Cogito
AP Calculus BC · Chapter 8 · Lesson 2
Logistic Growth
Growth with a ceiling.
12 problems · about 22 minutes · FUN-7.H
What this lesson teaches
The student analyses logistic differential equations and identifies carrying capacity.
- The logistic equation adds a carrying capacity to exponential growth.
- Growth is fastest at half the capacity, which is the inflection point.
- The solution levels off at the capacity.
Warm Up
Straightforward practice. Get the method working first.
5 problemsCarrying capacity 3000. At what population is growth fastest?
Answer 1500
Why 1500.
What does the bracket in the logistic equation do?
Answer It shrinks toward zero as P approaches the capacity, slowing growth.
Why It applies the brake near the ceiling.
Carrying capacity 800. At what population is growth fastest?
Answer 400
Why Half.
Carrying capacity 2000. Fastest growth at what population?
Answer 1000
Why Half.
dP/dt = 0.2P(1 − P ÷ 500). Carrying capacity?
Answer 500
Why The denominator.
Build It Up
The same ideas with more to keep track of.
3 problemsAs P approaches L, what does dP/dt approach?
Answer 0
Why The bracket vanishes.
How many equilibrium solutions does a logistic equation have?
Answer 2
Why Zero and the capacity.
When P is far below L, does it behave like exponential growth? 1 yes, 0 no.
Answer 1
Why The bracket is nearly 1.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsSort each statement by which model it describes.
Answer Exponential: Growth continues without limit, The rate is proportional to the amount alone · Logistic: Growth slows near a ceiling, The solution levels off at a carrying capacity
Why Only one of them has a ceiling.
Carrying capacity 1200. Fastest growth at what population?
Answer 600
Why Half.
The Capacity: dP/dt = 0.1P(1 − P ÷ 1000). What is the carrying capacity?
Answer 1000
Why 1000.
The Fastest Point: Same equation. At what population is growth fastest?
Answer 500
Why 500.