Short on time? This page is the recap. For the graphs and worked examples, use the full study guide. BC-only material. Prerequisites (Topics 7.1, 7.4 and 7.8, and concavity from Unit 5) are on the Calculus BC roadmap.
Recap
- Logistic growth: the rate is jointly proportional to the amount y and to the room left, a − y.
- So dy/dt = ky(a − y) with k > 0. The constant a is the carrying capacity.
- Growth starts almost exponentially, slows down, and levels off at a.
- You can read everything you need from the equation and the initial condition. No solving needed.
Key relationships
| Feature | How to find it | Example: dy/dt = 0.03y(90 − y) |
|---|---|---|
| Carrying capacity | a in ky(a − y) | 90 |
| Constant solutions | y = 0 and y = a | y = 0, y = 90 |
| Long-run limit | a if y(0) > 0; 0 if y(0) = 0 | 90 |
| y when changing fastest | a/2 | 45 |
| Greatest rate | ka²/4 = k(a/2)(a/2) | 0.03 × 45 × 45 = 60.75 |
| Second derivative | d²y/dt² = k(a − 2y) · dy/dt | Inflection at y = 45 |
| Other form | ry(1 − y/a) with r = ka | 2.7y(1 − y/90) |
Behaviour by starting value
| y(0) | What y does | Concavity |
|---|---|---|
| 0 or a | Stays constant | None |
| Between 0 and a/2 | Increases to a; S-shaped | Up, then down after y = a/2 |
| Between a/2 and a | Increases to a | Down |
| Above a | Decreases to a | Up |
Mistakes to avoid
- Reading a from the wrong place. Rewrite as ky(a − y) first. For 0.6y − 0.002y², factor: 0.002y(300 − y), so a = 300.
- “Fastest at a.” The rate is 0 at a. Fastest is at a/2.
- Answering with a time when the question asks for the value of y.
- Forgetting the chain rule in d²y/dt²: differentiate with respect to t, so dy/dt appears.
- Assuming growth. If y(0) > a, the quantity falls.
Quick self-check
- For dy/dt = 0.03y(90 − y) with y(0) = 10, find the limit as t → ∞ and the value of y when it grows fastest. (90; 45)
- Find the carrying capacity of dP/dt = 0.6P − 0.002P². (Factor: 0.002P(300 − P), so 300.)
- A logistic quantity starts at y(0) = 0. What is its limit? (0: it is a constant solution.)
- For dy/dt = ky(a − y) with y(0) > a, is y increasing or decreasing? (Decreasing, towards a.)
Next: practice questions.