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Cogito

AP Calculus BC · Chapter 8 · Lesson 2

Practice Worksheet

Logistic Growth

Growth with a ceiling.

12 problems · about 22 minutes · FUN-7.H

NameDate

Figure — use these to answer the problems

A logistic field flattens near the carrying capacity. Solution curves from below rise and level off there.
  • The equationdy/dx = a·y
  • Through(0, 1)
k, the growth constant1
Starting x0
Starting y1
Figure 1

Warm Up

Straightforward practice. Get the method working first.

5 problems

  1. Carrying capacity 3000. At what population is growth fastest?

    Answer

  2. What does the bracket in the logistic equation do?

    • It shrinks toward zero as P approaches the capacity, slowing growth.
    • It accelerates growth near the capacity.

  3. Carrying capacity 800. At what population is growth fastest?

    Answer

  4. Carrying capacity 2000. Fastest growth at what population?

    Answer

  5. dP/dt = 0.2P(1 − P ÷ 500). Carrying capacity?

    Answer

Build It Up

The same ideas with more to keep track of.

3 problems

  1. As P approaches L, what does dP/dt approach?

    Answer

  2. How many equilibrium solutions does a logistic equation have?

    Answer

  3. When P is far below L, does it behave like exponential growth? 1 yes, 0 no.

    Answer

Stretch Yourself

Mixed problems. Work out what kind of question it is before you start.

4 problems

  1. Sort each statement by which model it describes.

    Write each item under the heading it belongs to: Growth continues without limit · The rate is proportional to the amount alone · Growth slows near a ceiling · The solution levels off at a carrying capacity

    Exponential

    Logistic

  2. Carrying capacity 1200. Fastest growth at what population?

    Answer

  3. The Capacity

    dP/dt = 0.1P(1 − P ÷ 1000). What is the carrying capacity?

    Answer

  4. The Fastest Point

    Same equation. At what population is growth fastest?

    Answer