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 the average-rate formula gives 0/0 at a single instant, and why 0/0 is undefined rather than 0. (Guide: “Why one division cannot give the rate at an instant”; Practice Q2, Q4)
- I can describe the rate at an instant as the value that average rates approach as intervals containing the instant shrink. (Guide: “Shrinking the interval”; Practice Q3)
- I can explain how secant lines on a graph turn into a tangent line, and what each slope means. (Guide: Figure 1)
- I can describe the same rate verbally, graphically, numerically and with a formula. (Guide: representations table)
Calculation
- I can calculate an average rate of change over [a, b] and give it with units. (Guide: “Average rate of change”; Practice Q1, Q5(a), Q6(a))
- I can simplify (f(a + h) − f(a))/h for a simple polynomial and say what it approaches as h approaches 0. (Guide: Worked example 1; Practice Q6(b), Q6(c))
- I can estimate a rate at an instant from a table, using the shortest interval that contains it. (Guide: Worked example 2; Practice Q5(b))
Reasoning
- I can interpret a rate in context, naming the time, the quantity, increasing or decreasing, and the units. (Guide: Worked example 2; Practice Q5(c))
- I can explain why a long interval gives a poorer estimate than a short one containing the instant. (Guide: Worked example 2, step 4; Practice Q3)
- I can check whether averages from the left and from the right approach the same value, and explain what it means for the graph if they do not. (Guide: “Common misconceptions”; Practice Q7)
All ticked? Move on to Topic 1.2, Defining Limits and Using Limit Notation, or return to the Calculus AB roadmap or the Calculus BC roadmap.