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 10.10 and 10.11) are on the Calculus BC roadmap.
Understanding
- I can state the Lagrange error bound and say what n, M and f⁽ⁿ⁺¹⁾ stand for. (Guide: “The Lagrange error bound”; Practice Q1)
- I can explain that the bound is a guarantee, not the actual error, and that the true error is often smaller. (Guide: Worked example 2; Practice Q4)
- I can explain why a bigger M is safe but a smaller M than the true maximum is not. (Guide: “Choosing M”; Practice Q2, Q5(c))
Skills
- I can find a valid M for sin, cos, eˣ and for derivatives that only increase or only decrease on the interval. (Guide: “Choosing M”; Practice Q2, Q5(c), Q7(a))
- I can calculate the Lagrange error bound for a given polynomial and point. (Guide: Worked example 1; Practice Q1, Q6(c))
- I can use a bound to write an interval that must contain the true value. (Guide: Worked example 1(c); Practice Q4, Q6(d))
- I can find the smallest degree that the Lagrange bound guarantees for a required accuracy. (Guide: Worked example 2; Practice Q7(a))
- I can use the alternating series error bound for a Taylor estimate when its conditions hold. (Guide: “The alternating series error bound as an alternative”; Practice Q3, Q7(c))
Reasoning
- I can show that an error is less than a given number by stating the bound and then the comparison. (Guide: “When M is given”; Practice Q5(c), Q6(c))
- I can decide whether an estimate is an overestimate or an underestimate, using the sign of the first omitted term or of f⁽ⁿ⁺¹⁾. (Guide: Worked example 1 checks; Practice Q7(c))
- I can use an error interval to justify a conclusion, such as the sign of a function value. (Practice Q6(d))
- I can spot when Pₙ = Pₙ₊₁ and use the stronger bound. (Guide: sin 0.4 example; Practice Q7(b))
All ticked? Move on to Topic 10.13, Radius and Interval of Convergence of Power Series, or return to the Calculus BC roadmap.