Work through the list without notes or a calculator. 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 explain the difference between a general solution and a particular solution. (Guide: “From a family to one curve”; Practice Q6)
- I can explain why only one particular solution passes through a given point. (Guide: “From a family to one curve”; Practice Q6(c))
- I can explain why y₀ + ∫ (a to x) f(t) dt passes through (a, y₀) and solves dy/dx = f(x). (Guide: “The integral form”; Practice Q2)
- I can explain why a particular solution must live on one interval containing the initial point. (Guide: “Domain restrictions”; Figure 1)
Skills
- I can substitute an initial condition into the integrated equation to find the constant. (Guide: Worked examples 1 to 3; Practice Q4, Q5(a))
- I can choose the correct sign after a square root, using the initial value. (Guide: Worked example 1; Practice Q4)
- I can remove ln|y| and keep the sign of the initial value. (Guide: Worked example 2; Practice Q1, Q6(b))
- I can state the largest open interval on which a particular solution is valid. (Guide: Worked example 3; Practice Q3, Q7(b))
- I can write the integral form of a particular solution with the correct limit and starting value. (Guide: “The integral form”; Practice Q2)
- I can find a value of a particular solution, such as f(4) or g(1), exactly. (Practice Q4, Q6(d), Q7(c))
Reasoning
- I can check a particular solution by testing the initial point and differentiating. (Guide: Worked examples 1 to 3)
- I can explain why y = e^(G(x)) + C can pass through the right point and still be wrong. (Guide: Worked example 2; Practice Q1)
- I can interpret a particular solution and its domain in a context, with units. (Practice Q5)
- I can describe the behaviour of a solution near the edge of its domain. (Practice Q7(d); Figure 1)
All ticked? Move on to Topic 7.8, Exponential Models with Differential Equations, or return to the Calculus AB roadmap or the Calculus BC roadmap.