Laboratories in Mathematical Experimentation : A Bridge to Higher Mathematics

by Cobb, George; Davidoff, Giuliana; Durfee, Alan; Gifford, Janice; O'Shea, Donal; Peterson, Mark; Pollatsek, Harriet

• ISBN13: 9780387949222
• ISBN10: 0387949224
• Edition: 1st
• Format: Paperback
• Copyright: 1997-03-01
• Publisher: Springer Verlag

Summary

The text is composed of a set of sixteen laboratory investigations which allow the student to explore rich and diverse ideas and concepts in mathematics. The approach is hands-on, experimental, an approach that is very much in the spirit of modern pedagogy. The course is typically offered in one semester, at the sophomore (second year) level of college. It requires completion of one year of calculus. The course provides a transition to the study of higher, abstract mathematics. The text is written independent of any software. Supplements will be available on the projects' web site.

Table of Contents

Preface	vii	(8)
Introduction	xv
1 Iteration of Linear Functions	1	(15)
1.1 Introduction	1	(1)
1.2 What is iteration?	2	(1)
1.3 The mathematical ideas	3	(4)
1.4 Questions to explore	7	(4)
1.5 Discussion	11	(2)
1.6 Bibliography	13	(1)
Computer programs	14	(2)
2 Cyclic Difference Sets	16	(14)
2.1 Introduction	16	(1)
2.2 Arithmetic modulo 15	17	(2)
2.3 Cyclic difference sets modulo m	19	(2)
2.4 Questions to explore	21	(3)
2.5 Discussion	24	(1)
Computer programs	25	(5)
3 The Euclidean Algorithm	30	(24)
3.1 Introduction	30	(1)
3.2 The algorithm	31	(4)
3.3 Questions and discussion	35	(5)
3.4 Linear Diophantine Equations	40	(5)
3.5 Additional topic	45	(1)
Computer programs	46	(8)
4 Prime Numbers	54	(25)
4.1 Introduction	54	(1)
4.2 Listing prime numbers	55	(5)
4.3 Functions generating primes	60	(4)
4.4 Distribution of primes	64	(3)
4.5 Further reading	67	(1)
Computer programs	67	(12)
5 The Coloring of Graphs	79	(15)
5.1 Introduction	79	(2)
5.2 Introduction to the mathematical ideas	81	(9)
5.3 Questions to explore	90	(3)
5.4 Bibliography	93	(1)
6 Randomized Response Surveys	94	(25)
6.1 Introduction	94	(1)
6.2 Asking sensitive questions	95	(1)
6.3 Background	96	(1)
6.4 Questions to explore	97	(15)
Computer programs	112	(7)
7 Polyhedra	119	(4)
7.1 Introduction	119	(1)
7.2 Questions and discussion	120	(2)
7.3 Additional topic	122	(1)
8 The p-adic Numbers	123	(19)
8.1 Introduction	123	(2)
8.2 Absolute values on Q	125	(3)
8.3 The real numbers	128	(3)
8.4 The p-adic numbers	131	(11)
9 Parametric Curve Representation	142	(17)
9.1 Introduction	142	(1)
9.2 Symmetries and closed curves	143	(6)
9.3 Questions to explore	149	(3)
9.4 Polar representation of curves	152	(3)
9.5 Additional ideas to explore	155	(1)
Computer programs	155	(4)
10 Numerical Integration	159	(22)
10.1 Introduction	159	(1)
10.2 Standard numerical methods	160	(3)
10.3 Automating the standard methods	163	(3)
10.4 Questions to explore	166	(3)
10.5 Monte Carlo methods	169	(3)
10.6 Higher dimensions	172	(3)
Computer programs	175	(6)
11 Sequences and Series	181	(22)
11.1 Introduction	181	(1)
11.2 The mathematical ideas	182	(3)
11.3 The harmonic series	185	(6)
11.4 The natural logarithm	191	(4)
11.5 Euler's constant	195	(3)
11.6 Additional exercises and questions	198	(2)
Computer programs	200	(3)
12 Experiments in Periodicity	203	(16)
12.1 Introduction	203	(2)
12.2 Area accumulation using CALCWIN	205	(7)
12.3 A new type of function	212	(2)
12.4 Antiderivatives of periodic functions	214	(1)
12.5 Finding the periodic antiderivative	215	(2)
12.6 Further investigation	217	(2)
13 Iteration to Solve Equations	219	(7)
13.1 Introduction	219	(3)
13.2 Improving convergence	222	(1)
13.3 Questions to explore	223	(1)
Computer programs	224	(2)
14 Iteration of Quadratic Functions	226	(17)
14.1 Introduction	226	(1)
14.2 Some theory	226	(1)
14.3 Iterating f(x) = ax(1 - x)	227	(3)
14.4 The Feigenbaum diagram	230	(1)
14.5 Examining chaos	231	(3)
14.6 The tent and sawtooth functions	234	(1)
14.7 Conjugacy	235	(1)
14.8 Iterating other functions	236	(1)
14.9 Listening to chaos	236	(1)
14.10 Bibliography	237	(1)
Computer programs	237	(6)
15 Iterated Linear Maps in the Plane	243	(16)
15.1 Introduction	243	(1)
15.2 Multiplying matrices	244	(3)
15.3 An example to start	247	(5)
15.4 Questions to explore	252	(1)
15.5 Discussion	253	(2)
Computer programs	255	(4)
16 Euclidean Algorithm for Complex Integers	259	(16)
16.1 Introduction	259	(1)
16.2 Complex integers	260	(8)
16.3 Questions and discussion	268	(3)
Computer programs	271	(4)
Index	275