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 state what a critical point of an implicit relation is, including points where dy/dx does not exist. (Guide: “Critical points of an implicit relation”; Practice Q3)
- I can explain why the derivative tests still work on a piece of an implicit curve. (Guide: “Applying derivative tests to curves”)
- I can explain why N = D = 0 gives no conclusion from the dy/dx formula. (Guide: “When both N and D are 0”; Practice Q7(b), Q7(e))
Skills
- I can find horizontal tangents by solving N = 0 together with the curve’s equation, and check D. (Guide: Worked example 1, step 2; Practice Q1, Q6(b))
- I can find vertical tangents by solving D = 0 with the curve’s equation, and check N. (Guide: Worked example 1, step 3; Practice Q6(d), Q7(d))
- I can find d²y/dx² in terms of x, y and dy/dx, treating y as a function of x. (Guide: Worked example 2, step 6; Practice Q5(e))
- I can evaluate d²y/dx² at a horizontal tangent quickly by setting dy/dx = 0. (Guide: Worked example 1, step 5; Practice Q2, Q6(c))
- I can find where y is increasing or decreasing from the signs of x and y in dy/dx. (Guide: “Increasing, decreasing and concavity”; Practice Q4)
Reasoning
- I can justify a relative maximum or minimum of y with the first or second derivative test, naming the point and the evidence. (Guide: Worked examples 1 and 2; Practice Q2, Q7(c))
- I can explain why a sign of dy/dx holds, for example because a denominator is always positive. (Guide: Worked example 2, step 4; Practice Q5(b), Q5(d))
- I can argue that a single critical point gives an absolute maximum of y when y increases on one side and decreases on the other. (Guide: Worked example 2, step 5; Practice Q5(d))
All ticked? Move on to Topic 6.1, Exploring Accumulations of Change, or return to the Calculus AB roadmap or the Calculus BC roadmap.