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 state the First Derivative Test, including the condition that f is continuous at the critical point. (Guide: “The First Derivative Test”)
- I can explain why a change from increasing to decreasing gives a relative maximum, using the link to Topic 5.3. (Guide: “Why it works”, Figure 1)
- I can explain why f′(c) = 0 alone does not prove an extremum, and give an example of a critical point that is neither. (Guide: Worked example 1; Practice Q1, Q4)
- I can explain why a sign change of f′ at a jump or a gap in the domain proves nothing. (Guide: “When the test does not apply”; Practice Q6)
Skills
- I can find all critical points, including those where f′ does not exist, and build a sign chart for f′. (Guide: Worked examples 1 and 2; Practice Q3, Q7)
- I can classify each critical point as a relative maximum, a relative minimum or neither. (Practice Q1, Q7(b))
- I can find the value of a relative extremum by substituting into f, and keep it separate from its x-coordinate. (Practice Q2, Q7(c))
- I can locate relative extrema from a graph of f′, and ignore peaks and valleys of f′ that are not on the x-axis. (Guide: “Reading extrema from a graph of f′”; Practice Q5)
Reasoning
- I can write a justification that names the function, the x-value and the change of sign of f′. (Guide: “Writing a justification”; Practice Q5(a), Q5(b))
- I can explain why an argument based only on f′(c) = 0 is incomplete. (Practice Q7(d))
- I can decide whether a point is a relative extremum directly from nearby function values when the test does not apply. (Practice Q6(c))
All ticked? Move on to Topic 5.5, Using the Candidates Test to Determine Absolute (Global) Extrema, or return to the Calculus AB roadmap or the Calculus BC roadmap.