Hands on Mathematics for CS - Exercises

6. Session: December 1st

Applet text exercises ( Session 3 )

• Stefan Nesbigall, Sebastian Germesin

• Kamran Azam, Suleman Butt
• Hideo, Sato, Björn Gehl
• Sebastian Blohm, Daniel Beck
• Steffen Knapp, Christian Hümbert
• Sarmad Hussain, Noreen Nazar

• Michael Dietrich, Dominik Jednoralski (last-weeks exercise)

• Maciej Mrowiec, Khoa Chu

Exercises

Joint problem spaces: There are different ways of developing proofs. Teachers typically focus only on one kind of proof and omit other approaches. However, it is important to deal with different proofs in order to deeply understand the problem and to have a view on it from different sides. Therefore comparing proof attempts is a useful cognitive method.

For instance, for the particular zero sequence we know the following proofs:

• proof by logarithms
• proof by Bernoulli's inequality
• proof by squeeze theorem (Schachtelsatz; Einschließungskriterium )

1. First provide a proof by sqeeze theorem (Schachtelsatz). Then compare all three approaches. To do this, decompose all solutions into pieces. Insert links between pieces if you think they are related somehow. For example they might share a part of the solution path. Mark/Label links if they have different meanings. As a result you obtain a set of connected components.
   

   Do you know any other problem that can be solved in different ways? (**)

2. Imagine a scenario in which three students work at home and try to develop a proof for the same problem. Each of them uses a mathematical tool, which provides a peer to peer connection, such that all students can communicate with each other. This tool enables a private as well as a public workspace allowing to share ideas and concepts.
   

   Design a mudular tool for joint problem spaces. Explain its functionalities, fundamental data structures, visualizations, communication and interactions. Give free rein to your fantasy. What do you expect from such a tool? How would you integrated it into ActiveMath? (*)

Definition of limit on sequences:

• Structure and illustrate the proof for The limit of a convergent sequence is unique . What is given? What is unknown? What new knowledge can be derived from given assumption. Design a graphical Interface that supports learners in understanding the proof. (*)
  

• What do the following terms (formulae) express? Which of these terms characterize all sequences of real numbers , x_n [n], that approximate some "limit" x : xn-->x? Give examples and counter examples. Sketch pictures. (*)