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 why a differential equation usually has a family of solutions, and how one point selects a particular solution. (Guide: “Solutions are functions, and there are many”)
- I can say how the members of a family are related when dy/dx depends on x only or on y only. (Guide: shifts table; Practice Q4)
- I can explain why a horizontal line of flat segments is a solution but a sloping line of flat segments is not. (Guide: “Constant solutions”; Practice Q1)
Skills
- I can find the constant solutions of a differential equation and check them by substitution. (Practice Q1, Q5(c), Q6(a))
- I can sketch the particular solution through a given point, on both sides of the point. (Guide: “How to sketch a particular solution”; Practice Q5(a))
- I can find d²y/dx² by implicit differentiation and write it in terms of x and y. (Guide: Worked examples 1 and 2; Practice Q3, Q5(d), Q7(b))
- I can write a tangent line to a particular solution and use it to estimate a value. (Guide: Worked example 2; Practice Q6(c))
Reasoning
- I can describe the long-run behaviour of a solution from the field and justify it with the slopes. (Guide: Worked example 1(c); Practice Q2, Q5(b), Q6(d))
- I can decide whether a tangent-line estimate is an over- or underestimate using concavity. (Guide: Worked example 2(b); Practice Q6(c))
- I can justify a relative maximum or minimum of a particular solution with dy/dx and d²y/dx². (Practice Q5(d))
- I can show that a straight line is a solution and reason about whether nearby solutions approach it or move away. (Guide: Worked example 2; Practice Q7)
All ticked? Move on to Topic 7.6, Finding General Solutions Using Separation of Variables, or return to the Calculus AB roadmap or the Calculus BC roadmap.