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 4.6 and 7.1–7.4) are on the Calculus BC roadmap.
Understanding
- I can explain why each Euler step is a tangent-line approximation, and why several short steps usually beat one long one. (Guide: “The idea: follow tangent lines in short steps”; Practice Q5(a))
- I can explain how Euler’s method follows a slope field from a starting point. (Guide: “The idea”; Figure 1)
- I can say what happens to the error when the step size is made smaller, and why it does not become zero. (Guide: “How step size affects accuracy”; Practice Q7(d))
Skills
- I can carry out one Euler step from a given point, using the slope at that point. (Guide: “The procedure”; Practice Q1)
- I can carry out several steps in a table, updating both x and y each time. (Guide: Worked example 1; Practice Q2, Q7(b)–(c))
- I can work out the number of steps from the interval and the step size. (Guide: “The procedure”; Practice Q2)
- I can use a negative step to approximate a value to the left of the starting point. (Guide: “Stepping backwards”; Practice Q4)
- I can find d²y/dx² from a differential equation by implicit differentiation. (Guide: “Overestimate or underestimate?”; Practice Q5(c), Q6(c))
- I can keep full values between steps and round only the final answer. (Guide: Worked example 2; Practice Q6(b))
Reasoning
- I can decide whether an Euler estimate is an overestimate or an underestimate, using the sign of d²y/dx² and not whether y is increasing. (Guide: “Overestimate or underestimate?”; Practice Q3, Q5(d))
- I can predict whether a smaller step will raise or lower an estimate, given its concavity. (Practice Q6(d))
- I can state an Euler estimate in context, with units, and say what it means. (Guide: Worked example 2; Practice Q6)
All ticked? Move on to Topic 7.6, Finding General Solutions Using Separation of Variables, or return to the Calculus BC roadmap.