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. You need Topic 9.1 to be secure first; other prerequisites are on the Calculus BC roadmap.
Understanding
- I can explain, using the chain rule, why d²y/dx² = [d/dt (dy/dx)] ÷ (dx/dt). (Guide: “Deriving the correct method”)
- I can show with an example why dividing d²y/dt² by d²x/dt² does not give d²y/dx². (Guide: “The tempting shortcut that fails”; Practice Q1(C), Q2(C))
- I can explain why the sign of d/dt (dy/dx) and the concavity are opposite when dx/dt < 0. (Guide: “Watch the sign of dx/dt”; Practice Q7(c))
Skills
- I can simplify dy/dx as a function of t before differentiating it again. (Guide: Worked example 1; Practice Q5(a))
- I can find d²y/dx² in terms of t, using the quotient rule where needed. (Practice Q2, Q5(b), Q7(b))
- I can evaluate d²y/dx² at a given t, including when only dx/dt and dy/dx are given. (Practice Q1, Q3)
- I can find the values of t for which a parametric curve is concave up or concave down, including where dx/dt = 0. (Guide: Worked examples 2(b) and 3(c); Practice Q5(c))
- I can set up d²y/dx² and evaluate it with a calculator when one is allowed. (Practice Q6(b))
Reasoning
- I can decide whether a tangent-line estimate is an overestimate or an underestimate, and justify it with the sign of d²y/dx². (Guide: Worked example 2(c); Practice Q4, Q6(d))
- I can find and correct an error in someone else’s concavity argument. (Practice Q7(c))
- I can check a parametric second derivative by eliminating the parameter. (Guide: Worked example 1; Practice Q5(d), Q7(d))
- I can keep d²y/dx² (the shape of the path) separate from the second derivatives with respect to t used later for motion. (Guide: “Where this leads”)
All ticked? Move on to Topic 9.3, Finding Arc Lengths of Curves Given by Parametric Equations, or return to the Calculus BC roadmap.