Extend calculus beyond y = f(x). Differentiate and integrate parametric and vector-valued functions, solve motion problems in the plane, and work with polar coordinates — including the area enclosed by polar curves. This is a BC-only unit.
Six ways to master Unit 9 — pick whichever fits how you like to study.
All 9 topics from the College Board CED, in order. This entire unit is BC only.
Unit 9 extends calculus to curves that aren’t functions of the form y = f(x). For parametric equations x(t), y(t), the slope is dy/dx = (dy/dt)/(dx/dt), the second derivative is found by differentiating that with respect to t and dividing by dx/dt again, and arc length is ∫√((dx/dt)² + (dy/dt)²) dt.
Vector-valued functions r(t) = ⟨x(t), y(t)⟩ are differentiated and integrated componentwise, giving velocity, speed = |r′(t)|, and acceleration for planar motion. Finally, polar coordinates (r, θ) describe points by distance and angle; you convert with x = r cos θ, y = r sin θ, differentiate in polar form, and compute the area of a polar region with A = ½∫ r² dθ — and the area between two polar curves as a difference of such integrals.
On the exam this BC-only unit is 11–12% of the BC exam, and takes about 16 class periods (BC). The four mathematical practices below run through every topic: