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 Intermediate Value Theorem, naming both conditions and the conclusion. (Guide: “The theorem, part by part”; Practice Q3)
- I can explain, with a sketch, why a continuous graph cannot skip a height between f(a) and f(b). (Guide: “The idea” and Figure 1)
- I can explain why the theorem gives “at least one” value of c and never its location. (Guide: Figure 1; Practice Q3, Q6(c))
- I can tell “the theorem does not apply” apart from “there is no solution”. (Guide: “When the theorem does not apply”; Practice Q7(b), Q7(c))
Skills
- I can check continuity on a closed interval for a polynomial, a quotient or a sum of familiar functions, and spot a point where it fails. (Guide: “When the theorem does not apply”; Practice Q1)
- I can find the constant that makes a piecewise function continuous before using the theorem. (Guide: Worked example 3; Practice Q4)
- I can give the least number of solutions of f(x) = d from a table of a continuous function. (Guide: Worked example 2; Practice Q2, Q5(b))
- I can show that an equation such as g(x) = h(x) has a solution by applying the theorem to g(x) − h(x). (Guide: “Using the theorem to locate solutions”; Practice Q6(a))
- I can narrow a solution to a smaller interval by testing a point inside it. (Guide: Worked example 1; Practice Q6(b))
Reasoning
- I can write a complete three-part justification: continuity with a reason, the trapped value with numbers, and the theorem named. (Guide: “Writing a justification”; Practice Q5(a), Q6(a))
- I can explain why a value outside the range of the end values is not guaranteed. (Guide: Worked example 2(c); Practice Q5(c))
- I can combine the theorem with another fact, such as the degree of a polynomial, to show there are exactly n solutions. (Guide: Worked example 1)
All ticked? You have finished this unit. Move on to Unit 2, Differentiation: Definition and Fundamental Properties, starting with Topic 2.1, Defining Average and Instantaneous Rates of Change at a Point, or return to the Calculus AB roadmap.