Functions Are Everywhere

Five real-world functions and the shapes that describe them: an S-curve for learning, exponential growth from reinvesting, braking distance against the square of speed, profit against customers, and the link between a rate curve and integration.

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Whenever one quantity changes in relation to another, a function is quietly at work. Recognising the shape of that relationship — before calculating a single value — tells you how the whole story behaves.

Learning An S-shaped curve
building the basics approaching saturation
A student learning mathematics starts slowly while the fundamentals go in, accelerates once ideas connect, then slows again near saturation, when what’s left is the advanced, specialised material.
Investing Exponential growth
Reinvest the returns and each gain generates returns of its own. Under a constant growth rate that produces exponential growth — every increase larger than the one before it. Spotting the shape lets you predict the general behaviour before computing a single value.
Driving Braking distance and the square of speed
speed distance
Speed is the independent variable; braking distance, the dependent one. Under similar conditions, braking distance is roughly proportional to the square of speed — so doubling the speed produces roughly four times the braking distance, not twice.
Business Profit against number of customers
break-even loss profit
Customers is the independent variable; profit, the dependent one. Early on, fixed costs put the business at a loss. Past the break-even point, each additional customer starts adding to profit — in a simple model, almost linearly.
Connection Functions lead to integration
year 0 year 3
Suppose a curve shows the rate a business earns profit over time. The shaded area between that curve and the x-axis, from year to year , adds together every continuously changing profit amount across that stretch — the total accumulated profit.

That is the central idea of integration: using the area under a function to measure the total accumulation of a changing quantity.

Summary
  1. Learning follows an S-curve: slow, then fast, then slow again.
  2. Reinvested returns grow exponentially — each increase bigger than the last.
  3. Braking distance grows with the square of speed: double the speed, roughly quadruple the distance.
  4. Profit against customers starts as a loss, crosses break-even, then rises.
  5. The area under a rate curve, from integration, measures total accumulation over time.
288. Odd Functions
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