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 10.2, 10.11 and 10.13) are on the Calculus BC roadmap.
Understanding
- I can explain how a Taylor polynomial is a partial sum of the Taylor series, and why P₃ and P₄ for sin x are equal. (Guide: “From Taylor polynomials to Taylor series”; Practice Q5(b))
- I can state the difference between a Taylor series about x = c and a Maclaurin series. (Guide: “From Taylor polynomials to Taylor series”; Practice Q3)
- I can explain why the Maclaurin series for 1/(1 − x) is a geometric series and why it only works for −1 < x < 1. (Guide: “The four foundation series”)
Skills
- I can write the Maclaurin series for eˣ, sin x, cos x and 1/(1 − x), with general terms and intervals of convergence, from memory. (Guide: “The four foundation series”; Practice Q1)
- I can derive the eˣ, sin x and cos x series from their derivatives at 0. (Guide: “The four foundation series”)
- I can build a Taylor series about x = c from a pattern in the derivatives, including the general term. (Guide: Worked example 1; Practice Q5, Q6(a), Q7(a))
- I can write a series for a substituted function such as e^(−2x) from a known series. (Guide: “Building on the foundation series”)
- I can find the interval of convergence of a Taylor series using the ratio test and endpoint tests. (Guide: Worked example 1; Practice Q6(b))
Reasoning
- I can find f⁽ⁿ⁾(c) from a coefficient and use it to decide whether f is increasing, decreasing, concave up or down, or has a local extremum at c. (Guide: Worked example 2; Practice Q4, Q6(d))
- I can recognise a numerical series as a known Maclaurin series at a particular x and give its exact sum. (Guide: “Recognising the sum of a series”; Practice Q2)
- I can justify that a series converges for all real x using the ratio test. (Practice Q7(c))
All ticked? Move on to Topic 10.15, Representing functions as power series, or return to the Calculus BC roadmap.