age for ^^Undergraduates Gregory V. Bard Providence, Rhode Island AMERICAN MATHEMATICAL SOCIETY

Transcription

1 age for ^^Undergraduates Gregory V. Bard AMERICAN MATHEMATICAL SOCIETY Providence, Rhode Island

2 Contents Preface: How to Use This Book xv Acknowledgements xix Chapter 1. Welcome to Sage! Using Sage as a Calculator Using Sage with Common Functions Using Sage for Trigonometry Using Sage to Graph 2-Dimensionally Controlling the Viewing Window of a Plot Superimposing Multiple Graphs in One Plot Matrices and Sage, Part A First Taste of Matrices Complications in Converting Linear Systems Doing the RREF in Sage Basic Summary The Identity Matrix A Challenge to Practice by Yourself Vandermonde's Matrix The Semi-Rare Cases Making Your Own Functions in Sage Using Sage to Manipulate Polynomials Using Sage to Solve Problems Symbolically Solving Single-Variable Formulas Solving Multivariable Formulas Linear Systems of Equations Non-Linear Systems of Equations Advanced Cases Using Sage as a Numerical Solver Getting Help When You Need It Using Sage to Take Derivatives Plotting f(x) and f(x) Together 50 vii

3 1 viii Higher-Order Derivatives Using Sage to Calculate Integrals Sharing the Results of Your Work 60 Chapter 2. Fun Projects Using Sage Microeconomics: Computing a Selling Price Biology: Clogged Arteries and Poiseuille's Law Industrial Optimization: Shipping Taconite Chemistry: Balancing Reactions with Matrices B ackground Five Cool Examples The Slow Way: Deriving the Method Balancing the Quick Way Physics: Ballistic Projectiles Our First Example of Ballistic Trajectories Our Second Example of Ballistic Trajectories Counter-Battery Fire Your Challenge: A Multi-Stage Rocket Cryptology: Pollard's p - Attack on RSA Background: JB-Smooth Numbers and B\ The Theory behind the Attack Computing 2^ mod N Your Challenge: Make All This Happen in Sage Safeguards against Pollard's Factorial Attack Mini-Project on Electric Field Vector Plots 91 Chapter 3. Advanced Plotting Techniques Annotating Graphs for Clarity Labeling the Axes of Graphs Grids and Graphing Calculator-Style Graphs Adding Arrows and Text to Label Features Graphing an Integral Dotted and Dashed Lines Graphs of Some Hyperactive Functions Polar Plotting Examples of Polar Graphs Problems That Can Occasionally Happen Graphing an Implicit Function Contour Plots and Level Sets An Application to Thermodynamics Application to Microeconomics (Cobb-Douglas Equations) Parametric 2D Plotting Vector Field Plots Gradients and Vector Field Plots An Application from Physics 118

4 ix Gradients versus Contour Plot Log-Log Plots Rare Situations 124 Chapter 4. Advanced Features of Sage Using Sage with Multivariable Functions and Equations Working with Large Formulas in Sage Personal Finance: Mortgages Physics: Gravitation and Satellites Derivatives and Gradients in Multivariate Calculus Partial Derivatives Gradients Matrices and Sage, Part Defining Some Examples Matrix Multiplication and Exponentiation Right and Left System Solving Matrix Inverses Computing the Kernel of a Matrix Determinants Vector Operations Working with the Integers and Number Theory The gcd and the 1cm More about Prime Numbers About Euler's Phi Function The Divisors of a Number Another Meaning for tau Modular Arithmetic Further Reading in Number Theory Some Minor Commands of Sage Rounding, Floors, and Ceilings Combinations and Permutations The Hyperbolic Trigonometric Functions Calculating Limits Expressly Scatter Plots in Sage Making Your Own Regressions in Sage Computing in Octal? Binary? And Hexadecimal? Can Sage Do Sudoku? Measuring the Speed of Sage Huge Numbers and Sage Using Sage and L^K Matrices and Sage, Part Introduction to Eigenvectors Finding Eigenvalues Efficiently in Sage Matrix Factorizations Solving Linear Systems Approximately with Least Squares 173

