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 why a discontinuity can be removed only when the limit at that point exists. (Guide: “The idea: some breaks can be repaired” and Figure 1; Practice Q3)
- I can explain why redefining one value cannot remove a jump or a vertical asymptote. (Guide: “When it is impossible, or there are two answers”; Practice Q7(c))
- I can state the three values that must agree at a boundary of a piecewise function, and say which piece gives f(c). (Guide: “Piecewise functions: making the pieces meet”; Practice Q6(a))
- I can explain why a calculator graph that looks unbroken does not prove continuity. (Guide: “Using technology”; Practice Q5(c))
Skills
- I can find the value f(c) must take to remove a discontinuity, using factoring or a conjugate. (Guide: Worked example 1; Practice Q1, Q5(b))
- I can redefine f(c) when it is defined but does not equal the limit. (Guide: “Removing a discontinuity: define or redefine f(c)”)
- I can solve for one parameter that makes a piecewise function continuous at a boundary, and check both pieces. (Guide: Worked example 2 and Figure 2; Practice Q2)
- I can solve for two parameters using one equation from each boundary. (Guide: Worked example 3; Practice Q6)
- I can find a parameter that makes a 0/0 form possible, then the value that removes the discontinuity. (Practice Q7(a), Q7(b))
Reasoning
- I can decide, for each discontinuity of a rational function, whether it is removable, and justify my answer with a limit. (Guide: Worked example 1; Practice Q3)
- I can recognise when no parameter value works, and say so instead of forcing an answer. (Guide: “No value works”; Practice Q4)
- I can give every solution when the boundary condition is a quadratic in the parameter. (Guide: “More than one value works”)
- I can write a full justification that a piecewise function is continuous for all real x. (Guide: Worked example 3, step 6; Practice Q6(c))
All ticked? Move on to the Topic 1.14 study guide, Connecting Infinite Limits and Vertical Asymptotes, or return to the Calculus AB roadmap or the Calculus BC roadmap.