Calculus

(from Latin calculus 'small stone used for counting': the mathematics of change and accumulation)

What is calculus?

Calculus studies change and accumulation. It has two main halves: differentiation, which finds how fast something is changing at a single instant, and integration, which finds the total amount something has built up over time. These two halves turn out to be closely connected.

From slope to instantaneous rate of change

For a straight line, "slope" (rise ÷ run) tells you exactly how steep it is, and that steepness never changes. But a curve isn't a straight line — its steepness is different at every point. So what does "the slope at a single point on a curve" even mean?

Here's the idea: pick a point on the curve, and also a second, nearby point. Draw a straight line through both points (called a secant line), and calculate its slope — that gives you the average rate of change between those two points. Now imagine sliding the second point closer and closer to the first one. As the gap shrinks toward zero, the secant line's slope settles down and approaches a single specific number. That number is called the derivative at that point — the curve's instantaneous slope, the value all those average slopes are heading toward.

Working out a derivative directly: the curve y = x²

Let's find the slope of y = x² at a general point x, using the sliding-point idea directly, not just describing it.

Pick a small gap, h, and look at the two points (x, x²) and (x+h, (x+h)²). The average slope between them is:

[(x+h)² − x²] ÷ h

Expand (x+h)²: that's x² + 2xh + h². So the top of the fraction becomes:

(x² + 2xh + h² − x²) ÷ h = (2xh + h²) ÷ h

Since every term on top has a factor of h, we can cancel it: 2xh + h² divided by h simplifies to 2x + h.

Now let h shrink toward zero, exactly as described above. The 2x doesn't depend on h at all, so it stays put — but the h term vanishes as h shrinks to nothing. What's left is 2x. So the derivative of x² is 2x: at any point x, the curve y = x² has slope exactly 2x — for example, at x = 3, the curve's slope is exactly 6.

Integration: the reverse idea

If differentiation asks "how fast is this changing right now," integration asks the reverse question: "given how fast something has been changing, how much has accumulated in total?" Geometrically, this corresponds to finding the area between a curve and the horizontal axis. These two questions — instantaneous rate, and total accumulation — turn out to be exact opposites of one another, which is one of the central discoveries of calculus.

Why calculus matters

Anything that changes continuously — a car's speed, a population's growth, the spread of heat, the trajectory of a thrown ball — is naturally described using calculus. It is the mathematical language behind physics, engineering, economics, and much of modern science.