Calculus

Calculus is the mathematical study of continuous change, comprised of two major branches: differential calculus and integral calculus. Differential calculus focuses on instantaneous rates of change and the slopes of curves by breaking problems down into infinitesimally small parts, while integral calculus joins these small pieces together to determine accumulated quantities, such as the area under a curve or the volume of a solid. These two concepts are fundamentally linked by the Fundamental Theorem of Calculus, providing a powerful framework for modeling and understanding dynamic systems across physics, engineering, economics, and other sciences.

1. Foundations for Calculus
   1. Review of Functions
      1. Definition of a Function
         1. Mapping and Correspondence
         2. Domain, Codomain, and Range
         3. Well-Definedness
      2. Function Notation
         1. Evaluating Functions
         2. Piecewise Functions
         3. Interval Notation
      3. Domain and Range
         1. Determining Domain Algebraically
         2. Determining Range Algebraically
         3. Domain and Range from Graphs
         4. Restrictions and Excluded Values
      4. Graphs of Functions
         1. Plotting Points
         2. Interpreting Graphs
         3. Identifying Key Features
            1. Intercepts
            2. Maxima and Minima
            3. Symmetry
         4. The Vertical Line Test
      5. Combining Functions
         1. Arithmetic Operations
            1. Sums and Differences
            2. Products and Quotients
            3. Domain of Combined Functions
         2. Composition of Functions
            1. Notation and Evaluation
            2. Domain of Composed Functions
            3. Decomposing Functions
      6. Transformations of Functions
         1. Translations
            1. Horizontal Shifts
            2. Vertical Shifts
         2. Reflections
            1. Across the x-axis
            2. Across the y-axis
         3. Stretches and Compressions
            1. Vertical Scaling
            2. Horizontal Scaling
         4. Combining Transformations
      7. Inverse Functions
         1. Definition and Properties
         2. The Horizontal Line Test
         3. Finding Inverse Functions Algebraically
         4. Verifying Inverses
         5. Graphs of Inverse Functions
         6. Restricting Domains
   2. Essential Function Types
      1. Linear Functions
         1. Slope-Intercept Form
         2. Point-Slope Form
         3. Standard Form
         4. Applications and Modeling
      2. Polynomial Functions
         1. Degree and Leading Coefficient
         2. End Behavior
         3. Zeros and Multiplicity
         4. Factoring Techniques
         5. Graphing Polynomial Functions
         6. Remainder and Factor Theorems
      3. Power Functions
         1. Integer Powers
         2. Fractional Powers
         3. Negative Powers
         4. Graphs and Properties
      4. Rational Functions
         1. Domain and Vertical Asymptotes
         2. Horizontal and Oblique Asymptotes
         3. Holes and Removable Discontinuities
         4. Graphing Rational Functions
         5. End Behavior
      5. Radical Functions
         1. Square Root Functions
         2. Cube Root Functions
         3. Higher-Order Roots
         4. Domain Considerations
      6. Exponential Functions
         1. Definition and Properties
         2. The Natural Base e
         3. Exponential Growth and Decay Models
         4. Graphs of Exponential Functions
         5. Solving Exponential Equations
      7. Logarithmic Functions
         1. Definition as Inverse of Exponential
         2. Properties of Logarithms
         3. Natural Logarithm
         4. Common Logarithm
         5. Change of Base Formula
         6. Graphs of Logarithmic Functions
         7. Solving Logarithmic Equations
   3. Trigonometry Review
      1. Angle Measurement
         1. Degree Measure
         2. Radian Measure
         3. Converting Between Systems
         4. Coterminal Angles
      2. The Unit Circle
         1. Coordinates of Key Angles
         2. Reference Angles
         3. Symmetry Properties
      3. Right Triangle Trigonometry
         1. Definitions of Trigonometric Ratios
         2. Pythagorean Theorem
         3. Special Right Triangles
         4. Solving Right Triangles
      4. Trigonometric Functions
         1. Primary Functions
            1. Sine
            2. Cosine
            3. Tangent
         2. Reciprocal Functions
            1. Cosecant
            2. Secant
            3. Cotangent
         3. Function Values and Signs by Quadrant
      5. Graphs of Trigonometric Functions
         1. Sine and Cosine Functions
            1. Amplitude
            2. Period
            3. Phase Shift
            4. Vertical Shift
         2. Tangent and Cotangent Functions
         3. Secant and Cosecant Functions
      6. Trigonometric Identities
         1. Fundamental Identities
            1. Pythagorean Identities
            2. Reciprocal Identities
            3. Quotient Identities
         2. Sum and Difference Formulas
         3. Double-Angle Formulas
         4. Half-Angle Formulas
         5. Product-to-Sum Formulas
      7. Inverse Trigonometric Functions
         1. Definitions and Restricted Domains
         2. Principal Values
         3. Graphs of Inverse Functions
         4. Solving Trigonometric Equations
   4. Algebraic Prerequisites
      1. Factoring Techniques
         1. Common Factors
         2. Difference of Squares
         3. Perfect Square Trinomials
         4. General Trinomials
         5. Sum and Difference of Cubes
      2. Rational Expressions
         1. Simplifying Rational Expressions
         2. Operations with Rational Expressions
         3. Complex Fractions
      3. Radical Expressions
         1. Simplifying Radicals
         2. Operations with Radicals
         3. Rationalizing Denominators
      4. Solving Equations
         1. Linear Equations
         2. Quadratic Equations
         3. Rational Equations
         4. Radical Equations
         5. Exponential and Logarithmic Equations