Mathematics & Logic · Depth 2 · Intermediate · 2 min read
Calculus
The mathematics of change: finding an instantaneous rate of change (derivatives) and adding up infinitely many tiny pieces (integrals).
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What is calculus?
Calculus is the mathematics of change. Its central idea is the rate of change: how fast one quantity changes compared with another.[1] It grows out of two classic problems.
Problem 1: the tangent problem and derivatives
Imagine a curved graph of a car’s position over time. The tangent problem asks how to find the slope of the line that just touches the curve at a single point.[1] That slope is the derivative: the rate of change of the function at that exact point.[1] For a moving object, the derivative of its position is its instantaneous velocity, its speed at one precise moment.[1]
Problem 2: the area problem and integrals
The area problem asks how to find the area under a curve.[1] The trick is to fill the region with thin rectangles, add up their areas, and then make the rectangles narrower and narrower. As their widths shrink, the total approaches the true area.[1] That limiting value is the integral.
The idea underneath: limits
Both problems depend on limits: seeing what happens to a quantity as one value gets closer and closer to another, without necessarily reaching it.[1] Limits are what let calculus deal with “instantaneous” rates and infinitely thin slices.
Where calculus is used
Calculus is applied throughout physics, engineering and economics.[2] In physics, for example, some quantities, such as the work done by a force, can be interpreted as an area under a curve.[1] On this map, calculus is a prerequisite for the PID Controller, the control method used to keep drones level.
Going further
A first university course covers differentiation and integration of functions of one variable, and ends with infinite series.[2] MIT’s complete course and a free OpenStax textbook are listed below.
Real-life examples
Your speedometer
A speedometer shows your speed right now, not your average over the trip. That 'right now' rate of change is exactly what a derivative describes: instantaneous velocity.[1]
Work as an area
In physics, the work done by a changing force can be interpreted as an area under a curve, the kind of quantity integrals calculate.[1]
Getting to space
Calculus textbooks introduce the subject with engineering problems such as space travel, where speeds and forces change continuously.[1]
Beyond physics
Calculus is applied across physics, engineering and economics.[2]
Go deeper
2 of 2 topics below this one are written so far. The rest are uncharted: on the map, but not yet written. The Atlas is growing.
- DerivativesThe derivative measures how fast something is changing at a single instant: the slope of a curve at one point, found as a limit of average rates of change.
- IntegralsAn integral adds up thin slices: the area under a curve as a limit of sums of rectangles, tied to derivatives by the Fundamental Theorem of Calculus.
Connected across the map
- AlgebraUsing letters for unknown or changing numbers and solving equations step by step: a subject named after a treatise by al-Khwarizmi.
- GeometryThe mathematics of shape, size and space: points, lines, angles, triangles and circles, reasoned from a few assumptions since Euclid's Elements, about 300 BC.
- NumbersFrom counting numbers to fractions, irrationals and complex numbers: how each new kind fills a gap left by the last, and how zero and negatives arose.
- PID Controller
Learn more
Short descriptions are our own summaries. The resources belong to, and are run by, their publishers.
- Calculus Volume 1 (free textbook) ↗
by OpenStax
A free, peer-reviewed calculus textbook: limits, derivatives and integrals, with worked examples.
- 18.01SC Single Variable Calculus ↗
by MIT OCW
MIT’s complete single-variable calculus course with lecture videos, problem sets and exams, free to use.
Evidence & sources
Supported by extensive evidence and broad scientific consensus.
Why this level? Derivatives, integrals and limits are standard, proven mathematics, explained here from a peer-reviewed textbook and MIT.
This is a Knowledge Atlas editorial classification of the sources we could find, not a certificate of truth. How we evaluate knowledge
Sources
Based on 2 sources from 2 institutions: OpenStax, MIT OCW.
Show all 2 sourcesHide the list
- ScholarlyOpenStax (Rice University)· Academic publisherCalculus Volume 1, 2.1 A Preview of CalculusOpened and checked against this page on 28 Sept 2026 · License: CC BY-NC-SA 4.0
- MIT OpenCourseWare· University18.01SC Single Variable Calculus (course description)Opened and checked against this page on 28 Sept 2026 · License: CC BY-NC-SA 4.0