--- title: Calculus description: Calculus is the mathematical study of continuous change, focusing on limits, derivatives, and integrals. Learners will understand how to calculate rates of change, find areas under curves, and model dynamic physical systems. category: mathematics subcategory: calculus difficulty: beginner, intermediate, advanced url: /subject/calculus --- # Calculus Calculus is the mathematical study of continuous change, focusing on limits, derivatives, and integrals. Learners will understand how to calculate rates of change, find areas under curves, and model dynamic physical systems. ## Available Resources 4 Books • 14 Courses • 5 Websites ## Websites ### 1. Calculus II: Series & Sequences **Author:** Paul Dawkins Chapter of Paul Dawkins's free Calculus II notes covering sequences, infinite series, partial sums, the integral, comparison, alternating, ratio and root tests, absolute convergence, power series and Taylor series, with worked examples. Learners can decide convergence and represent functions as power series. **Difficulty:** Intermediate | **Price:** Free **Link:** https://tutorial.math.lamar.edu/classes/calcii/seriesintro.aspx **Tags:** sequences, infinite-series, convergence-tests, power-series, taylor-series ### 2. Paul's Online Math Notes - Calculus I Lamar University lecture notes covering limits, derivatives, and integrals, with worked examples and graded practice problem sets. Readers finish able to compute derivatives and integrals of standard functions and apply them to optimization and related-rates problems. **Difficulty:** Beginner | **Price:** Free **Link:** https://tutorial.math.lamar.edu/Classes/CalcI/CalcI.aspx **Tags:** calculus, limits, derivatives, integrals, lecture-notes, practice-problems ### 3. Khan Academy - Calculus Applications Khan Academy's Calculus 1 course covers derivatives, integrals, and their applications to optimization, related rates, and area/volume problems, with practice exercises and video explanations for each topic. **Difficulty:** Beginner | **Price:** Free **Link:** https://www.khanacademy.org/math/calculus-1 **Tags:** calculus, derivatives, integrals, optimization ### 4. brilliant.org/calculus Interactive, problem-driven calculus lessons on Brilliant.org cover topics like limits, derivatives, integrals, and series, with guided practice, hints, and step-by-step solutions to build understanding. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://brilliant.org/calculus **Tags:** websites, mathematics-statistics, mathematics ### 5. desmos.com Desmos is a free online graphing calculator and math exploration platform that lets you plot functions and visualize Calculus concepts like derivatives and integrals. It also provides interactive activities and classroom resources to support learning and teaching. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://desmos.com **Tags:** websites, mathematics-statistics, mathematics ## Courses ### 1. Calculus One (Full Course) Full-length recorded lectures from Professor Leonard's first-semester calculus class, covering limits, continuity, derivatives and their applications, and introductory integration. Each topic is worked example by example at classroom pace, suited to self-study alongside a textbook. **Difficulty:** Intermediate | **Price:** Free **Link:** https://www.youtube.com/playlist?list=PLF797E961509B4EB5 **Tags:** calculus, limits, derivatives, integrals, video-lectures ### 2. Complex Variables with Applications (MIT 18.04) **Author:** Jeremy Orloff Complex analytic functions and their applications: Cauchy's theorem and integral formula, Taylor and Laurent series, residues for evaluating hard integrals, harmonic functions, conformal maps, two-dimensional fluid flow, and Laplace and Fourier transforms. Provides 14 lecture note files plus problem sets and exams with solutions. **Difficulty:** Intermediate | **Price:** Free **Link:** https://ocw.mit.edu/courses/18-04-complex-variables-with-applications-spring-2018/ **Tags:** complex-analysis, analytic-functions, residue-theorem, laurent-series, conformal-mapping ### 3. Real Analysis (MIT 18.100A) **Author:** Casey Rodriguez Rigorous foundations of calculus: real numbers, convergence of sequences and series, continuity, differentiability, the Riemann integral, sequences and series of functions, uniform convergence and interchange of limits. Includes 25 lecture videos, notes, problem sets and exams, and teaches how to read and construct proofs. **Difficulty:** Intermediate | **Price:** Free **Link:** https://ocw.mit.edu/courses/18-100a-real-analysis-fall-2020/ **Tags:** real-analysis, proofs, sequences-and-series, uniform-convergence, riemann-integral ### 4. Essence of Calculus Master integration with Grant Sanderson! This calculus course unlocks the essence of integration through intuitive visuals. **Difficulty:** Beginner | **Price:** Free **Link:** https://www.youtube.com/playlist?list=PLZHQObOWTQDMsr9K-rj53DwVRMYO3t5Yr **Tags:** calculus, integration, visual-intuition, 3blue1brown, derivatives ### 5. Calculus Revisited: Complex Variables, Differential Equations, and Linear Algebra (MIT RES.18-008) **Author:** Herbert Gross Third part of a 1972 lecture series, covering complex variables, ordinary differential equations and linear algebra at the sophomore level. Provides 20 lecture videos (about 11.5 hours), study guides, supplementary notes and problem sets with solutions. Assumes single and multivariable calculus. **Difficulty:** Intermediate | **Price:** Free **Link:** https://ocw.mit.edu/courses/res-18-008-calculus-revisited-complex-variables-differential-equations-and-linear-algebra-fall-2011/ **Tags:** differential-equations, complex-variables, linear-algebra, ordinary-differential-equations ### 6. Calculus Revisited: Multivariable Calculus (MIT RES.18-007) **Author:** Herbert Gross A lecture series first released in 1971 reviewing multivariable calculus: vector arithmetic and vector functions, partial derivatives, multiple and line integrals, and matrix and linear algebra ideas. Includes 26 lecture videos, study guides, supplementary notes and problem solutions, suited to first-time learners and review. **Difficulty:** Intermediate | **Price:** Free **Link:** https://ocw.mit.edu/courses/res-18-007-calculus-revisited-multivariable-calculus-fall-2011/ **Tags:** multivariable-calculus, vector-calculus, partial-derivatives, multiple-integrals, linear-algebra ### 7. Highlights of Calculus (MIT RES.18-005) **Author:** Gilbert Strang Eighteen short videos on how calculus works and why it matters: functions and their slopes, the key ideas of differential calculus, and applications to real-life problems, with summary slides and practice problems. Suits high school students or anyone starting or returning to calculus. **Difficulty:** Beginner | **Price:** Free **Link:** https://ocw.mit.edu/courses/res-18-005-highlights-of-calculus-spring-2010/ **Tags:** calculus, derivatives, differential-calculus, calculus-intuition ### 8. Real Analysis (MIT 18.100B) **Author:** Tobias Colding Proof-based analysis covering real numbers, limits, convergence, series, continuity, metric spaces, differentiation, Riemann integration and differential equations, with emphasis on proving theorems rigorously. Provides 25 lecture videos (about 33 hours), 26 lecture note files, problem sets and exams. **Difficulty:** Intermediate | **Price:** Free **Link:** https://ocw.mit.edu/courses/18-100b-real-analysis-spring-2025/ **Tags:** real-analysis, metric-spaces, riemann-integral, mathematical-proofs, sequences-and-series ### 9. Calculus Open Textbook (MIT RES.18-001) **Author:** Gilbert Strang The full open textbook, 3rd edition, covering single-variable and multivariable calculus in depth: derivatives, integrals, exponentials, infinite series, vectors, partial derivatives, multiple integrals and vector calculus, with many applications. An accompanying study guide and instructor's manual support self-study or teaching a complete sequence. **Difficulty:** Beginner | **Price:** Free **Link:** https://ocw.mit.edu/courses/res-18-001-calculus-fall-2023/ **Tags:** calculus, derivatives, integrals, infinite-series, multivariable-calculus, vector-calculus ### 10. Single Variable Calculus (MIT 18.01SC) **Author:** David Jerison MIT's self-paced 18.01SC covers differentiation, applications of derivatives, integration, techniques of integration, and infinite series, with problem sets, recitation videos, and exams with solutions. Completing it gives the standard first-semester calculus foundation. **Difficulty:** Beginner | **Price:** Free **Link:** https://ocw.mit.edu/courses/18-01sc-single-variable-calculus-fall-2010/ **Tags:** single-variable-calculus, derivatives, integrals, infinite-series ### 11. Multivariable Calculus (MIT 18.02SC) **Author:** Denis Auroux Differential, integral and vector calculus in several variables: vectors and matrices, partial derivatives, optimization, double and triple integrals, line integrals, and the theorems of Green, Stokes and Gauss. Includes 106 lecture and recitation videos, notes, and problems and exams with