Discovering Mathematics with Maple

1 A Tour of Maple V.- 1.1 Introduction.- 1.2 Maple’s User Interface.- 1.3 Just a Maple Session.- 1.3.1 Numerical Calculations.- 1.3.2 Symbolic Calculations.- 1.4 Worksheets and Assignments.- 1.4.1 Worksheet la. Numbers.- 1.4.2 Assignments la. Numbers.- 1.4.3 Worksheet lb. Variables and Names.- 1.4.4 Assignments lb. Variables and Names.- 1.5 Exercises.- 1.6 Survey of Used Maple Expressions.- 2 Functions and Sequences.- 2.1 Introduction.- 2.2 Selected Concepts from Analysis.- 2.3 A Tricky Probability Function.- 2.3.1 Mathematical Functions.- 2.3.2 Investigating a Probability Function.- 2.3.3 Investigating a Strange Sequence.- 2.4 Worksheets and Assignments.- 2.4.1 Worksheet 2a. Functions.- 2.4.2 Assignments 2a. Functions.- 2.4.3 Worksheet 2b. Sequences.- 2.4.4 Assignments 2b. Sequences.- 2.5 Exercises.- 2.6 Survey of Used Maple Expressions.- 3 Matrices and Vectors.- 3.1 Introduction.- 3.2 Selected Concepts from Linear Algebra.- 3.3 A Least Squares Filter.- 3.3.1 Matrices and Vectors.- 3.3.2 Removing Noise from a Data Set.- 3.4 Worksheets and Assignments.- 3.4.1 Worksheet 3a. Matrices.- 3.4.2 Assignments 3a. Matrices.- 3.4.3 Worksheet 3b. Vectors.- 3.4.4 Assignments 3b. Vectors.- 3.5 Exercises.- 3.6 Survey of Used Maple Expressions.- 4 Counting and Summation.- 4.1 Introduction.- 4.2 Selected Concepts from Discrete Mathematics.- 4.3 Binomial Coefficients.- 4.3.1 Maple’s binomial Function.- 4.3.2 Two Binomial Identities.- 4.3.3 The Binomial Series.- 4.3.4 The De Moivre-Laplace Limit Theorem.- 4.4 Worksheets and Assignments.- 4.4.1 Worksheet 4a. Symbolic Summation.- 4.4.2 Assignments 4a. Symbolic Summation.- 4.4.3 Worksheet 4b. Counting and Random Numbers.- 4.4.4 Assignments 4b. Counting and Random Numbers.- 4.5 Exercises.- 4.6 Survey of Used Maple Expressions.- 5 Derivative and Integral.- 5.1 Introduction.- 5.2 Further Concepts from Analysis.- 5.3 Riemann’s Non-differentiable Function.- 5.3.1 Investigating the Riemann Function.- 5.3.2 Finding a Primitive Function.- 5.4 Worksheets and Assignments.- 5.4.1 Worksheet 5a. Differentiation.- 5.4.2 Assignment 5a. Differentiation.- 5.4.3 Worksheet 5b. Integration.- 5.4.4 Assignments 5b. Integration.- 5.5 Exercises.- 5.6 Survey of Used Maple Expressions.- 6 Vector Spaces and Linear Mappings.- 6.1 Introduction.- 6.2 Further Concepts from Linear Algebra.- 6.3 Diagonalization and the SVD.- 6.3.1 A Simple Orthogonal Projection.- 6.3.2 Singular Value Decomposition.- 6.4 Worksheets and Assignments.- 6.4.1 Worksheet 6a. Linear Mappings.- 6.4.2 Assignments 6a. Linear Mappings.- 6.4.3 Worksheet 6b. Eigenvectors and Eigenspaces.- 6.4.4 Assignments 6b. Eigenvectors and Eigenspaces.- 6.5 Exercises.- 6.6 Survey of Used Maple Expressions.- A Exercises in Experimentation.- A.1 Upper Bound for a Parameterized Function.- A.2 Matrix Differentiation.- A.3 Chaotic Sequences.- B Hints and Answers.- B.1 Exercises 1.5.- B.2 Exercises 2.5.- B.3 Exercises 3.5.- B.4 Exercises 4.5.- B.5 Exercises 5.5.- B.6 Exercises 6.5.- C Quick Guide to Selected Maple Commands.- C.1 Linear Algebra.- C.1.1 Elementary Matrix Algebra.- C.1.2 Elementary Vector Algebra.- C.1.4 Eigenvalues and Eigenvectors.- C.2 Mathematical Analysis.- C.2.1 Writing Your Own Procedures.- C.2.2 Plotting the Graph of a Function.- C.2.3 Computing Limits.- C.2.4 Differential Calculus.- C.2.5 Integration.- C.2.6 Solving Equations.- C.2.7 Solving Differential/Difference Equations.- C.3 Statistics.- C.3.1 Probabilities.- C.3.2 Generating Random Numbers.- C.3.3 Analysis of Statistical Data.- C.4 Linear Programming.- C.5 Mathematical Finance.