AoPS Calculus

Lessons

Module 1: Sets/Functions and Limits/Continuity

Numbers and Intervals

Graphs of Functions

Trigonometric Functions

Basic Trigonometric Identities

Exponentials and Logarithms

Extension: Relationship between Trigonometric Functions and Exponentials

Extension: Proofs of some continuity results

Module 2: The Derivative and Applications of the Derivative

Intuitive introduction to the Derivative

Definition of the Derivative

Basic Derivative Computations

The Chain Rule for Derivatives

Rolle's Theorem and the Mean Value Theorem

Implicit Differentiation

Summary of Derivative Computation

Extension: Proof of the Chain Rule

Extension: Proof the Mean Value Theorem

Graphical Interpretation of the Derivative

Extrema and Optimization

Tangent Line Approximation

Newton's Method

Module 3: Integration and Infinity

Area Under a Curve

The Fundamental Theorem of Calculus

Definition and Basic Examples and Antiderivatives of Common Functions

The Chain Rule and Integration by Parts

Substitution Methods and Partial Fractions

A Monster Example

Area of Regions in the Plane and Volumes

Length of a Curve and Average Value of a Function

Approximation Techniques

Extension: Formal definition of logarithms and exponentials

Extension: Simpson's Rule

Limits towards Infinity and Limits of Infinity

Rational indeterminate forms and L'Hopital's Rule

Exponential Indeterminate Forms

Improper Integrals

Extension: Proof of L'Hopital's Rule

Module 4: Series, Plane Curves, and Differential Equations

Infinite Sequences

Infinite Series

Series Convergence Tests

Alternating Series

Taylor Polynomials

Extension: A strange formula for pi

Parametric Curves

Polar Coordinates

Areas in polar coordinates

Definitions and basic examples of differential equations

Second-order linear differential equations

Extension: Euler's Method

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