What is calculus? The mathematics of change, in plain words

Calculus is the branch of mathematics that deals with continuous change. It has two halves. Differential calculus asks how fast something is changing at a single instant, like the reading on a car's speedometer. Integral calculus asks how much change has built up over time, like the miles added on the odometer. The fundamental theorem of calculus shows that the two are opposites, each undoing the other, and together they describe almost anything that moves, grows or flows.

In short

• A derivative is a rate of change at an instant (speed from position); an integral is a total built up from a rate (distance from speed).
• Both work by taking a quantity you can't compute directly and approaching it through ever smaller pieces, a process called a limit.
• Newton and Leibniz invented calculus independently in the late 1600s; it now underlies physics, engineering, economics and machine learning.

What problems does calculus solve?

Ordinary algebra handles quantities that change at a steady rate. If a car drives at exactly 60 kilometers an hour, then distance is speed times time, and two hours gives 120 kilometers. Real things rarely change steadily. A car speeds up and slows down, a population grows faster as it grows, a cooling cup of coffee loses heat quickly at first and then slowly.

Calculus handles two questions about anything that changes unevenly. The first: how fast is it changing right now? The second: given how fast it changes at every moment, how much has it changed in total? A car's dashboard shows both. The speedometer answers the first question, and the odometer answers the second. Calculus is the mathematics that turns one of those dials into the other.

Why can't algebra find the speed at an instant?

Because speed is distance divided by time, and at a single instant both are zero. Here is the problem calculus was built for. A ball dropped from a height falls 4.9t² meters in t seconds (ignoring air resistance). How fast is it moving at exactly 2 seconds?

Algebra can give an average speed over any stretch of time: distance covered divided by time taken. So take shorter and shorter stretches starting at 2 seconds:

Time interval	Distance fallen in it	Average speed
2 to 3 seconds	24.5 m	24.5 m/s
2 to 2.1 seconds	2.009 m	20.09 m/s
2 to 2.01 seconds	0.19649 m	19.649 m/s
2 to 2.001 seconds	0.0196049 m	19.6049 m/s

The averages close in on 19.6 meters per second. You can never set the interval to zero, since that would mean dividing 0 by 0, but you can see exactly where the averages are heading. That destination, the value the averages approach as the interval shrinks toward zero, is called a limit, and it is the idea every part of calculus rests on. The speed at an instant is defined as that limit, and it is the derivative of the distance. For this ball, a short calculation shows the speed is always 9.8t, which is 19.6 at t = 2.

Notice what was needed: not a cleverer formula, but a new kind of step, approaching a value through infinitely many better approximations. Making sense of that "infinitely many" without contradiction took mathematicians until the 19th century, and it is one of the places where the mathematics of infinity stops being abstract.

What are derivatives and integrals?

A derivative is the instantaneous rate of change of a function: how much the output changes per unit change of input, at one particular point. On a graph it is the steepness, or slope, of the curve at that point. Position's derivative is speed; speed's derivative is acceleration; the derivative of a company's total cost with respect to output is, roughly, the cost of making one more unit.

An integral is the accumulated total of a rate. On a graph it is the area under the curve. If you plot a car's speed against time, the area under that plot between two moments is the distance covered between them. At a steady 60 kilometers an hour for 2 hours the area is a rectangle, 60 × 2 = 120. When the speed varies, the integral slices the time into thin strips, treats the speed as steady within each, adds up the strips and takes the limit as they get thinner. The integral note works that through with numbers.

What is the fundamental theorem of calculus?

It says that differentiation and integration undo each other. Start with the odometer reading as time goes on; its rate of change is the speedometer reading. Start with the speedometer reading; adding it up over time gives back the odometer reading. In symbols: if F is a function whose derivative is f, then the integral of f from a to b is simply F(b) − F(a).

That turns a hard problem into an easy one. Adding up infinitely many thin strips sounds endless, but the theorem says you only need a function whose rate of change is the one you have, and then a subtraction. For the falling ball, the speed is 9.8t, and the function whose rate of change is 9.8t is 4.9t². So the distance fallen between 1 and 2 seconds is 4.9 × 4 − 4.9 × 1 = 14.7 meters, with no strips at all.

Who invented calculus?

Pieces of it are ancient. Archimedes, around 250 BCE, found areas and volumes by filling shapes with ever smaller pieces, the same idea as an integral. But calculus as a single method, with derivatives and integrals linked, was created twice in the late 17th century.

Isaac Newton worked out his "method of fluxions" in 1665 and 1666, as a young man away from Cambridge while plague closed the university. He used it for his laws of motion and gravity, but published it only partly and late. Gottfried Wilhelm Leibniz, a German then living in Paris, developed his own version in the mid-1670s and published first, in 1684. Leibniz's notation, dy/dx for a derivative and the long S, ∫, for an integral (from the Latin summa), is the one still used.

	Isaac Newton	Gottfried Wilhelm Leibniz
Worked it out	1665–1666	1674–1676
First published	Partly in the Principia, 1687; fluxions in 1704	"Nova Methodus", 1684
Way of thinking	Quantities flowing in time	Sums of infinitely small differences
Notation that survived	A dot over a letter, still used in physics	dy/dx and ∫, used everywhere

A bitter dispute over who came first followed. In 1712 a committee of the Royal Society, of which Newton was president, sided with Newton, and Newton himself wrote its report without asking for Leibniz's account. Historians now agree that the two invented calculus independently: Newton earlier, Leibniz first in print, as the history of the controversy sets out. Leibniz's notation won anyway. As early as 1696 the first textbook of differential calculus, by the Marquis de l'Hôpital, credited Newton's work but preferred Leibniz's symbols for convenience.

Where is calculus used?

Almost everywhere quantities change. Newton's laws of motion are statements about derivatives: force sets acceleration, the rate of change of velocity, and calculus turns that into the path of a thrown ball or a planet. Engineers use integrals to find the force of water on a dam and the load on a bridge. In medicine, the total exposure to a drug is the area under the curve of its concentration in the blood over time.

In economics, "marginal" cost and revenue are derivatives, and in deep learning, training a network means repeatedly taking derivatives of its error and nudging millions of numbers downhill. Weather forecasts, spacecraft trajectories and the spread of heat through an engine are all worked out by solving equations written in derivatives.

The most common misconception is that calculus is about infinitely small numbers. Leibniz did think in terms of "infinitesimals", and they remain a useful way to picture it, but the calculus taught today is built on limits and on a careful definition of the real numbers, made rigorous in the 1800s by Augustin-Louis Cauchy, Karl Weierstrass, Richard Dedekind and others, with no infinitely small numbers in it. The picture works; the precise version is about approaching, never arriving.

Check yourself

• Without looking back, explain why algebra alone can't give the speed of a falling ball at exactly 2 seconds, and how a limit gets around it.
• In the speedometer and odometer picture, which dial is the derivative of which, and which is the integral?
• What does the fundamental theorem of calculus let you do instead of adding up infinitely many thin strips?

Sources

• The rise of calculus — MacTutor History of Mathematics, University of St Andrews
• A preview of calculus — OpenStax, Calculus Volume 1, Rice University, 2016
• The fundamental theorem of calculus — OpenStax, Calculus Volume 1, Rice University, 2016
• Leibniz–Newton calculus controversy — Wikipedia