johann Bernoulli, address the most brilliant mathematicians in the world. Nothing is more attractive to intelligent people than an honest, challenging problem, whose possible solution will bestow fame and remain as a lasting monument.

Acta Eruditorum, June 1696

I, Johann Bernoulli, address the most brilliant mathematicians in the world. Nothing is more attractive to intelligent people than an honest, challenging problem, whose possible solution will bestow fame and remain as a lasting monument. Following the example set by Pascal, Fermat, etc., I hope to gain the gratitude of the whole scientific community by placing before the finest mathematicians of our time a problem which will test their methods and the strength of their intellect.

Acta Eruditorum, June 1696

I, Johann Bernoulli, address the most brilliant mathematicians in the world. Nothing is more attractive to intelligent people than an honest, challenging problem, whose possible solution will bestow fame and remain as a lasting monument. Following the example set by Pascal, Fermat, etc., I hope to gain the gratitude of the whole scientific community by placing before the finest mathematicians of our time a problem which will test their methods and the strength of their intellect. If someone communicates to me the solution of the proposed problem, I shall publicly declare him worthy of praise.

Brachistochrone Problem

Given two points A and B in a vertical plane, what is the curve traced out by a point acted on only by gravity, which starts at A and reaches B in the shortest time.

Galileo Galilei

"If one considers motions with the same initial and terminal points then the shortest distance between them being a straight line, one might think that the motion along it needs least time. It turns out that this is not so.”- Discourses on Mechanics (1588)

Galileo’s curves of quickest descent, 1638

Curve of Fastest Descent

Solutions and Commentary

June 1696: Problem proposed in ActaBernoulli: the “lion is known by its claw” when reading anonymous Royal Society paperMay 1697: solutions in Acta Eruditorum from Bernoulli, Bernoulli, Newton, Leibniz, l’Hospital1699: Leibniz reviews solutions from Acta

The bait…

...there are fewer who are likely to solve our excellent problems, aye, fewer even among the very mathematicians who boast that [they]... have wonderfully extended its bounds by means of the golden theorems which (they thought) were known to no one, but which in fact had long previously been published by others.

The Lion

... in the midst of the hurry of the great recoinage, did not come home till four (in the afternoon) from the Tower very much tired, but did not sleep till he had solved it, which was by four in the morning.

The Lion

I do not love to be dunned and teased by foreigners about mathematical things ...

Showed that the path is that of an inverted arch of a cycloid.

CYCLOID

The curve traced out by a point P on the circumference of a circle as the circle rolls along a straight line is called a cycloid.

Find parametric equations for the cycloid if:

– The circle has radius r and rolls along the x-axis.

– One position of P is the origin.

CYCLOIDS

We choose as parameter the angle of rotation θ of the circle (θ = 0 when P is at the origin).

Suppose the circle has rotated through θ radians.

CYCLOIDS

As the circle has been in contact with the line, the distance it has rolled from the origin is:| OT | = arc PT = rθ

– Thus, the center of the circle is C(rθ, r).

CYCLOIDS

Let the coordinates of P be (x, y). Then, from the figure, we see that:

– x = |OT| – |PQ| = rθ – r sin θ = r(θ – sinθ)

– y = |TC| – |QC| = r – r cos θ = r(1 – cos θ)

CYCLOIDS

Day 2, Session 4

Use Desmos to explore various cycloidsFamous Curves I

Session 4 – Famous Curves II

Hypocycloid – follow a point on a wheel as it rolls around the inside of another wheel

Session 5 – Famous Curves III

Hypotrochoid – follow a point on a spoke of a wheel as it rolls around the inside of another wheel

Session 5 – Famous Curves III

Epitrochoid – follow a point on a spoke of a wheel as it rolls around the outside of another wheel

Session 5

Wankel Engine – Famous Curves III activityDesign Time – with show off