Calculus is the mathematics of motion or change. It was developed because mathematicians around the turn of the 18th century were grappling with how to model the obvious, observable motions of the planets against the background of (relatively speaking) fixed stars in the night sky. There was a lot of suspicion that this motion must be governed by the same physical "laws" that describe the motion of moving bodies here on Earth, such as the motion of a thrown ball, which always traces out a *parabolic* arc. And the arc of that ball, as we know, is completely due to the invisible force of gravity — the same force that controls the motion of all of the stars and planets.

The mathematics necessary to explain all of the data that was piling up was discovered in within a few years of 1690, nearly simultaneously by Sir Isaac Newton of England, and by Gottfried Leibniz, a German.

Newton and Leibniz found a way to use the mathematics of infinitesimals – very small (vanishingly small) changes in a variable, to solve the problem of the slope of a curve and to make it easy to calculate the area under a complicated curve in the plane.

Slopes are related to rates of change. For example, speed is the change in displacement (measured by some coordinate, like the x-axis) over time, or Δx/Δt. There are many examples of slope as the model of an important rate of change, from physics to finance, economics to engineering. With calculus, we can find the rate of change at a moment in time for a system in which that rate itself is changing. That's a huge leap forward.

The idea of area under a curve is equally important because it allows us to calculate complicated sums over time or any other variable and get the total exactly right. In physics, for example, the integral of a force function over the distance coordinate (that is, the area under a force curve) gives us the amount of work done.

Finally, Newton, Leibniz and mathematicians who followed them were able to show that the slopes problem, **differential calculus**, was the inverse of the areas problem, **integral calculus**. So it all ties together.

Sir Isaac Newton (1642-1727), Englishman

Gottfried Wilhelm Leibniz (1646-1716), German

Images: Wikipedia Commons

That's a tricky question, and I have two answers.

**First, **you *ought* to learn calculus because it is a **beautiful** subject, the poetry of mathematics, I think. Learning calculus will **open the doors** to many other deeper subjects in mathematics, including differential equations, multi-variable mathematics and complex analysis, all of which will take you to places you probably can't yet imagine. And calculus is **relevant** to many fields of work and study, including physics, chemistry, electronics, engineering, finance and economics. I love calculus.

**But ... **very few people will actually need to *use* calculus in their lives, and I sometimes wonder how it became the pinnacle of achievement in our high school curricula. In my view, most people would benefit far more from an excellent grounding in **probability and statistics**. Most of the information that comes to us daily as professionals and citizens is presented as statistics or probability. How likely is it that you'll actually get eaten by a mountain lion if you go outside? How likely is it that you'll contract the Ebola virus? What does the margin of error of a political poll mean? Why do we have to count every single individual in our census, and is that even possible?

This isn't to say that it's not worth learning calculus even if you're pretty sure you won't ever use it. There is a lot of value in pushing yourself to explore new things, and there is a kind of logic in calculus that could well overlap with other interests you might have. Sometimes the most important discoveries come when we push our limits beyond what's comfortable.

The word calculus comes from the Latin words * calx* for

The word was used to mean "**small stone used for counting**," and referred to small stones or pebbles used to keep track of large counts.

This makes sense in the mathematics of calculus, especially when we think of integral calculus, the counting of complicated areas under curves by counting an infinite number of infinitesimally small chunks of area. Think about the definition of **calculus** as you work through the sections in the calculus site map below.

This is roughly the order in which I teach calculus, though the subjects can be arranged in a number of ways that make sense.

- Calculus, a brief history (this page)
- Limits
- Definitions
- Continuity of functions
- Trigonometric limits
- L'Hopital's rule
- Epsilon-delta proofs

- The derivative
- Existence theorems
- Mean value theorem
- Intermediate value theorem
- Rolle's theorem

- The integral
- Definite integrals (antiderivatives)
- Differential equations
- Fundamental theorem of calculus (FTOC)
- Integration by substitution
- Calculating the area between curves
- The integral as a sum
- Integration by parts
- Integration by rational decomposition
- Integration by trigonometric substitution
- Numerical integration
- Volumes of integration: Disk method
- Volumes of integration: Washer method
- Volumes of integration: Shell method
- Volumes: Slices

- Infinite series
- Definitions
- Power series & radius of convergence
- p-series
- Taylor / McLaurin series
- Divergence test
- Comparison test for convergence
- Integral test for convergence
- Ratio test & root test for convergence
- Fourier series

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