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, with a sketch, why a horizontal tangent on a concave-up curve marks a relative minimum and on a concave-down curve marks a relative maximum. (Guide: “The idea” and Figure 1; Practice Q1)
- I can explain why the second derivative test is a shortcut to the first derivative test. (Guide: “Why the test works”)
- I can explain why f″(c) = 0 gives no conclusion, using examples such as x⁴, −x⁴ and x³. (Guide: Figure 2; Practice Q4)
- I can state the conditions of the one-critical-point rule and explain why it works. (Guide: “From relative to absolute”; Practice Q7(c))
Skills
- I can find the critical points of a function and evaluate f″ at each one, including trigonometric, logarithmic and rational functions. (Guide: Worked examples 1 and 3; Practice Q2, Q5)
- I can read a table of f′ and f″ values and say which points are definitely extrema. (Practice Q3)
- I can switch to the first derivative test when f″(c) = 0 or f′(c) does not exist, and finish the classification. (Guide: Worked example 2; Practice Q6(c))
- I can find an absolute extremum on an interval without endpoints, using the second derivative test and the one-critical-point rule. (Guide: Worked example 3; Practice Q5(c), Q7(a), Q7(b))
Reasoning
- I can write a justification that names f′(c) = 0 and the sign of f″(c), using the names f′ and f″. (Guide: “Writing the justification”; Practice Q2, Q6(b))
- I can correct the claim “f″(c) = 0, so there is no extremum”. (Practice Q6(d))
- I can explain why a relative extremum is not always absolute when an interval contains more than one critical point. (Practice Q7(c))
All ticked? Move on to Topic 5.8, Sketching Graphs of Functions and Their Derivatives, or return to the Calculus AB roadmap or the Calculus BC roadmap.