Work through the list without notes. Use a calculator only where the linked question allows one. 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, using “rate × short time”, why the integral of a rate over an interval is the net change in the amount. (Guide: “The big idea”)
- I can explain the difference between net change and total change, and write an integral for each. (Guide: Worked example 1 extension; Practice Q4)
- I can explain why the derivative of an amount function Q(x) = Q(a) + ∫ (a to x) Q′(t) dt is the rate Q′(x). (Guide: “From net change to an amount function”; Practice Q7(a))
- I can give the units of an integral from the units of the rate and of the input. (Guide: “Units”; Practice Q1)
Calculation
- I can find an amount at a later time from a starting value and a rate given as a formula, a graph or a table. (Guide: Worked examples 1 and 2; Practice Q2, Q5)
- I can write the amount when there is a rate in and a rate out, with the subtraction the right way round. (Guide: Worked example 2; Practice Q3, Q6(c))
- I can use a calculator to evaluate an applied integral and to solve rate in = rate out, keeping stored values. (Guide: Worked example 2; Practice Q6)
Reasoning
- I can decide whether an amount is increasing or decreasing at a given time by comparing the rate in with the rate out. (Guide: Worked example 2(c); Practice Q6(b))
- I can find and justify the greatest or least amount with the candidates test, including the endpoints. (Guide: “Greatest and least amounts”; Practice Q6(d), Q7(c))
- I can write a complete interpretation of a definite integral: the quantity, the interval and the units. (Guide: Worked example 3; Practice Q1, Q5(a), Q7(b))
- I can tell ∫ (a to b) P(t) dt apart from ∫ (a to b) P′(t) dt = P(b) − P(a) in context. (Guide: Worked example 3; Practice Q5(b))
All ticked? Move on to Topic 8.4, Finding the Area Between Curves Expressed as Functions of x, or return to the Calculus AB roadmap or the Calculus BC roadmap.