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E-raamat: Exploring Mathematics with CAS Assistance

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Exploring Mathematics with CAS Assistance is designed as a textbook for an innovative mathematics major course in using a computer-algebra system (CAS) to investigate, explore, and apply mathematical ideas and techniques in problem solving. The book is designed modularly with student investigations and projects in number theory, geometry, algebra, single-variable calculus, and probability. The goal is to provoke an inquiry mindset in students and to arm them with the CAS tools to investigate low-entry, open-ended questions in a variety of mathematical arenas. Because of the modular design, the individual chapters could also be used selectively to design student projects in a number of upper-division mathematics courses. These projects could, in fact, lead into undergraduate research projects. The existence of powerful computer-algebra systems has changed the way mathematicians perform research; this book enables instructors to put some of those new methods and approaches into their undergraduate instruction.

Prerequisites include a basic working knowledge of discrete mathematics and single-variable calculus. Programming experience and some basic familiarity with elementary probability and statistics are beneficial but not required. The book takes a software-agnostic approach and emphasizes algorithmic structure of solution methods by systematically providing their step-by-step verbal descriptions or suitable pseudocode that can be implemented in any CAS.
Introduction xi
Acknowledgments xv
Part 1 Algebra & Geometry
1(96)
1 Computer Algebra Systems and Elements of Algorithmics
3(16)
1.1 Terms and notation
4(1)
1.2 About data types and data structures
5(3)
1.3 Elements of algorithmics and algorithmic problem solving
8(8)
1.4 Glossary
16(3)
2 Topics in Classical Geometry
19(16)
2.1 Review: Matrices, vectors, and lines
19(6)
2.2 Rigid transformations of the plane
25(2)
2.3 Complex numbers in classical geometry
27(3)
2.4 Three centers of a triangle
30(3)
2.5 Glossary
33(2)
3 More Topics in Classical Geometry
35(12)
3.1 Lab 2: The Euler line
35(2)
3.2 The Simson line
37(3)
3.3 Conies
40(5)
3.4 Glossary
45(2)
4 Topics in Elementary Number Theory
47(18)
4.1 Number of primes and the Riemann Hypothesis
49(1)
4.2 Algorithms from elementary number theory
50(6)
4.3 Pythagorean triples
56(2)
4.4 Lab 4: Plotting legs of primitive Pythagorean triples
58(1)
4.5 Linear Diophantine equations in two variables
59(3)
4.6 Lab 5: Industrial application of an LDE in three variables
62(1)
4.7 Glossary
63(2)
5 Topics in Algebra: Solving Univariate Algebraic Equations
65(14)
5.1 Roots of univariate polynomials
66(3)
5.2 Geometry of cubic equations: Counting the number of real roots
69(5)
5.3 Lab 6: Solving cubic equations using Vieta's substitution
74(2)
5.4 Nonnegative univariate polynomials
76(2)
5.5 Glossary
78(1)
6 Topics in Algebra: Bivariate Systems of Polynomial Equations
79(18)
6.1 Linear systems of two equations
80(2)
6.2 Nonlinear systems of polynomial equations: Motivating example
82(3)
6.3 Solving nonlinear polynomial systems
85(9)
6.4 Implicitization of plane curves
94(2)
6.5 Glossary
96(1)
Part 2 Calculus and Numerics
97(94)
7 Derivatives
99(14)
7.1 Review: Definitions, notation, and terminology
100(4)
7.2 Convexity of a univariate function
104(1)
7.3 Some facts about functions and derivatives
105(5)
7.4 Lab 8: Constructing a square circumscribed about ellipse
110(2)
7.5 Glossary
112(1)
8 Definite Integrals
113(16)
8.1 Review: Some basic concepts and facts of univariate integral calculus
114(2)
8.2 Area of a region bounded by a simple closed curve
116(5)
8.3 Lab 9: Submergence depth of a body of revolution in equilibrium
121(2)
8.4 Solving some ordinary differential equations
123(4)
8.5 Glossary
127(2)
9 Approximating Zeros of Functions by Iteration Methods
129(14)
9.1 Fixed point iteration method
130(5)
9.2 Newton's method
135(2)
9.3 Lab 11: Kepler's Equation and deriving Kepler's Second Law
137(3)
9.4 Lab 12: Exploration of the logistic maps
140(1)
9.5 Glossary
141(2)
10 Polynomial Approximations
143(16)
10.1 Taylor polynomials
145(2)
10.2 Interpolating polynomials in the Lagrange form
147(3)
10.3 Piecewise polynomial interpolation: Splines
150(3)
10.4 Approximating large data sets: Regression
153(3)
10.5 Two real-life applications of the LS method
156(2)
10.6 Glossary
158(1)
11 Trigonometric Approximation
159(16)
11.1 Short review of trigonometric functions
160(4)
11.2 Fourier series
164(3)
11.3 About the accuracy of trigonometric approximations
167(3)
11.4 Celebrated classical application of Fourier series
170(3)
11.5 Glossary
173(2)
12 Fourier Analysis in Music and Signal Processing
175(16)
12.1 Introduction and background
175(2)
12.2 Fourier series and periodic signals
177(3)
12.3 The Fourier transform for non-periodic signals
180(2)
12.4 The Discrete Fourier Transform
182(5)
12.5 Fourier series in signal processing
187(1)
12.6 Glossary
188(3)
Part 3 Probability and Statistics
191(48)
13 Probability and Statistics Basics
193(16)
13.1 Review: Some basic concepts of probability
194(3)
13.2 Some discrete probability distributions
197(4)
13.3 About continuous probability distributions
201(3)
13.4 Law of Large Numbers
204(2)
13.5 Central Limit Theorem
206(2)
13.6 Glossary
208(1)
14 Computer Simulation of Statistical Sampling
209(16)
14.1 Random number generation
209(2)
14.2 Lab 17: CLT and LLN in action: Life expectancy in the world population
211(2)
14.3 Sampling from non-uniform distributions (optional)
213(2)
14.4 Monte Carlo methods for finding integrals and areas
215(7)
14.5 Lab 18: Buffon's needle problem
222(1)
14.6 Glossary
223(2)
15 Simple Random Walks
225(14)
15.1 Simple random walks on integers
226(5)
15.2 Lab 19: The gambler's ruin problem
231(2)
15.3 Random walk on the square lattice
233(1)
15.4 Lab 20: Drunken sailor problem
234(1)
15.5 Glossary
235(2)
A Data for Lab 17 in
Chapter 14
237(2)
Bibliography 239(2)
Index 241
Lydia S. Novozhilova, Western Connecticut State University, Danbury, CT.

Robert D. Dolan, University of Connecticut, Storrs, CT.
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