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AP® Calculus BC · Unit 9 · Topic 9.1

Defining and Differentiating Parametric Equations: Topic Checklist (Calculus BC 9.1)

Specific “I can…” statements for parametric curves and dy/dx, each linked to the guide section and practice question that shows whether you can really do it.

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 2.5–2.7 and 3.1) are on the Calculus BC roadmap.

Understanding

  • I can explain what x = f(t), y = g(t) means, make a table of values, and show the direction of travel on a sketch. (Guide: “What a parametric curve is”)
  • I can explain, using the chain rule, why dy/dx = (dy/dt) ÷ (dx/dt), and why dx/dt must not be zero. (Guide: “Deriving dy/dx with the chain rule”)
  • I can explain why one point on a parametric curve can have two different tangent lines. (Guide: Worked example 1(c))

Skills

  • I can find dy/dx as a function of t and evaluate it at a given t. (Practice Q1, Q5(a), Q7(a))
  • I can write the equation of a tangent line, taking the point and the slope from the same value of t. (Guide: Worked example 1(b); Practice Q2, Q5(b))
  • I can find the slope at a given point (x, y) by first finding the value of t, and rejecting values that do not fit both coordinates. (Guide: Worked example 1(c); Practice Q4)
  • I can find horizontal tangents (dy/dt = 0, dx/dt ≠ 0) and vertical tangents (dx/dt = 0, dy/dt ≠ 0). (Guide: “Horizontal and vertical tangents”; Practice Q3, Q5(c), Q7(b))
  • I can use a calculator to evaluate dy/dx or solve dy/dt = 0 when a calculator is allowed. (Practice Q6)

Reasoning

  • I can say why no conclusion follows when dx/dt and dy/dt are both zero, and investigate further. (Guide: “Why the both-zero case needs care”; Practice Q3)
  • I can justify that a curve has no vertical (or horizontal) tangents by showing a derivative is never zero. (Practice Q5(d), Q6(c))
  • I can interpret dy/dx for a path in context, with units, and keep it separate from the direction of motion. (Guide: Worked example 2; Practice Q6(b))
  • I can check a parametric slope by another method, such as eliminating t or implicit differentiation. (Practice Q5, Q7(d))

All ticked? Move on to Topic 9.2, Second Derivatives of Parametric Equations, or return to the Calculus BC roadmap.

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