5 X Computing Taylor or MacLaurin Polynomials Examples of Taylor Polynomials An Application: Understanding How g Changes Minimizations and Lagrange Multipliers Unconstrained Optimization Constrained Optimization by Lagrange Multipliers A Lagrange Multipliers Example in Sage Some Applied Problems Infinite Sums and Series Verifying Summation Identities The Geometric Series Using Sage to Guide a Summation Proof Continued Fractions in Sage Systems of Inequalities and Linear Programming A Simple Example Convenient Features in Practice The Polyhedron of a 3-Variable Linear Program Integer Linear Programs and Boolean Variables Further Reading on Linear Programming Differential Equations Some Easy Examples An Initial-Value Problem Graphing a Slope Field The Torpedo Problem: Working with Parameters Laplace Transforms Transforming from f(t) to L(s) Computing the Laplace Transform "the Long Way" Transforming from L(s) to f(t) Vector Calculus in Sage Notation for Vector-Valued Functions Computing the Hessian Matrix Computing the Laplacian The Jacobian Matrix The Divergence Verifying an Old Identity The Curl of a Vector-Valued Function A Challenge: Verifying Some Curl Identities Multiple Integrals 216 Chapter 5. Programming in Sage and Python Repetition without Boredom: The For Loop Using Sage to Generate Tables Carefully Formatting the Output Arbitrary Lists Loops with Accumulators 223

6 xi Using Sage to Find a Limit, Numerically For Loops and Taylor Polynomials If You Already Know How to Program Writing Subroutines An Example: Working with Coinage A Challenge: A Cash Register An Example: Designing Aquariums A Challenge: A Cylindrical Silo Combining Loops and Subroutines Another Challenge: Totaling a Sequence Loops and Newton's Method What Is Newton's Method? Newton's Method with a For Loop Testing the Code Numerical versus Exact Representations Working with Optional and Mandatory Parameters Returning a Value and Nesting Subroutines A Challenge: Finding a Point of Agreement A Challenge: Finding Parallel Tangent Lines A Challenge: Halley's Method An Introduction to Control Flow Verbosity Control Theoretical Interlude: When Newton's Method Goes Crazy Stopping Newton's Method Early The List of Comparators A Challenge: How Many Primes Are Less Than x? More Concepts in Flow Control Raising an "Exception" The If-Then-Else Construct Easy Challenge: Registrar's End of Semester Run, Part A Harder Challenge: Registrar's End of Semester Run, Part While Loops versus For Loops A Question about Factorials A Challenge: Finding the Next Prime Number Newton's Method with a While Loop The Impact of Repeated Roots Factorization by Trial Division A Mini-Challenge: Trial Division, Stopping Early A Challenge: An Upgraded Cash Register How Arrays and Lists Work Lists of Points and Plotting A Challenge: Making the Plot Command Operations on Lists Looping through an Array 274

7 xii A Challenge: A Company Profit Report Averaging Some Numbers Mini-Challenge: Averaging, Both with and without Dropping Mini-Challenge: Doubling Counting the Highest Quiz A Challenge: Registrar's End of Semester Run, Part A Utility: Returning Only Real, Rational, or Integer Roots Where Do You Go from Here? Further Resources on Python Programming What Has Been Left Out? 280 Chapter 6. Building Interactive Webpages with Sage The Six-Stage Process for Building Interacts The Tangent-Line Interact 287 Stage 0: Develop a Concept 287 Stage 1: Design a Sage Subroutine 288 Stage 2: Polish This Subroutine 290 Stage 3: Convert to an Interactive Subroutine 291 Stage 4: Insert into a Web Template 292 Stage 5: Flesh-Out the Webpage A Challenge to the Reader The Optimal Aquarium Interact Selectors and Checkboxes The Definite Integral Interact 297 Appendix A. What to Do When Frustrated! 299 Appendix B. Transitioning to SageMathCloud 305 B.l. What Is SageMathCloud? 305 B.2. Getting Started in SageMathCloud 306 B.3. Other Cloud Features 307 Appendix C. Other Resources for Sage 309 Appendix D. Linear Systems with Infinitely Many Solutions 313 D.l. The Opening Example 313 D.2. Another Example 316 D.3. Exploring Deeper 318 D.3.1. An Interesting Re-Examination of Unique Solutions 318 D.3.2. Two Equations and Four Unknowns 319 D.3.3. Two Equations and Four Unknowns, but No Solutions 319 D.3.4. Four Equations and Three Unknowns 320 D.4. Misconceptions 320 D.5. Formal Definitions of REF and RREF 321 D.6. Alternative Notation 322 D.7. Geometric Interpretations in Three Dimensions 322

8 xiii D.7.1. Visualization of These Cases 323 D.7.2. Geometric Interpretations in Higher Dimensions 324 D.8. Dummy-Variable Notation 324 D.9. Solutions to Challenges 325 Appendix E. Installing Sage on Your Personal Computer 327 Appendix F. Index of Commands by Section and by Name 331 List of References 351