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 9.1 and 9.2, the product rule and unit-circle values) are on the Calculus BC roadmap.
Understanding
- I can explain what the polar point (r, θ) means, including when r is negative, and give two other polar names for the same point. (Guide: “What polar coordinates are”; Practice Q1)
- I can explain why a polar curve r = f(θ) is a parametric curve with parameter θ. (Guide: “A polar curve is a parametric curve”)
- I can explain the difference between dr/dθ, dy/dθ and dy/dx, and why tan θ is not the slope of the curve. (Guide: “Three derivatives, three meanings”; Practice Q3, Q7(b))
Skills
- I can convert points and equations between polar and rectangular form, checking the quadrant. (Guide: “Polar curves and equations”; Practice Q1, Q2)
- I can find dx/dθ and dy/dθ with the product rule and use them to find dy/dx at a given angle. (Guide: Worked example 1(b); Practice Q3, Q5)
- I can write the equation of a tangent line to a polar curve. (Practice Q5, Q6(c))
- I can find d²y/dx² for a polar curve and use its sign to state the concavity. (Guide: Worked example 1(c); Practice Q7(c))
- I can find horizontal and vertical tangents and check that the other derivative is non-zero. (Guide: Worked example 1(d); Practice Q6(b))
- I can use a calculator in radian mode to evaluate polar derivatives at a non-exact angle. (Guide: Worked example 2; Practice Q7)
Reasoning
- I can decide whether a point is moving towards or away from the pole by comparing the signs of r and dr/dθ. (Guide: Worked example 2(a); Practice Q4, Q6(a), Q7(a))
- I can find the greatest distance from the pole on an interval using dr/dθ = 0 and the endpoints. (Guide: Worked example 2(d); Practice Q7(d))
- I can explain why the point furthest from the pole need not be the point furthest east. (Guide: Worked example 2(e))
All ticked? Move on to Topic 9.8, Finding the Area of a Polar Region or the Area Bounded by a Single Polar Curve, or return to the Calculus BC roadmap.