solutions, built for independent study. **Difficulty:** Intermediate | **Price:** Free **Link:** https://ocw.mit.edu/courses/18-02sc-multivariable-calculus-fall-2010/ **Tags:** multivariable-calculus, partial-derivatives, multiple-integrals, vector-calculus, line-integrals ### 12. Introducción al cálculo **Author:** Jaume Pujol, Mercè Villanueva Spanish-language course from the Universitat Autònoma de Barcelona that reviews numbers and functions, polynomial, exponential, logarithmic and trigonometric functions, then introduces differentiation, integration and complex numbers. Prepares students for a first-semester university calculus course in science or engineering. **Difficulty:** Beginner | **Language:** English | **Duration:** 6-8 horas de trabajo por módulo | **Price:** Free **Link:** https://www.coursera.org/learn/introduccion-al-calculo **Tags:** precalculus, functions, trigonometry, differentiation, integration, spanish-language ### 13. Calculus for Engineers **Author:** Jeffrey R. Chasnov Hong Kong University of Science and Technology course presenting university calculus for engineering and science students in 59 short lectures. Reviews precalculus, then covers derivatives, integrals, Taylor series and applications, leaving learners able to set up and solve standard single-variable calculus problems. **Difficulty:** Beginner | **Language:** English | **Price:** Free **Link:** https://www.coursera.org/learn/calculus-for-engineers **Tags:** calculus, derivatives, integrals, taylor-series, engineering-mathematics ### 14. Introduction to Calculus **Author:** David Easdown University of Sydney course on Coursera taught by David Easdown. Over five weeks it reviews precalculus functions, introduces tangents, limits and the derivative, then develops differential and integral calculus with applications, preparing learners for university mathematics in science, engineering and commerce. **Difficulty:** Beginner | **Language:** English | **Price:** Free **Link:** https://www.coursera.org/learn/introduction-to-calculus **Tags:** calculus, limits, derivatives, integration, precalculus ## Books ### 1. Calculus **Author:** Michael Spivak Short companion primer by Michael Spivak that moves from lines and slope through nonlinear and trigonometric functions to derivatives, velocity, maxima and minima, Rolle's theorem, the Mean Value Theorem, integrals and the fundamental theorem, building intuition for why calculus works before formal coursework. **Difficulty:** Beginner | **Language:** English | **Price:** Paid **Link:** https://www.amazon.com/dp/0914098918?tag=edmonddante07-20 **Tags:** calculus, calculus-intuition, derivatives, integration, mean-value-theorem ### 2. Calculus: Early Transcendentals **Author:** Jon Rogawski, Colin Adams, Robert Franzosa Rogawski, Adams and Franzosa's fourth-edition university textbook, sold here as SaplingPlus online access, covering limits, derivatives, integration, sequences and series, and multivariable calculus with the exercise sets used in standard first-year and second-year university courses. **Difficulty:** Intermediate | **Language:** English | **Price:** Paid **Link:** https://www.amazon.com/dp/1319050743?tag=edmonddante07-20 **Tags:** books, mathematics-statistics, calculus ### 3. Infinite Powers **Author:** Steven Strogatz Steven Strogatz traces how calculus developed from Archimedes through Newton and Leibniz to modern applications in medicine, GPS, and physics. Written without technical prerequisites, it explains why the infinity principle works and where calculus shows up today. **Difficulty:** Intermediate | **Language:** English | **Price:** Paid **Link:** https://www.amazon.com/dp/0358299284?tag=edmonddante07-20 **Tags:** books, mathematics-statistics, mathematics ### 4. Calculus Made Easy **Author:** Silvanus P. Thompson, Martin Gardner Silvanus Thompson's 1910 primer, revised and annotated by Martin Gardner in 1998, teaching differentiation and integration through plain language and worked examples rather than formal proofs, with three added chapters and practice problems for self-study. **Difficulty:** Beginner | **Language:** English | **Price:** Paid **Link:** https://www.amazon.com/dp/0312185480?tag=edmonddante07-20 **Tags:** calculus, differentiation, integration, calculus-intuition --- *This content is part of Dantes.io - Your Treasure Map to Knowledge* *Curated by humans at Dantes.io. Personal study use welcome; republishing this curation requires permission (team@dantes.io).* View this page online: https://dantes.io/subject/calculus