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.
Author(s): Lydia S. Novozhilova, Robert D. Dolan
Series: Classroom Resource Materials, 69
Publisher: American Mathematical Society
Year: 2022
Language: English
Pages: 259
City: Providence
Front Cover
Title page
Copyright
Contents
Introduction
Acknowledgments
Part 1. Algebra & Geometry
Chapter 1. Computer Algebra Systems and Elements of Algorithmics
1.1. Terms and notation
1.2. About data types and data structures
1.3. Elements of algorithmics and algorithmic problem solving
1.4. Glossary
Chapter 2. Topics in Classical Geometry
2.1. Review: Matrices, vectors, and lines
2.2. Rigid transformations of the plane
2.3. Complex numbers in classical geometry
2.4. Three centers of a triangle
2.5. Glossary
Chapter 3. More Topics in Classical Geometry
3.1. Lab 2: The Euler line
3.2. The Simson line
3.3. Conics
3.4. Glossary
Chapter 4. Topics in Elementary Number Theory
4.1. Number of primes and the Riemann Hypothesis
4.2. Algorithms from elementary number theory
4.3. Pythagorean triples
4.4. Lab 4: Plotting legs of primitive Pythagorean triples
4.5. Linear Diophantine equations in two variables
4.6. Lab 5: Industrial application of an LDE in three variables
4.7. Glossary
Chapter 5. Topics in Algebra: Solving Univariate Algebraic Equations
5.1. Roots of univariate polynomials
5.2. Geometry of cubic equations: Counting the number of real roots
5.3. Lab 6: Solving cubic equations using Vieta’s substitution
5.4. Nonnegative univariate polynomials
5.5. Glossary
Chapter 6. Topics in Algebra: Bivariate Systems of Polynomial Equations
6.1. Linear systems of two equations
6.2. Nonlinear systems of polynomial equations: Motivating example
6.3. Solving nonlinear polynomial systems
6.4. Implicitization of plane curves
6.5. Glossary
Part 2. Calculus and Numerics
Chapter 7. Derivatives
7.1. Review: Definitions, notation, and terminology
7.2. Convexity of a univariate function
7.3. Some facts about functions and derivatives
7.4. Lab 8: Constructing a square circumscribed about ellipse
7.5. Glossary
Chapter 8. Definite Integrals
8.1. Review: Some basic concepts and facts of univariate integral calculus
8.2. Area of a region bounded by a simple closed curve
8.3. Lab 9: Submergence depth of a body of revolution in equilibrium
8.4. Solving some ordinary differential equations
8.5. Glossary
Chapter 9. Approximating Zeros of Functions by Iteration Methods
9.1. Fixed point iteration method
9.2. Newton’s method
9.3. Lab 11: Kepler’s Equation and deriving Kepler’s Second Law
9.4. Lab 12: Exploration of the logistic maps
9.5. Glossary
Chapter 10. Polynomial Approximations
10.1. Taylor polynomials
10.2. Interpolating polynomials in the Lagrange form
10.3. Piecewise polynomial interpolation: Splines
10.4. Approximating large data sets: Regression
10.5. Two real-life applications of the LS method
10.6. Glossary
Chapter 11. Trigonometric Approximation
11.1. Short review of trigonometric functions
11.2. Fourier series
11.3. About the accuracy of trigonometric approximations
11.4. Celebrated classical application of Fourier series
11.5. Glossary
Chapter 12. Fourier Analysis in Music and Signal Processing
12.1. Introduction and background
12.2. Fourier series and periodic signals
12.3. The Fourier transform for non-periodic signals
12.4. The Discrete Fourier Transform
12.5. Fourier series in signal processing
12.6. Glossary
Part 3. Probability and Statistics
Chapter 13. Probability and Statistics Basics
13.1. Review: Some basic concepts of probability
13.2. Some discrete probability distributions
13.3. About continuous probability distributions
13.4. Law of Large Numbers
13.5. Central Limit Theorem
13.6. Glossary
Chapter 14. Computer Simulation of Statistical Sampling
14.1. Random number generation
14.2. Lab 17: CLT and LLN in action: Life expectancy in the world population
14.3. Sampling from non-uniform distributions (optional)
14.4. Monte Carlo methods for finding integrals and areas
14.5. Lab 18: Buffon’s needle problem
14.6. Glossary
Chapter 15. Simple Random Walks
15.1. Simple random walks on integers
15.2. Lab 19: The gambler’s ruin problem
15.3. Random walk on the square lattice
15.4. Lab 20: Drunken sailor problem
15.5. Glossary
Appendix A. Data for Lab 17 in Chapter 14
Bibliography
Index
Back Cover
