BC-only material. Work through the list without notes. If you cannot do a statement, follow its link, then come back. Guide = study guide; Practice = practice questions. Prerequisites (Topics 6.7, 6.8, 6.9, 7.7 and 9.4) are on the Calculus BC roadmap.
Understanding
- I can explain why a vector-valued function is integrated one component at a time. (Guide: “Integrate one component at a time”)
- I can explain why the constant of integration for a vector is a vector ⟨C₁, C₂⟩ with two separate parts. (Guide: “Two constants, not one”; Practice Q2)
- I can explain why ∫ from a to b of r′(t) dt equals the net change r(b) − r(a), and why it is not the position. (Guide: “Net change and the position formula”; Practice Q2)
- I can recognise that a pair of rates dx/dt and dy/dt is the same problem as a rate vector. (Guide: “Parametric form is the same problem”; Practice Q7)
Skills
- I can evaluate a definite integral of a vector-valued function and give the answer as a vector. (Practice Q1)
- I can find a particular solution r(t) from a rate vector and an initial condition given at any time, not only t = 0. (Guide: Worked example 1; Practice Q2, Q5)
- I can use substitution inside a component when it is needed. (Practice Q5)
- I can work from an acceleration vector to a position vector, using the starting velocity and then the starting position. (Guide: Worked example 2; Practice Q3, Q6)
- I can use a graphing calculator and the formula r(b) = r(a) + ∫ from a to b of r′(t) dt when a component has no elementary antiderivative. (Guide: Worked example 3; Practice Q4, Q7)
- I can find a position at an earlier time by subtracting the accumulated change. (Practice Q4, Q7(b))
Reasoning
- I can check a particular solution by differentiating it and substituting the given time. (Guide: “Checking your answer”; Practice Q5(c))
- I can explain an error in a student’s choice of constant. (Practice Q6(d))
- I can use the sign of a rate on an interval to justify whether a coordinate increases or decreases. (Practice Q7(b), Q7(c))
All ticked? Move on to Topic 9.6, Solving Motion Problems Using Parametric and Vector-Valued Functions, or return to the Calculus BC roadmap.