BC-only material. Work through the list without notes. If you cannot do a statement, follow its link, then come back. Guide = study guide; Practice = practice questions. Prerequisites (Topics 7.1, 7.4 and 7.8) are on the Calculus BC roadmap.
Understanding
- I can explain how a logistic model differs from an exponential model, and why logistic growth levels off. (Guide: “From exponential to logistic growth”)
- I can explain what “jointly proportional” means and turn such a sentence into dy/dt = ky(a − y). (Guide: Worked example 2; Practice Q4)
- I can say what the carrying capacity means in context. (Guide: Worked example 1(a); Practice Q6(b))
Skills
- I can rewrite a logistic equation given in another form (such as by − cy² or ry(1 − y/a)) as ky(a − y) and read off k and a. (Guide: “Other forms you will meet”; Practice Q2, Q6(a))
- I can find the constant solutions and the long-run limit, using the initial condition. (Guide: “Reading the equation without solving it”; Practice Q1, Q5(a))
- I can find the value of y when it is changing fastest, and the greatest rate, with units. (Guide: Worked example 1(b); Practice Q2, Q5(b), Q7(d))
- I can find d²y/dt² for a logistic equation and use its sign to describe concavity. (Guide: “Concavity”; Practice Q5(c))
- I can find a constant of proportionality from one known rate. (Guide: Worked example 2(a))
Reasoning
- I can describe how a solution behaves when it starts above the carrying capacity, at it, or at zero. (Guide: Figure 2 and Worked example 1(e); Practice Q3, Q5(d))
- I can explain why the relative growth rate in a logistic model is not constant. (Guide: Worked example 1(d); Practice Q6(d))
- I can use concavity to explain why an Euler estimate for a logistic model is too low or too high. (Practice Q7(b))
All ticked? Move on to Topic 8.1, Finding the Average Value of a Function on an Interval, or return to the Calculus BC roadmap.