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 that a function differentiable at a point is continuous there, and explain why. (Guide: “Differentiable means continuous”; Practice Q1)
- I can use the contrapositive: a discontinuity at a means f′(a) does not exist. (Guide: “The contrapositive: a fast test”; Practice Q5(b), Q6(c))
- I can explain why a point missing from the domain of f is missing from the domain of f′. (Guide: the r(x) example; Practice Q3)
- I can explain, with an example, why continuity does not guarantee differentiability. (Guide: “The converse is false”; Practice Q7(d))
Skills
- I can recognise a corner, a cusp, a vertical tangent and a discontinuity on a graph. (Guide: Figure 1; Practice Q2, Q5)
- I can find both one-sided limits of the difference quotient, using the correct value of f(a). (Guide: Worked examples 1 and 2; Practice Q6(b), Q7(b))
- I can rewrite an absolute-value function piece by piece near the point where its inside changes sign. (Guide: Worked example 1; Practice Q7(b))
- I can find constants that make a piecewise function continuous and differentiable at a join, or show none exist. (Guide: Worked example 2; Practice Q4)
Reasoning
- I can write a justification that names the one-sided limits or quotes the theorem, not just “it’s sharp”. (Guide: “Writing a justification”; Practice Q5, Q6)
- I can check continuity before comparing slopes at the join of a piecewise function. (Guide: “The two-step method”; Practice Q6)
- I can explain why a calculator’s numerical derivative may report a number where no derivative exists. (Guide: “A warning about calculators”; Practice Q7(c))
All ticked? Move on to Topic 2.5: Applying the Power Rule, or return to the Calculus AB roadmap or the Calculus BC roadmap.