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 the difference between an average rate of change over an interval and an instantaneous rate of change at a point. (Guide: “Representations”; Practice Q4, Q6(d))
- I can explain why (f(a + h) − f(a))/h and (f(x) − f(a))/(x − a) give the same number when x = a + h. (Guide: “Average rate of change: two difference quotients” and Figure 1)
- I can state the definition of f′(a) in both limit forms, including “provided the limit exists”. (Guide: “The derivative at a point”)
- I can explain the difference between f(a) and f′(a). (Guide: “Common misconceptions”; Practice Q2, Q4)
Skills
- I can evaluate an average rate of change with either difference quotient, including with a negative h (an interval to the left of a). (Guide: Worked example 1; Practice Q5(a), Q6(a), Q7(b))
- I can write f(a + h) correctly for a polynomial or a rational function and simplify the difference quotient. (Guide: Worked example 1; Practice Q1, Q5(b))
- I can find f′(a) exactly from the h-form, cancelling h only after noting h ≠ 0. (Guide: Worked example 1; Practice Q5(c), Q6(b))
- I can find f′(a) from the x-form by factoring or by using a conjugate. (Guide: Worked example 2; Practice Q5(d))
- I can state an instantaneous rate in context, with its sign, size and units. (Guide: Worked example 2; Practice Q6(c))
Reasoning
- I can recognise a given limit as f′(a) and name both the function and the point, checking the subtracted value is f(a). (Guide: Worked example 3; Practice Q2, Q3)
- I can show that f′(a) does not exist when the one-sided limits of the difference quotient differ. (Guide: “When the limit does not exist”; Practice Q7)
All ticked? Move on to Topic 2.2, Defining the Derivative of a Function and Using Derivative Notation, or return to the Calculus AB roadmap or the Calculus BC roadmap.