An Introduction to Calculus

An Introduction to Calculus — Moosa Academy

Before any of the rules of calculus are worth learning, three ideas have to mean something: what a function describes, what an integral measures, and what a derivative measures. A single example — a car speeding up and slowing down — holds all three at once.

Concept The function: a described behaviour

A car starts from rest, accelerates smoothly to a top speed of 120 km/h, then slows in the same way until it stops. Plot its speed against time and you get a smooth arch.

That curve is the function. It does not give one number — it describes the speed at every instant of the journey, and how that speed changes.

Concept The integral: the area beneath

Shade the region under the speed curve from the start up to any moment. That shaded area is exactly the distance the car has travelled by then.

Early in the journey the area is small — little distance covered

As time passes the area grows — the distance grows with it

By the end, the whole area under the arch is the total distance

This is what integration does in general: the function gives a value at each instant, and the integral accumulates those instants into a total. Speed at every moment adds up to distance overall.

Concept The derivative: the slope at an instant

Now look at the steepness of the same curve. Its slope at any moment is the acceleration — the rate at which the speed is changing right then.

Climbing — the speed is rising, so the acceleration is positive

At the peak — the speed is momentarily not changing, so the acceleration is zero

Falling — the speed is dropping, so the acceleration is negative

The peak is worth pausing on. The car is at its fastest there, yet the rate of change is zero — the tangent line lies flat. Fastest and "not changing" describe different things, and the derivative measures the second.

Theorem Why the definition needs a limit

Take two points on the curve, at $x$ and at $x + h$ . The line through them has slope:

\frac{f(x+h) - f(x)}{h}

But this is an average rate across a gap, not the rate at a single instant. So shrink the gap:

h

gets smaller, the line through the two points swings closer to the tangent

In the limit as

h \to 0

, it becomes the tangent exactly

f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}

The limit is not a technicality bolted on afterwards. It is there because "the rate at one exact instant" cannot be measured across any gap, however small — it can only be approached.

Note The three ideas in one picture

Function — a behaviour that changes. Here: the speed at each moment.

Integral — the area under the curve. Here: the distance travelled.

Derivative — the slope of the curve. Here: the acceleration.

The function describes; the integral gathers it up; the derivative takes it apart instant by instant. Integration and differentiation are opposite operations — which is why one undoes the other.

Summary

A function describes a behaviour that changes — the car's speed at every instant.
The integral is the area under the curve — here the distance travelled.
The derivative is the slope of the curve — here the acceleration.
At the peak the slope is zero: top speed, but no change at that instant.
The derivative uses a limit because a rate at one exact instant can only be approached.
Integration and differentiation are opposite operations.