This calculus course combines OpenStax Calculus Volume 1, Volume 2, and Volume 3 into a single sequence, covering everything from functions and limits through vector calculus. It is built for high school and lower-division undergraduate students, with a focus on clear explanations and practical study strategies.
OpenStax publishes Calculus as three separate volumes, and each later volume opens with a short review of material from the one before it. Volume 2 begins by repeating Volume 1's integration chapters, and Volume 3 begins by repeating Volume 2's chapter on parametric equations and polar coordinates. To avoid covering the same material twice, this project merges the three volumes into one continuous sequence and keeps each repeated chapter only once, at its first appearance. Chapter and section numbers below are renumbered to run continuously across the combined project rather than restarting with each volume.
This course consists of 17 chapters, organized progressively from foundational concepts through multivariable calculus:
- Review of Functions
- Basic Classes of Functions
- Trigonometric Functions
- Inverse Functions
- Exponential and Logarithmic Functions
- A Preview of Calculus
- The Limit of a Function
- The Limit Laws
- Continuity
- The Precise Definition of a Limit
- Defining the Derivative
- The Derivative as a Function
- Differentiation Rules
- Derivatives as Rates of Change
- Derivatives of Trigonometric Functions
- The Chain Rule
- Derivatives of Inverse Functions
- Implicit Differentiation
- Derivatives of Exponential and Logarithmic Functions
- Related Rates
- Linear Approximations and Differentials
- Maxima and Minima
- The Mean Value Theorem
- Derivatives and the Shape of a Graph
- Limits at Infinity and Asymptotes
- Applied Optimization Problems
- L'Hopital's Rule
- Newton's Method
- Antiderivatives
- Approximating Areas
- The Definite Integral
- The Fundamental Theorem of Calculus
- Integration Formulas and the Net Change Theorem
- Substitution
- Integrals Involving Exponential and Logarithmic Functions
- Integrals Resulting in Inverse Trigonometric Functions
- Areas between Curves
- Determining Volumes by Slicing
- Volumes of Revolution: Cylindrical Shells
- Arc Length of a Curve and Surface Area
- Physical Applications
- Moments and Centers of Mass
- Integrals, Exponential Functions, and Logarithms
- Exponential Growth and Decay
- Calculus of the Hyperbolic Functions
- Integration by Parts
- Trigonometric Integrals
- Trigonometric Substitution
- Partial Fractions
- Other Strategies for Integration
- Numerical Integration
- Improper Integrals
- Basics of Differential Equations
- Direction Fields and Numerical Methods
- Separable Equations
- The Logistic Equation
- First-order Linear Equations
- Sequences
- Infinite Series
- The Divergence and Integral Tests
- Comparison Tests
- Alternating Series
- Ratio and Root Tests
- Power Series and Functions
- Properties of Power Series
- Taylor and Maclaurin Series
- Working with Taylor Series
- Parametric Equations
- Calculus of Parametric Curves
- Polar Coordinates
- Area and Arc Length in Polar Coordinates
- Conic Sections
- Vectors in the Plane
- Vectors in Three Dimensions
- The Dot Product
- The Cross Product
- Equations of Lines and Planes in Space
- Quadric Surfaces
- Cylindrical and Spherical Coordinates
- Vector-Valued Functions and Space Curves
- Calculus of Vector-Valued Functions
- Arc Length and Curvature
- Motion in Space
- Functions of Several Variables
- Limits and Continuity
- Partial Derivatives
- Tangent Planes and Linear Approximations
- The Chain Rule
- Directional Derivatives and the Gradient
- Maxima/Minima Problems
- Lagrange Multipliers
- Double Integrals over Rectangular Regions
- Double Integrals over General Regions
- Double Integrals in Polar Coordinates
- Triple Integrals
- Triple Integrals in Cylindrical and Spherical Coordinates
- Calculating Centers of Mass and Moments of Inertia
- Change of Variables in Multiple Integrals
- Vector Fields
- Line Integrals
- Conservative Vector Fields
- Green's Theorem
- Divergence and Curl
- Surface Integrals
- Stokes' Theorem
- The Divergence Theorem
- Second-Order Linear Equations
- Nonhomogeneous Linear Equations
- Applications
- Series Solutions of Differential Equations
Each chapter includes:
- Video Lectures: Explanations of core concepts with worked examples
- Study Guides: Chapter summaries and key formulas
- Start with Chapter 1 if you need a refresher on functions, or start with Chapter 2 if you are already comfortable with them
- Watch the video lectures for each section
- Read the study guides to reinforce key ideas and formulas
- Work through problems by hand rather than only watching, since calculus is a skill built through practice
This course emphasizes:
- Building a solid foundation in limits and derivatives before moving to integration
- Connecting each new technique back to the definitions it comes from, rather than memorizing rules in isolation
- Progressing naturally from single-variable to multivariable calculus
- Practicing enough worked problems to build real fluency, not just recognition
- Comfort with algebra and trigonometry
- Familiarity with functions, graphs, and function notation
- Appendix C in the source material (Review of Pre-Calculus) is a good refresher if needed
This course is built on materials from:
- Calculus Volume 1, Volume 2, and Volume 3 by OpenStax (Rice University)
- Licensed under Creative Commons Attribution 4.0 International License (CC BY 4.0)
- Access free materials at https://openstax.org
This project is licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).
Source materials based on OpenStax Calculus Volume 1, Volume 2, and Volume 3. Access for free at https://openstax.org.
Have questions or suggestions? Open an issue in this repository or visit the main Curious2Study hub.
Last updated: 2026