Worked example
Logistic Growth: The S-Curve of Limited Resources
Unlimited growth is exponential, but real populations run into limits: food, space, or market size. The logistic function 10 / (1 + 9·exp(-x)) starts out looking exponential, then bends over and levels off at a carrying capacity — here, 10. The result is the famous S-curve seen in bacterial colonies, product adoption, and the spread of ideas.
The curve has an inflection point where it switches from accelerating to decelerating — the moment growth is fastest, exactly halfway to the carrying capacity. Before that point the curve bends upward (growth feeding on itself); after it, the curve bends downward as the limit bites. Finding that inflection point is one of the most useful things calculus can do for a model.
Plotted expressions
- y = 10 / (1 + 9*exp(-x))
The link preloads these exact expressions into the calculator — no typing needed.
What to notice
- The horizontal asymptote y = 10 is the carrying capacity the curve approaches but never exceeds.
- The steepest part of the S is the inflection point — where growth is fastest.
- Try 10 / (1 + 9*exp(-2*x)) to see how a faster growth rate steepens the middle of the S without changing its ceiling.