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 what it means for a function to be a solution of a differential equation on an interval. (Guide: “What ‘solution’ means here”)
- I can explain why agreement at one input value does not prove a solution, but disagreement at one value disproves it. (Guide: “Why one point is not enough”; Practice Q7(c))
- I can explain why a differential equation can have infinitely many solutions, using a family with a constant C. (Guide: “Infinitely many solutions” and Figure 1; Practice Q4, Q5(d))
- I can explain why a formula is a solution only on intervals where it is defined and differentiable. (Guide: Worked example 3; Practice Q6(c))
Skills
- I can verify a first-order solution by setting out the left side and right side separately. (Guide: Worked example 1; Practice Q1, Q3)
- I can verify a second-order solution, finding y″ with the chain rule. (Guide: “Second-order equations”; Practice Q5)
- I can find the values of r that make y = e^(rx) a solution. (Guide: Worked example 2; Practice Q2)
- I can find unknown constants in a given form by matching coefficients. (Practice Q7(a))
- I can show a family with an arbitrary constant solves an equation for every value of the constant. (Guide: Worked examples 1 and 3; Practice Q6(b), Q7(b))
Reasoning
- I can spot a constant solution and say whether the family includes it. (Guide: Worked example 3; Practice Q6(d))
- I can explain why adding a constant to a solution may give a non-solution. (Guide: Worked example 2; Practice Q5(c))
- I can check that a solution’s shape matches what the equation says about the rate. (Guide: Worked example 1 and Figure 1)
All ticked? Move on to Topic 7.3, Sketching Slope Fields, or return to the Calculus AB roadmap or the Calculus BC roadmap.