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 describe concave up and concave down in terms of how the slope changes, using tangent lines. (Guide: Figure 1)
- I can explain why concavity is different from increasing or decreasing. (Guide: “Which way does the graph bend?”; Practice Q1)
- I can explain why f″ > 0 means concave up. (Guide: “The second derivative decides concavity”)
- I can state the two conditions for a point of inflection. (Guide: “Points of inflection”; Practice Q6(c))
Skills
- I can find f″, locate where it is zero or undefined, and build a sign chart. (Guide: Worked example 1; Practice Q1, Q6)
- I can find f″ from a given f′ and use it to locate inflection points. (Practice Q4)
- I can read concavity and inflection points of f from a graph of f′. (Guide: Worked example 2; Practice Q5(a), Q5(b))
- I can find f″ at a point from the slope of a graph of f′. (Practice Q5(d))
- I can give a point of inflection as a coordinate pair. (Guide: Worked example 1; Practice Q6(c))
Reasoning
- I can explain why f″(c) = 0 alone does not prove an inflection point, using x⁴ as an example. (Guide: table in “Points of inflection”; Practice Q2, Q7(c))
- I can tell the difference between where f′ crosses zero (extrema of f) and where f′ turns (inflection points of f). (Guide: Worked example 2(c); Practice Q5(c))
- I can explain why a gap in the domain, such as x = 0 for 1/x, is not a point of inflection. (Guide: “Points of inflection”; Practice Q6(c))
- I can interpret the signs of f′ and f″ in context with the words “increasing at a decreasing rate”. (Guide: “Concavity in context”; Practice Q3)
All ticked? Move on to Topic 5.7, Using the Second Derivative Test to Determine Extrema, or return to the Calculus AB roadmap or the Calculus BC roadmap.