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 Extreme Value Theorem and explain why it needs a closed interval and continuity. (Guide: “The Extreme Value Theorem, part by part”; Practice Q2)
- I can explain the difference between a global and a local extremum, and point to each on a graph. (Guide: Figure 1; Practice Q6(d))
- I can explain why an endpoint can be a global extremum but, with the open-interval definition, not a local one. (Guide: “Global versus local extrema”)
- I can state the definition of a critical point, including the condition that f is defined there. (Guide: “Critical points”; Practice Q3)
- I can explain why a critical point is only a candidate for a local extremum, with an example. (Guide: Figure 2; Practice Q4)
Skills
- I can find critical points where f′(x) = 0 by factoring. (Guide: Worked example 1; Practice Q1)
- I can find critical points where f′(x) does not exist, by writing f′ as one fraction or checking corners. (Guide: Worked example 2; Practice Q3, Q5)
- I can decide whether the Extreme Value Theorem applies to a function on a given interval. (Guide: Worked example 3; Practice Q2)
- I can show that a critical point is not an extremum by comparing nearby values with f(c). (Guide: Worked example 1; Practice Q5(c))
Reasoning
- I can justify the existence of a maximum or minimum value using the Extreme Value Theorem, naming the condition. (Practice Q6(a), Q7(b))
- I can explain why a table of values does not locate a maximum exactly. (Practice Q6(b))
- I can argue that an interior global extremum of a differentiable function has f′(c) = 0. (Practice Q6(d))
- I can argue that a function continuous on [a, b] with no critical points inside (a, b) has its maximum and minimum at the endpoints. (Practice Q7(c))
All ticked? Move on to Topic 5.3, Determining Intervals on Which a Function Is Increasing or Decreasing, or return to the Calculus AB roadmap or the Calculus BC roadmap.