1. Using Technology to Uncover the Mathematics August 3-6, 2015 Dave Brownslides available at Professor, Ithaca Collegehttp://faculty.ithaca.edu/dabrown/geneva/ dabrown@ithaca.edu

2. Recursion Modeling 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.

3. 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?

4. Day 2, Session 1 Implicit Curves – Technology as exploration – Intro to Desmos – Desmos.com Desmos.com – Play a little

5. Day 2, Session 1 Implicit Curves – Technology as exploration – Is this the graph of a function? Why or why not? This is the curve y 2 =x 3 +x 2 -3x+2 What does this mean?

6. Day 2, Session 1 Implicit Curves Plotting and exploration using parameters On to Activities!

7. Day 2, Session 2 Day 1 – Described plane curves via 1.Explicitly: y as a function of x -> y=f(x) y=3x+2 2.Implicitly: relation between x and y x 2 +y 2 =1 xy+x 3 =y 4 – The first is easy to plot and visualize – The second requires understanding what points in Cartesian plane satisfy the relation.

8. Curve vs Parametric We see Billy’s path, but what are we missing?

9. Figure Skating

10. 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?

11. 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!

12. 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.

13. Explorations Activity 1 – Inch worm races Discussion Activity 2 – Non-linear inch worm Discussion Activity 3 – Multiple inch worms and collisions Discussion

14. Day 2, Session 3 Linear motion – Ant on the Picnic Table Activity

15. Day 2, Session 3 Example

16. Day 2, Session 3 Exploration with graphing calculator x=A cos t, y=B sin t, with A,B any numbers Try A=3, B=2; Try A=1, B=1 Explore 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?

17. Day 2, Session 3 Parameter Elimination Activity Ferris Wheel Activity

18. 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?

19. Day 2, Session 3 Trammel of Archimedes

20. Day 2, Session 4 Brachistochrone Problem and the cycloid Fascinating history

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

22. Acta Eruditorum, June 1696 I, Johann Bernoulli, address the most brilliant mathematicians in the world.

23. 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.

24. 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.

25. 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.

26. 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.

27. 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)

28. Galileo’s curves of quickest descent, 1638

29. Curve of Fastest Descent

30. Solutions and Commentary June 1696: Problem proposed in Acta Bernoulli: the “lion is known by its claw” when reading anonymous Royal Society paper May 1697: solutions in Acta Eruditorum from Bernoulli, Bernoulli, Newton, Leibniz, l’Hospital 1699: Leibniz reviews solutions from Acta

31. 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.

32. 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.

33. 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.

34. 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.

35. 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

36. 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

37. 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

38. 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

39. Day 2, Session 4 Use Desmos to explore various cycloids Famous Curves I

40. Session 4 – Famous Curves II Hypocycloid – follow a point on a wheel as it rolls around the inside of another wheel

41. Session 5 – Famous Curves III Hypotrochoid – follow a point on a spoke of a wheel as it rolls around the inside of another wheel

42. Session 5 – Famous Curves III Epitrochoid – follow a point on a spoke of a wheel as it rolls around the outside of another wheel

43. Session 5 Wankel Engine – Famous Curves III activity Design Time – with show off