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 a graph of f is really a graph of g′ when g(x) = ∫ (a to x) f(t) dt. (Guide: “The big idea”)
- I can say which feature of f gives each feature of g: sign for increasing, slope for concavity, area for values. (Guide: “How features of f show up in g”; Practice Q1)
- I can explain why a zero of f without a sign change gives no extremum of g. (Guide: feature table; Practice Q3, Q7(b))
- I can explain why a corner in f can still give a point of inflection of g. (Guide: Worked example 1(c))
Skills
- I can find the intervals where g increases or decreases from a graph, a formula or a sign description of f. (Guide: Worked examples 1 and 2; Practice Q2, Q5(b))
- I can find the points of inflection of g from where f changes between increasing and decreasing. (Guide: Worked examples 1(c) and 2(b); Practice Q5(c), Q7(c))
- I can find values of g with signed areas, including values to the left of the starting point. (Guide: Worked example 1(d); Practice Q5(a))
- I can use the candidates test to find the absolute extrema of g on a closed interval. (Guide: Worked example 1(d); Practice Q4, Q5(d))
- I can estimate a value of an accumulation function from a table with a trapezoidal sum. (Guide: Worked example 3(d); Practice Q6(d))
Reasoning
- I can justify every claim about g with a reason about g′ = f or g″ = f′, not about g itself. (Guide: “How to justify a claim about g”)
- I can use the Intermediate Value Theorem, with stated facts about f, to locate and count the critical points of g from a table. (Guide: Worked example 3(b); Practice Q6(b))
- I can interpret the rate and the accumulated amount in context, with units. (Practice Q6(a))
All ticked? Move on to Topic 6.6, Applying Properties of Definite Integrals, or return to the Calculus AB roadmap or the Calculus BC roadmap.