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 say which derivative reveals each key feature of a graph: increase, extrema, concavity and inflection. (Guide: “What a sketch has to show”)
- I can match each sign pair of f′ and f″ to one of the four basic shapes. (Guide: shapes table; Practice Q2)
- I can explain why a zero of f′ or f″ is not enough on its own, and what extra evidence is needed. (Guide: Worked example 1; Practice Q1)
- I can explain why a peak of the f′ graph is a point of inflection of f, not a maximum. (Guide: Figure 2; Practice Q6)
Skills
- I can build a combined sign chart for f′ and f″ from a formula and calculate f at every key point. (Guide: Worked example 1; Practice Q7)
- I can read where f increases, has extrema and changes concavity from a graph of f′. (Guide: Worked example 2; Practice Q6)
- I can draw conclusions about f from a table of f′ values plus information about f′ between them. (Guide: Worked example 3; Practice Q4)
- I can describe the graph of f′ from the features of f. (Guide: “Going the other way”; Practice Q3)
- I can sketch f with every extremum and inflection point labelled, using the correct bend in each interval. (Practice Q5(d), Q6(d), Q7(d))
Reasoning
- I can justify each feature by naming the sign change or behaviour of f′ or f″ that causes it. (Practice Q5(a), Q5(b), Q6)
- I can handle a critical point where f′ does not exist, such as a cusp, and say which test still works there. (Practice Q7(b))
All ticked? Move on to Topic 5.9, Connecting a Function, Its First Derivative, and Its Second Derivative, or return to the Calculus AB roadmap or the Calculus BC roadmap.