Calculus How To: Welcome!

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What is Calculus?

Calculus is the study of rates of change. However, if you’ve never taken calculus before, “rates of change” might not have too much meaning to you. In a nutshell, it’s just what it sounds like: studying how rates (of motion, particles, vehicles…) change over time.

The word calculus comes from the Latin word for “pebble”, used for counting and calculations. Although you won’t be using small pebbles in modern calculus, you will be using tiny amounts—very tiny amounts; Calculus is a system of calculation that uses infinitely small (or infinitely large) amounts to deal with change. What kinds of “change”? It’s extremely useful for studying things that change rapidly (like acceleration and velocity), and it’s responsible for a myriad of modern miracles, like:

  • Putting people in space,
  • Building particle accelerators,
  • Making sure that off ramp is just the right angle so that cars can safely exit.

Calculus emerged in the 17th century as a “shortcut” to the results obtained by the method of exhaustion. Calculus proved suitable for finding solutions to finding areas, lengths, and volumes of curves figuring out local properties like curvature, normals, tangents, and normals—what we now recognize as problems involving differentiation and integration. Calculus found its outstanding success partly because it replaced tedious exhaustion arguments with short, routine calculations. Calculus gave mathematicians a set of rules to follow not tied down with long, logical justifications.[1].

Calculus Sub Categories

Calculus is divided into two separate categories: differentiation and integration. The branch that deals with differentiation is called differential calculus and the branch that deals with integration is called integral calculus. The two work hand in hand and basically do the opposite from each other (or “undo” each other), much like addition and subtraction work together in basic arithmetic.


Differentiation is a tool where you can find an object’s velocity and acceleration based on the formula for that object’s position. Likewise, if you know the object’s velocity you can find that object’s acceleration. With integration, the opposite is true: you can find an object’s position if you know the object’s velocity or acceleration. This is illustrated in the following diagram:

Finding Slopes

If you’ve studied algebra. In algebra, you found the slope of a line using the slope formula (slope = rise/run). Here, you’ll be studying the slope of a curve. The slope of a curve isn’t as easy to calculate as the slope of a line, because the slope is different at every point of the curve (and there are technically an infinite amount of points on the curve!).

That’s where differentiation comes in: it’s a set of tools that allows you to find the slope of a tangent line at any point on any curve. This slope is called a derivative.
what is calculus

You can find the slope of a line at any point on a line by using two points (a and b on the top left picture) and the slope formula. However, you can’t use the same formula to calculate the slope of a point on a curve. Points a and b on the top right picture shows that the two points have very different tangent lines (shown in red). In order to find the slope of the tangent line at these points, you need calculus.

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[1] Stillwell, J. Mathematics and Its History. 3rd Edition. Springer.

Stephanie Glen. "Calculus How To: Welcome!" From Calculus for the rest of us!

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