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 what it means for a function to be continuous on an interval. (Guide: “From one point to a whole interval”)
- I can state the extra conditions needed at the endpoints of a closed interval [a, b]. (Guide: “Open and closed intervals”; Practice Q7(b))
- I can list the six function families that are continuous on their domains, and say where each common example is continuous. (Guide: family table; Practice Q1)
- I can explain why “continuous on its domain” does not mean “continuous on every interval”, using 1/x as an example. (Guide: “Common misconceptions”; Practice Q7(a))
Skills
- I can find the domain of a formula with denominators, even roots and logarithms, and write it as a union of intervals. (Guide: Worked example 1; Practice Q2, Q5(a))
- I can choose the correct bracket at each endpoint of an interval of continuity. (Guide: Worked example 1, step 5; Practice Q5(b))
- I can test a piecewise boundary using the left limit, the right limit and the function value. (Guide: Worked example 2; Practice Q3, Q6(b), Q6(c))
- I can find where a trigonometric quotient fails to be defined and compare those values with a given interval. (Guide: Worked example 3; Practice Q4)
Reasoning
- I can justify intervals of continuity by naming the family facts and the combination rules I used. (Guide: “A procedure you can rely on”; Practice Q5(b), Q6(a))
- I can decide which side’s interval a boundary point belongs to, using one-sided continuity. (Guide: Figure 2; Practice Q3, Q6(d))
- I can explain why an asymptote of a formula does not matter outside the interval where that formula is used. (Guide: Worked example 2, step 1)
- I can show that one excluded value inside an interval is enough to make the answer “not continuous”. (Guide: Worked example 3; Practice Q4, Q5(c))
All ticked? Move on to Topic 1.13, Removing Discontinuities, or return to the Calculus AB roadmap or the Calculus BC roadmap.