1. Introduction to Calculus

    1. What is Calculus?
    2. Prologue and Historical Context
    3. Understanding the Gradient function
  2. Limits

    1. Introduction to Limits
    2. Determining Limits Graphically
    3. Formal Definition of a Limit at a Point
    4. Determining One Sided Limits
    5. Determining When a Limit does not Exist
    6. Determining Limits Algebraically
    7. Infinite Limits and Vertical Asymptotes
    8. Limits for The Squeeze Theorem
    9. Limits at Infinity and Horizontal Asymptotes
    10. Continuous Functions
    11. Intemediate Value Theorem
    12. Definition of Continuity at a Point
    13. Classifying Topics of Discontinuity (removable vs. non-removable)
  3. Derivatives

    1. Rate of Change of a Function
    2. Average Rate of Change Over an Interval
    3. Instantaneous Rate of Change at a Point
    4. Slope of a Curve at a Point
    5. Tangent Line to a Curve
    6. Normal Line to a Tangent
    7. Average Velocity
    8. Instantaneous Velocity
    9. Limit Definition of Derivative
    10. First Principles Example 1: x²
    11. First Principles Example 2: x³
    12. First Principles Example 3: square root of x
    13. Standard Notation and Terminology
    14. Differentiable vs. Non-differentiable Functions

  4. Basic Differentiation Rules

    1. Power Rule
    2. Sum Rule
    3. Product Rule
    4. Proof of the Product Rule
    5. Quotient Rule
    6. Proof of Quotient Rule
    7. Chain Rule
    8. Implicit Differentiation
    9. Summary of Differentiation Rules
  5. Differentiating Trigonometric Functions

    1. Limits Involving Trigonometric Functions
    2. Intuitive Approach to the derivative of y=sin(x)
    3. Derivative Rules for y=cos(x) and y=tan(x)
    4. Differentiating sin(x) from First Principles
    5. Special Limits Involving sin(x), x, and tan(x)
    6. Graphical Relationship Between sin(x), x, and tan(x), using Radian Measure
    7. Derivatives of y=sec(x), y=cot(x), y= csc(x)
    8. Differentiating Inverse Trigonometric Functions
  6. Differentiating Exponential Functions

    1. From First Principles
    2. Differentiating Exponential Functions with Calculators
    3. Differentiating Exponential Functions with Base e
    4. Differentiating Exponential Functions with Other Bases

  7. Differentiating Logarithmic Functions

    1. Differentiating Logarithmic Functions with Base e
    2. Differentiating Logarithmic Functions without Base e
    3. Overview of Different Functions
  8. Graphing with the First Derivative

    1. Interpreting the Sign of the First Derivative (Increasing and Decreasing Functions)
    2. Identifying Stationary Points (Critical Points) for a Function
    3. Identifying Turning Points (Local Extrema) for a Function
    4. Classifying Critical Points and Extreme Values for a Function
    5. Mean Value Theorem for Continuous Functions
  9. Graphing with the Second Derivative

    1. Relationship between First and Second Derivatives of a Function
    2. Analyzing Concavity of a Function
    3. Notation for the Second Derivative
    4. Application of the Second Derivative (Acceleration)
    5. Determining Points of Inflection for a Function
    6. First Derivative Test vs Second Derivative Test for Local Extrema
    7. The special case of x⁴
    8. Critical Points of Inflection
    9. Examples of Curve Sketching

  10. Applications of Derivatives

    1. Introduction
    2. Solving Optimization Problems
    3. Using the Tangent Line to Approximate Function Values
    4. Using Newton's Method to Approximate Solutions to Equations
    5. Using Implicit Differentiation to Solve Related Rates Problems
  11. Introduction to Integration

    1. Sigma Notation
    2. Integration: the Area Problem
    3. Formal Definition of the Definite Integral
    4. Definite and indefinite integrals
    5. Integrals of Polynomial functions
    6. Determining Basic Rates of Change Using Integrals
    7. Integrals of Trigonometric Functions
    8. Integrals of Exponential Functions
    9. Integrals of Rational Functions
    10. Basic Properties of Definite Integrals
    11. The Fundamental Theorem of Calculus
  12. Techniques of Integration

    1. Evaluating the Constant of Integration
    2. Integration by Substitution
    3. Integration by Parts
    4. Integration by Trigonometric Substitution
    5. Integral by Partial Fractions

  13. Using Integrals to Find Areas and Volumes

    1. Symmetrical Areas
    2. Calculating Areas using Integrals
    3. Calculating Volume using Integrals
    4. Deriving Formulae Related to Circles using Integration
    5. Definite Integrals with Substitution
  14. Methods of Approximating Integrals

    1. RAM (Rectangle Approximation Method/Riemann Sum)
    2. Integration Using the Trapezoidal Rule
    3. Integration Using Simpson's Rule
    4. Analyzing Approximation Error
    5. Integration Using Euler's Method
  15. Applications of Definite Integrals

    1. The Average Value of a Function
    2. Solving Separable Differential Equations
    3. Slope Fields
    4. Exponential Growth and Decay Models
    5. Logistic Growth Models
    6. Net Change: Motion on a Line
    7. Determining the Surface Area of a Solid of Revolution
    8. Determining the Length of a Curve
    9. Determining the Volume of a Solid of Revolution
    10. Determining Work and Fluid Force

  16. Parametric Functions

    1. Introduction to Parametric Equations
    2. Derivative of Parametric Functions
    3. Determining the Length of a Parametric Curve (Parametric Form)
    4. Determining the Surface Area of a Solid of Revolution
    5. Determining the Volume of a Solid of Revolution
  17. Polar Curves

    1. Introduction to Polar Coordinates
    2. Determining the Slope and Tangent Lines for a Polar Curve
    3. Calculating Polar Areas
    4. Determining the Length of a Polar Curve
    5. Determining the Surface Area of a Solid of Revolution
    6. Determining the Volume of a Solid of Revolution
  18. Tests of Convergence / Divergence

    1. Indeterminate Forms and de L'hospital's Rule
    2. Infinite Sequences
    3. Infinite Series
    4. Partial Sums of Infinite Series
    5. Geometric Series
    6. Harmonic Series
    7. Nth Term Test for Divergence of an Infinite Series
    8. Integral Test for Convergence of an Infinite Series
    9. Alternating Series Test (Leibniz's Theorem) for Convergence of an Infinite Series
    10. Direct Comparison Test for Convergence of an Infinite Series
    11. Limit Comparison Test for Convergence of an Infinite Series
    12. Ratio Test for Convergence of an Infinite Series
    13. Root Test for for Convergence of an Infinite Series
    14. Strategies to Test an Infinite Series for Convergence

  19. Power Series

    1. Introduction to Power Series
    2. Determining the Radius and Interval of Convergence for a Power Series
    3. Differentiating and Integrating Power Series
    4. Constructing a Taylor Series
    5. Constructing a Maclaurin Series
    6. Lagrange Form of the Remainder Term in a Taylor Series
    7. Applications of Power Series
    8. Power Series Representations of Functions
    9. Power Series and Exact Values of Numerical Series
    10. Power Series and Estimation of Integrals
    11. Power Series and Limits
    12. Product of Power Series
    13. Binomial Series
    14. Power Series Solutions of Differential Equations