Slope and Area Undo Each Other

Speedometer and Odometer

A car pulls away, cruises and brakes, all in 12 seconds. The edge of the brass template is its speed plotted against time.

The speedometer reads the speed at this moment: the curve’s height now. The odometer reads the distance covered so far: the area under the curve from 0 up to now.

A pen on the carriage draws the distance against time as it goes (scale on the right, in metres). That curve’s slope at each point is the speed at that moment: the faster the distance grows, the faster the car is going. On the road along the back, a model car moves as the odometer reads.

Zoom In, and It Is a Slope

The curve is y = x². A straightedge rests on two rollers, one at x and one at x + h. Its slope is (f(x + h) − f(x)) / h, the slope of a secant.

Bring the rollers together and h shrinks: the secant turns towards the tangent. For x² the secant’s slope is exactly 2x + h, closer to 2x the smaller h is.

With the carriage moved aside and the template lifted off, the glass keeps the curve and its tangent. Magnify the curve around a point and it looks more and more like a straight line, until it can hardly be told from the tangent. The derivative is that line’s slope: at x, the derivative of y = x² is 2x.

Area Is Many Thin Strips

The area under a curve can be cut into strips dx wide, each taken as a rectangle: width dx, height the curve’s value at one point. Add up the rectangles and you have a Riemann sum.

The height can be taken at a strip’s left edge, its right edge or its middle. As dx shrinks, all three sums close in on the true area. With the left or right edge the error is roughly proportional to dx; with the middle it is proportional to dx², and falls far faster.

As the strips grow ever thinner, the sums approach a limit: the integral ∫ f(x) dx.

How Fast the Area Grows

Push the carriage a little further, by dx, and the area gains one more strip: dx wide and very nearly f(x) tall, the curve’s height there. So the area grows by dA ≈ f(x)·dx.

Divide by dx: the rate at which the area grows as the right edge moves, dA/dx, equals f(x), and the smaller dx, the closer the match. This is the fundamental theorem of calculus.

The integrator on the carriage does this with metal. A disc turns as the carriage moves; a steel ball sits on it at a distance from the centre proportional to the curve’s height; a cylinder resting on the ball turns faster the further out the ball is. The cylinder’s total turn is proportional to the area, and the counter reads it as the area.

The pen draws the area curve A(x), and the glass shows its tangent at the carriage: that slope always equals the height of the first curve. Integrate, then differentiate, and you are back at f.

A Square Gains Two Strips

A square with side x has area x². Lengthen the side by dx, and the area gains two strips, x long and dx wide, plus a small dx × dx square in the corner.

The gain is 2x·dx + dx². When dx is small, the corner is tiny next to the strips and can be dropped. So the area grows at the rate 2x: the derivative of x².

Sine on the Unit Circle

On the unit circle, the point turned through an angle θ from the horizontal axis stands at height sin θ.

Move a little further round the arc, by dθ: the point moves at right angles to the radius. The upward part of that small step is cos θ·dθ, so the height grows at the rate cos θ.

So the derivative of sin is cos. The straightedge’s slope on the sine curve always equals the across coordinate of the matching point on the circle.

Growing at Its Own Value

eˣ is the one function that at every point grows at a rate equal to its own value, and equals 1 at x = 0.

The straightedge’s slope on eˣ is exactly the height at that point. Piling up area from x = −2.5, the area curve is eˣ − e⁻²·⁵, just 0.08 below eˣ; start further left and the gap shrinks, and from −∞ the two coincide.

The exponential of a complex number keeps the same rule: when the velocity is i times the position, the point turns steadily round the unit circle. That is Euler’s formula.