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

  3. Derivatives

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

  4. Basic Differentiation Rules

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

  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. Determining Points of Inflection for a Function
    5. First Derivative Test vs Second Derivative Test for Local Extrema
    6. The special case of x⁴
    7. Critical Points of Inflection
    8. Application of the Second Derivative (Acceleration)
    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. The Fundamental Theorem of Calculus
    11. Basic Properties of Definite Integrals

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

  14. Methods of Approximating Integrals

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

  15. Applications of Definite Integrals

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

  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. Determining the Length of a Polar Curve
    4. Determining the Surface Area of a Solid of Revolution
    5. Determining the Volume of a Solid of Revolution
    6. Calculating Polar Areas

  18. Power Series

    1. Introduction to Power Series
    2. Differentiating and Integrating Power Series
    3. Constructing a Taylor Series
    4. Constructing a Maclaurin Series
    5. Lagrange Form of the Remainder Term in a Taylor Series
    6. Determining the Radius and Interval of Convergence for a Power 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

  19. Tests of Convergence / Divergence

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