Preface
To theย Student
Aย Note onย Computer Algebra Systems
Current Computer Algebra Systems
Calculators withย Computer Algebra Capabilities
Contents
1 Numerical Differentiation
1.1 Taylorโs Theorem
1.1.1 Forms of the Remainder
1.1.2 Big O'' Notation
1.2 Proving Taylor's Theorem
1.3 Maple and Taylor Expansions
1.4 Tailored Expressions for the First Derivative
1.4.1 Two Useful Taylor Expansions
1.4.2 Forward Difference Approximation
1.4.3 Backward Difference Approximation
1.4.4 Centered Difference Approximation
1.4.5 Comparing the Error Terms
1.5 Higher Derivatives
1.6 Next Steps
1.7 Taylor's Statement of the Theorem
1.8 Centered Difference Coefficients Chart
1.9 Links and Others
2 Numerical Integration
2.1 Methods of Elementary Calculus
2.1.1 Rectangle Methods
2.1.2 Trapezoid Sums
2.1.3 Simpson's Rule
2.1.4 A Maple Comparison
2.1.5 A Chart of the Methods
2.2 The Basic Integration Methods and Maple
2.3 Gaussian Quadrature
2.3.1 In Search of Improvements
2.3.2 Patterns
2.3.3 Testing Gauss
2.3.4 An Error Bound for Gaussian Quadrature
2.3.5 Values of Gaussian Weights and Nodes
2.4 Gauss-Kronrod Quadrature
2.4.1 Aleksandr Kronrod's Idea
2.4.2 Gauss-Kronrod Quadrature in Practice
2.5 Transforming the Integral Over [a,b] to [-1,1]
2.6 Easy, but Hard: A Class Quadrature Project
2.7 Next Steps
2.8 A Menagerie of Test Integrals
2.9 Legendre and Stieltjes Polynomials for GK7,15
2.10 Links and Others
3 Projects
3.1 Polynomials and Roots from Data Using Numerical Derivatives
3.1.1 Monic Polynomials and Their Derivatives
3.1.2 The Roots
3.1.3 The Project
3.2 Space Shuttle Acceleration
3.2.1 Space Shuttle Acceleration: The Situation
3.2.2 The Project
3.3 Measuring Capacitance and Inductance with Derivatives
3.3.1 Capacitance and Inductance Equations
3.3.2 The Project: Measuring Component Values
3.4 Marginal Revenue From Data, Cost, and Profit
3.4.1 Marginal Revenue and Marginal Cost
3.4.2 The Project
3.5 Estimating Speed and Acceleration from GPS Data
3.5.1 The Global Positioning System and Distance
3.5.2 The Project
3.6 Fornberg's Algorithm
3.6.1 The Algorithm
3.6.2 The Project
3.7 Euler's Method of Solving Differential Equations
3.7.1 Euler's Method Graphically
3.7.2 Euler's Method Algebraically
3.7.3 The Project
3.8 A Dog's Life
3.8.1 Rate of Aging Data
3.8.2 The Project
3.9 Computing the Cost at a Gas Pump
3.9.1 Calculating the Cost of the Gas Pumped
3.9.2 The Project
3.10 The Fourier Power Spectrum
3.10.1 Calculating the Power Spectrum of a Signal
3.10.2 The Project: Produce a Spectrum Plot of the Square Wave
3.10.3 Extra for Experts
3.11 One Function for All, The Normal Distribution
3.11.1 The Project
3.12 Monte Carlo Integration
3.12.1 Algebraic: The Definite Integral From an Average
3.12.2 Geometric: The Definite Integral From Areas
3.12.3 The Project
3.13 Computing a Transition Curve
3.13.1 Transition Curves
3.13.2 The Project: Computing a Transition Curve for a Roadbed
3.14 Finding the Factorial of -1/2
3.14.1 The Gamma Function
3.14.2 The Project
3.15 Theory: Plugging the Hole in thePower Rule''
3.15.1 The Integral ``Power Rule''
3.15.2 The Project
Further Readings
Index