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 explain why the graph of f′ carries two kinds of information about f: its sign gives direction and its rising or falling gives concavity. (Guide: “One story told by three graphs”; Practice Q1)
- I can say what a graph of f″ tells me about f and f′, and what it cannot tell me. (Guide: “Reading a graph of f″”; Practice Q2)
- I can explain why the height of f′ is a slope of f and never a value of f. (Guide: “Reading a graph of f′”)
Skills
- I can find the intervals where f increases, decreases, is concave up and is concave down from a graph of f′. (Guide: Worked example 1; Practice Q3, Q6(b))
- I can locate and classify relative extrema of f from where f′ crosses the axis. (Guide: Worked example 1(b); Practice Q6(a))
- I can locate inflection points of f as turning points of f′ or sign changes of f″. (Guide: Worked examples 1 and 3; Practice Q4, Q6(c))
- I can find a value of f″ as the slope of a straight part of the graph of f′. (Guide: Worked example 1(e); Practice Q6(d))
- I can decide which of three graphs is f, f′ and f″ by lining up zeros with turning points. (Guide: Worked example 2; Practice Q5(a))
Reasoning
- I can justify an extremum or inflection point with a sentence that names f′ or f″ and says how it changes. (Guide: “Justifying a conclusion”; Practice Q5(b), Q6(a))
- I can explain why a zero of f″, or a point where f″ does not exist, is not automatically an inflection point. (Guide: “Common misconceptions”; Practice Q4, Q6(e))
- I can use sign facts about f″ and a few values of f′ to work out where f is increasing or decreasing. (Practice Q7)
All ticked? Move on to Topic 5.10, Introduction to Optimization Problems, or return to the Calculus AB roadmap or the Calculus BC roadmap.