Calculus I, II and III Course

Lesson 1 • Understanding Limits Intuitively

Introduces the limit concept through tables, graphs, and informal definitions. Connects intuitive understanding to formal notation.

Lesson 2 • Review of Functions and Graphs

Covers domain, range, function notation, and graph transformations. Establishes the language used throughout all three calculus courses.

Lesson 3 • Continuity and Its Consequences

Defines continuity formally and identifies types of discontinuities. Introduces the Intermediate Value Theorem as a key theoretical result.

Lesson 1 • Implicit Differentiation and Related Rates

Differentiates implicitly defined curves and applies derivatives to dynamic geometric problems. Bridges algebraic and applied differentiation.

Lesson 2 • Product, Quotient, and Chain Rules

Extends differentiation to products, quotients, and compositions of functions. Enables differentiation of virtually all elementary expressions.

Lesson 3 • The Derivative as a Rate of Change

Defines the derivative via the limit of a difference quotient. Links instantaneous rate of change to slope of the tangent line.

Lesson 4 • Derivatives of Transcendental Functions

Covers derivatives of exponential, logarithmic, and all trigonometric functions. Completes the toolkit for Calculus I differentiation.

Lesson 1 • Extreme Values and Critical Points

Identifies local and global extrema using the first derivative. Introduces the Extreme Value Theorem for closed intervals.

Lesson 2 • Optimization Problems

Translates real-world constraints into objective functions and finds optimal values. Demonstrates calculus as a practical decision-making tool.

Lesson 3 • Linear Approximation and Differentials

Uses tangent lines to approximate function values and quantify error. Prepares students for numerical methods and error analysis.

Lesson 4 • Curve Sketching and Graph Analysis

Synthesizes limits, continuity, and derivative information into accurate sketches. Develops systematic graph analysis as a professional skill.

Lesson 1 • Antiderivatives and Indefinite Integrals

Defines antidifferentiation and builds the basic integral table. Establishes notation and the constant of integration.

Lesson 2 • Integration by Substitution

Applies the chain rule in reverse to evaluate composite integrands. Extends the range of integrable functions significantly.

Lesson 3 • The Fundamental Theorem of Calculus

Proves and applies both parts of the Fundamental Theorem. Unifies differentiation and integration as inverse processes.

Lesson 1 • Physical Applications of Integration

Applies integrals to work, fluid pressure, and center of mass problems. Demonstrates integration as a universal accumulation tool.

Lesson 2 • Volumes of Solids of Revolution

Generates volumes by rotating plane regions using disk, washer, and shell methods. Develops three-dimensional geometric reasoning.

Lesson 3 • Arc Length and Surface Area

Derives and applies formulas for curve length and surface area of revolution. Extends integration to geometric measurement beyond area.

Lesson 1 • Partial Fraction Decomposition

Decomposes rational functions into simpler fractions for integration. Covers distinct, repeated, and irreducible quadratic factors.

Lesson 2 • Improper Integrals

Extends integration to unbounded intervals and discontinuous integrands using limits. Introduces convergence and divergence of integrals.

Lesson 3 • Integration by Parts

Applies the product rule in reverse to integrate products of functions. Introduces the LIATE heuristic for choosing factors.

Lesson 1 • Power Series and Radius of Convergence

Represents functions as power series and determines intervals of convergence. Prepares students for Taylor series construction.

Lesson 2 • Alternating Series and Absolute Convergence

Analyzes alternating series and distinguishes absolute from conditional convergence. Completes the convergence test toolkit.

Lesson 3 • Sequences and Their Limits

Defines sequences formally and determines convergence using limit techniques. Establishes the foundation for series analysis.

Lesson 4 • Infinite Series and Convergence Tests

Introduces partial sums and applies divergence, integral, comparison, and ratio tests. Provides a systematic toolkit for determining series behavior.

Lesson 1 • Multiple Integrals

Evaluates double and triple integrals in Cartesian, polar, cylindrical, and spherical coordinates. Computes volume, mass, and center of mass in 3D.

Lesson 2 • Partial Derivatives and Gradient

Defines partial derivatives and the gradient vector for multivariable functions. Connects single-variable differentiation to higher dimensions.

Lesson 3 • Vector Fields and Integral Theorems

Introduces line integrals, surface integrals, and the major theorems of vector calculus. Unifies the course with Green's, Stokes', and Divergence theorems.

Lesson 4 • Optimization of Multivariable Functions

Finds critical points and classifies extrema using the second derivative test. Applies Lagrange multipliers for constrained optimization.