Work through the list without notes. Use a calculator only where a statement says so. 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 how averaging equally spaced samples of f leads to (1/(b − a)) ∫ (a to b) f(x) dx. (Guide: “From averaging a list to averaging a function”)
- I can describe the average value as the height of a rectangle with the same signed area as the region under f. (Guide: Figure 1)
- I can explain why a continuous function reaches its average value somewhere in the interval, and why that point need not be the midpoint. (Guide: Figure 1 section; Practice Q5(c), Q7(d))
- I can state the units of an average value in context. (Guide: “The definition”; Practice Q6)
Calculation
- I can find the average value of a function given by a formula, by hand. (Guide: Worked example 1; Practice Q1, Q5)
- I can find a value c where f(c) equals the average value, and reject values outside the interval. (Guide: Worked examples 1 and 3; Practice Q5(b))
- I can estimate an average value from a table with unequal gaps, using only the interval asked for. (Guide: Worked example 2; Practice Q2)
- I can find an average value from a graph by adding signed areas. (Guide: Worked example 3)
- I can use a graphing calculator to find an average value correct to three decimal places. (Guide: “With technology”; Practice Q4, Q6)
- I can turn an average value back into an integral, and combine averages over adjoining intervals. (Practice Q3, Q7(b))
Reasoning
- I can tell average value apart from average rate of change, by formula, units and picture. (Guide: comparison table, Figure 2; Practice Q6(c))
- I can explain why the average value of f′ equals the average rate of change of f. (Guide: “The link”; Practice Q7(a))
- I can explain why (f(a) + f(b))/2 is not, in general, the average value; it is guaranteed to equal it only when f is linear. (Guide: Worked example 1; Practice Q1, Q7(c))
All ticked? Move on to Topic 8.2, Connecting Position, Velocity, and Acceleration of Functions Using Integrals, or return to the Calculus AB roadmap or the Calculus BC roadmap.