Here are 100 chapter titles for a book on Rational Functions, progressing from beginner to advanced:
I. Foundations (1-20)
- Introduction to Rational Functions: What are They?
- Definition and Notation: Numerator and Denominator
- Domain of a Rational Function: Restrictions and Exclusions
- Evaluating Rational Functions: Substituting Values
- Simplifying Rational Functions: Reducing to Lowest Terms
- Equivalent Rational Functions: Recognizing Equality
- Operations on Rational Functions: Addition and Subtraction
- Finding a Common Denominator: Essential for Addition/Subtraction
- Operations on Rational Functions: Multiplication and Division
- Complex Fractions: Simplifying Nested Fractions
- Polynomial Division: Long Division and Synthetic Division (Review)
- Improper Fractions: Rational Functions Where Degree(Numerator) >= Degree(Denominator)
- Mixed Fractions: Expressing Improper Fractions
- Converting Between Improper and Mixed Fractions
- Graphing Linear Functions (Review): Lines
- Graphing Quadratic Functions (Review): Parabolas
- Introduction to Asymptotes: Vertical, Horizontal, and Slant
- Identifying Vertical Asymptotes: Zeros of the Denominator
- Identifying Horizontal Asymptotes: Comparing Degrees
- Review and Preview: Looking Ahead
II. Intermediate Techniques (21-40)
- Identifying Slant Asymptotes: When the Numerator's Degree is One Greater
- Graphing Rational Functions: Putting It All Together
- Holes in Graphs: Removable Discontinuities
- Finding Holes: Common Factors in Numerator and Denominator
- Graphing Rational Functions with Holes
- Intercepts of Rational Functions: x-intercepts and y-intercepts
- Zeros of Rational Functions: Where the Numerator is Zero
- Solving Rational Equations: Clearing the Denominator
- Extraneous Solutions: Checking for Validity
- Applications of Rational Functions: Modeling Real-World Phenomena
- Direct Variation: Rational Functions as Proportions
- Inverse Variation: Rational Functions with x in the Denominator
- Joint Variation: Multiple Variables
- Combined Variation: A Mix of Direct and Inverse
- Partial Fraction Decomposition: Breaking Down Complex Rational Functions
- Decomposing into Partial Fractions: Distinct Linear Factors
- Partial Fractions: Repeated Linear Factors
- Partial Fractions: Irreducible Quadratic Factors
- Applications of Partial Fractions: Calculus and Engineering
- Review and Practice: Intermediate Techniques
III. Advanced Topics (41-60)
- Limits and Rational Functions: Behavior Near Asymptotes
- Limits at Infinity: Horizontal Asymptotes Revisited
- Continuity of Rational Functions: Points of Discontinuity
- Differentiability of Rational Functions: The Quotient Rule
- Derivatives of Rational Functions: Applications to Graphing
- Increasing and Decreasing Intervals: Using the First Derivative
- Local Maxima and Minima: Critical Points
- Concavity and Inflection Points: Using the Second Derivative
- Optimization Problems: Maximizing or Minimizing Quantities
- Related Rates: Rates of Change in Connected Quantities
- Integration of Rational Functions: Partial Fractions in Calculus
- The Fundamental Theorem of Algebra (Review): Polynomial Roots
- Complex Numbers and Rational Functions: Complex Roots and Poles
- Rational Functions in Complex Analysis: Poles and Residues
- Laurent Series: Representing Functions with Positive and Negative Powers
- Conformal Mappings: Transformations Using Rational Functions
- Linear Fractional Transformations: Möbius Transformations
- The Riemann Sphere: Extending the Complex Plane
- Rational Functions and Polynomials: Connections and Differences
- Review and Practice: Advanced Topics
IV. Special Topics and Applications (61-80)
- Rational Functions in Signal Processing: Digital Filters
- Z-Transforms: Representing Discrete-Time Signals
- Transfer Functions: System Analysis
- Rational Functions in Control Theory: System Stability
- Rational Functions in Electrical Engineering: Circuit Analysis
- Impedance and Admittance: Rational Functions in AC Circuits
- Rational Functions in Cryptography: Error-Correcting Codes
- Reed-Solomon Codes: Using Rational Functions for Data Integrity
- Rational Functions in Computer Graphics: Bézier Curves and Surfaces
- Rational Bézier Curves: Generalizing Polynomial Bézier Curves
- Rational Functions in Economics: Modeling Economic Behavior
- Supply and Demand: Rational Functions in Market Analysis
- Rational Functions in Physics: Describing Physical Phenomena
- Optics: Lens Equations and Magnification
- Fluid Dynamics: Flow Rates and Pressure
- Rational Functions in Chemistry: Reaction Rates
- Enzyme Kinetics: Michaelis-Menten Equation
- Rational Functions in Biology: Population Growth Models
- Rational Functions in Statistics: Probability Distributions
- Advanced Applications: A Survey
V. Deeper Dive and Extensions (81-100)
- Rational Functions and Algebraic Geometry: Algebraic Curves
- Projective Geometry: Extending Euclidean Geometry
- The Riemann-Roch Theorem: Relating Functions and Divisors
- Elliptic Curves: Rational Functions and Number Theory
- Modular Forms: Functions with Special Symmetry Properties
- Continued Fractions: Representing Numbers as Rational Functions
- Padé Approximants: Approximating Functions with Rational Functions
- Rational Function Interpolation: Approximating Functions through Points
- Rational Splines: Smooth Curves Made of Rational Functions
- Rational Maps: Functions Between Algebraic Varieties
- The Jacobian Variety: Connecting Curves and Rational Functions
- Automorphisms of Rational Function Fields: Galois Theory Connections
- Rational Functions and Dynamical Systems: Iterating Functions
- Julia Sets and the Mandelbrot Set: Complex Dynamics
- Rational Functions and Coding Theory: Advanced Topics
- Rational Functions and Cryptography: Elliptic Curve Cryptography
- Computational Aspects of Rational Functions: Algorithms and Software
- History of Rational Functions: A Detailed Account
- Open Problems and Future Directions in Rational Function Research
- Research Topics in Rational Functions: A Guide for Exploration