Work through the list without notes. Use a calculator only where a statement needs decimal values. If you cannot do a statement, follow its link, then come back. Guide = study guide; Practice = practice questions. This topic is shared by Calculus AB and Calculus BC.
Understanding
- I can turn “the rate of change of a quantity is proportional to the quantity” into dy/dt = ky, and tell it apart from “proportional to time”. (Guide: “From a sentence to a differential equation”; Practice Q1)
- I can say what y, t, dy/dt and k mean in a given context, including the units of k. (Guide: “From a sentence to a differential equation”; Practice Q6(b))
- I can explain why k is a relative rate, so the actual rate dy/dt changes as y changes. (Guide: “What y₀ and k mean”; Practice Q4)
- I can explain why the doubling time or half-life does not depend on the starting amount. (Guide: Figure 1; Practice Q4)
Skills
- I can derive y = y₀e^(kt) from dy/dt = ky by separating variables, keeping the constant as a factor. (Guide: “Solving dy/dt = ky”; Practice Q2, Q5(c))
- I can find k from two data values, a doubling time or a half-life. (Guide: Worked examples 1 and 2; Practice Q3, Q6(b), Q7(a))
- I can use the model to find a value, an instantaneous rate with units, and the time to reach a given amount. (Guide: Worked example 1; Practice Q6)
- I can use a decay model for velocity to find acceleration, position and the total distance travelled. (Guide: Worked example 3; Practice Q7)
Reasoning
- I can confirm a proposed solution by differentiating it and checking the initial value, and find the error in a wrong one. (Guide: “Confirm the solution before you use it”; Practice Q5)
- I can judge whether an answer is reasonable from the sign of k, the units and the long-run behaviour. (Guide: “Confirm the solution”; Practice Q5(d), Q7(d))
- I can explain why rounding k early changes later answers. (Guide: Worked example 1 tip; Practice Q6(c))
All ticked? BC students continue to Topic 7.9, Logistic Models with Differential Equations. Everyone then moves on to Topic 8.1, Finding the Average Value of a Function on an Interval, or returns to the Calculus AB roadmap or the Calculus BC roadmap.