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 3.6 and 4.6, and factorials) are on the Calculus BC roadmap.
Understanding
- I can explain why the coefficient of (x − a)ⁿ is f⁽ⁿ⁾(a)/n!, by differentiating the polynomial n times. (Guide: “The coefficient formula, and where n! comes from”)
- I can explain that the tangent line is the first-degree Taylor polynomial. (Guide: “From a tangent line to a better polynomial”)
- I can describe how the approximation usually improves as the degree rises, and why it is best near the centre. (Guide: “Seeing the approximation improve”; Practice Q6(c))
- I can tell the difference between “nth-degree” and “first n nonzero terms”. (Guide: “Degree versus number of terms”)
Skills
- I can find a single coefficient from a derivative value, dividing by the right factorial. (Practice Q1)
- I can build a Taylor polynomial for a given function by finding and evaluating its derivatives at the centre. (Guide: Worked example 1; Practice Q2, Q5(a))
- I can build a Taylor polynomial from a table of derivative values. (Guide: Worked example 2; Practice Q6(a))
- I can use a Taylor polynomial to estimate a function value, taking care with the sign of (x − a). (Practice Q4, Q5(b), Q6(b), Q7(d))
- I can read derivative values back from a given Taylor polynomial using f⁽ᵏ⁾(a) = k! × coefficient. (Guide: “Reading backwards”; Practice Q3, Q6(d), Q7(a))
Reasoning
- I can use derivative values read from a Taylor polynomial to justify a relative maximum or minimum. (Practice Q7(b))
- I can find the Taylor polynomial for f′ from the one for f. (Practice Q7(c))
- I can interpret a coefficient in context, with units. (Guide: Worked example 2(c))
All ticked? Move on to Topic 10.12, Lagrange Error Bound, or return to the Calculus BC roadmap.