Calculus

Limits and Continuity

  • Understanding the Concept of a Limit

  • Evaluating Limits (Numerically, Graphically, Algebraically)

  • One-Sided and Infinite Limits

  • Vertical Asymptotes and Continuity

  • The Intermediate Value Theorem

Introduction to Derivatives

  • Definition of the Derivative (Limit Definition)

  • Basic Rules: Power, Product, Quotient, Chain

  • Higher-Order Derivatives

  • Derivatives of Polynomial, Rational, Trig, Exponential, and Logarithmic Functions

Advanced Differentiation Techniques

  • Implicit Differentiation

  • Related Rates

  • Motion Along a Line (Velocity and Acceleration)

Analyzing Functions with Derivatives

  • Interpreting Derivatives: Slope and Rate of Change

  • Increasing/Decreasing Intervals

  • Local Extrema

  • First and Second Derivative Tests

  • Concavity and Points of Inflection

Optimization and Curve Sketching

  • Optimization Problems

  • Curve Sketching Using Derivatives

Antiderivatives and Basic Integration

  • Antiderivatives and Indefinite Integrals

  • Basic Integration Rules

  • u-Substitution

Definite Integrals and the FTC

  • Area Under a Curve

  • The Definite Integral

  • The Fundamental Theorem of Calculus

  • Accumulation Functions

Applications of Integration

  • Area Between Curves

  • Net Change and Total Accumulation

  • Displacement and Distance Traveled

  • Average Value of a Function

  • Volume of Solids: Disk, Washer, and Shell Methods

Differential Equations

  • Solving Basic Differential Equations

  • Separation of Variables

  • Slope Fields

  • Exponential Growth and Decay

Sequences and Series (Optional/Advanced)

  • Introduction to Sequences and Series

  • Convergence and Divergence

  • Geometric Series, p-Series, Harmonic Series

  • Integral, Comparison, and Ratio Tests

  • Taylor and Maclaurin Series