A Function, Its Integral and Its Derivative

A Function, Its Integral and Its Derivative — Moosa Academy

Take one function and stack two more on top of it: its integral above, its derivative below. All three describe the same motion, each answering a different question. Once the three are lined up, a striking pattern appears between them.

Concept Three layers, one motion

The integral — accumulation. How much has built up so far? Here: the distance travelled.

The function — the thing itself. Here: the car's velocity at each moment.

The derivative — rate of change. How fast is it changing now? Here: the acceleration.

Nothing physical differs between the three — the same car, the same journey. Only the question changes.

Example A car that reverses, then returns

The velocity starts negative — the car is reversing. It slows, stops at the middle, then drives forward and slows to a stop again.

Below the axis — moving backwards, fastest in reverse at the trough

Crossing zero — momentarily stopped

Above the axis — moving forwards, fastest at the peak

The two regions have equal area, one negative and one positive, so they cancel: the car ends exactly where it began.

Concept What the integral sees

The integral watches the running total — the position of the car:

While velocity is negative, the total keeps falling — the car moves further back

At the moment velocity is zero, the total stops changing — it has reached its lowest point

While velocity is positive, the total climbs back up

⟹ the position curve has its minimum exactly where the velocity is zero

Notice what happened: a zero of the velocity became a turning point of the integral. That is the pattern.

Concept What the derivative sees

The derivative ignores where the curve sits and looks only at whether it is rising or falling:

Wherever the velocity is increasing,

f'(x) > 0

— even while the velocity itself is negative

Wherever the velocity is decreasing,

f'(x) < 0

At a peak or a trough,

f'(x) = 0

— no change at that instant

Being below the axis and being on the way down are different things. The derivative only reports the second. This also explains why the derivative of a constant is zero: sliding the whole curve up or down changes its position but never its steepness.

Theorem The pattern that links all three

Peaks and troughs of $f$ are zeros of $f'$

Zeros of $f$ are peaks and troughs of its integral

Both statements say the same thing, one layer apart. Reading down the stack, each turning point on one layer sits directly above a zero on the layer below:

Velocity at its trough ⟹ acceleration is zero there

Velocity at zero ⟹ position is at its minimum there

This is why sketching one of the three curves tells you a great deal about the other two, without computing anything.

Summary

Integral, function and derivative are three layers describing one motion.
The integral accumulates: velocity builds up into distance travelled.
The derivative measures steepness: velocity changes into acceleration.
Negative area cancels positive area, so the car returns to where it started.
Peaks and troughs of f are exactly the zeros of f′.
Zeros of f are exactly the peaks and troughs of its integral.