Aim for FiveCalculus BC

Free response without a calculator (Part B)

BC: Logistic growth and Euler's method

  • Unit 7
  • 9 points
  • About 15 minutes
  • BC only

A multi-part problem you solve by hand, often from a graph, a table, an equation or a differential equation. You show your work, use exact values, and justify answers with calculus reasons such as a sign change in a derivative or the conditions of a theorem. On the exam: 4 questions in Part B of the free-response section (60 minutes, no calculator). The whole free-response section is 6 questions in 90 minutes and counts for 50% of the score. AB and BC use the same format; on BC, Part B usually includes a series question and a question on BC-only topics.

The question

The number of fish in a pond is modeled by a function P that satisfies the logistic differential equation dP/dt = P(600 − P)/1000, where t is measured in weeks. At time t = 0, there are 100 fish in the pond.

Suggested time: 15 minutes

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Part (a)

2 points

For what values of P is the fish population increasing? Find lim (t→∞) P(t).

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Part (b)

2 points

For what value of P is the population growing the fastest? Justify your answer.

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Part (c)

2 points

Use Euler's method, starting at t = 0 with two steps of equal size, to approximate P(2).

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Part (d)

3 points

It can be shown that 100 ≤ P(t) < 300 for 0 ≤ t ≤ 2. Is the approximation in part (c) an overestimate or an underestimate of P(2)? Give a reason for your answer.

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