Parametric, Polar & Vector-Valued Functions

What this unit covers

The topics below follow the published Calculus BC course framework for Unit 9. This unit is worth 11–12% of the exam, so budget your time against that rather than against how long the unit takes to teach.

Lessons in this unit

• Parametric Derivatives15 min · 3 objectivesCompute dy/dx for a parametrically defined curve · Find the second derivative d²y/dx² of a parametric curve · Locate horizontal and vertical tangents from the parametric derivatives
• Area in Polar Coordinates15 min · 3 objectivesCompute the area enclosed by a polar curve · Find the area between two polar curves · Determine the correct limits of integration in θ
• Vector-Valued Functions & Motion15 min · 3 objectivesDifferentiate a vector-valued position function to get velocity and acceleration · Compute the speed of a particle moving in the plane · Find the position from velocity using integration and an initial condition
• Parametric Arc Length13 min · 3 objectivesCompute the arc length of a parametric curve · Relate parametric arc length to the integral of speed · Distinguish arc length (distance traveled) from displacement

Formulas in Unit 9

Every term in Unit 9

All 12 terms we publish for Parametric, Polar & Vector-Valued Functions, with definitions. Reading them through is the fastest way to find the ones you cannot define — then drill those in cram mode until you can produce them without the prompt.

Parametric derivative

dy/dx = (dy/dt)/(dx/dt), defined wherever dx/dt ≠ 0. Not the derivative of y with respect to t.

Second derivative parametrically

d²y/dx² = [d/dt(dy/dx)]/(dx/dt). Differentiate the first derivative with respect to t, then divide by dx/dt again.

Vertical and horizontal tangents parametrically

Horizontal where dy/dt = 0 and dx/dt ≠ 0; vertical where dx/dt = 0 and dy/dt ≠ 0.

Vector-valued position

r(t) = ⟨x(t), y(t)⟩ with velocity ⟨x′, y′⟩ and acceleration ⟨x″, y″⟩. Speed is |v| = √((x′)² + (y′)²), a scalar.

Distance traveled by a particle in the plane

∫√((x′)² + (y′)²)dt — the integral of speed, which equals arc length along the path.

Finding position from velocity

Integrate each component separately and use the initial position to fix each constant.

Polar to Cartesian

x = r cos θ and y = r sin θ, with r² = x² + y² and tan θ = y/x going back.

Area in polar coordinates

A = ½∫r²dθ over the correct θ interval. The one-half and the square are both essential.

Finding the limits for a polar area

Determine the θ values that trace the region exactly once. Tracing twice doubles the answer, a common error with roses.

Area between two polar curves

A = ½∫(r_outer² − r_inner²)dθ, taking the difference of the squares rather than squaring the difference.

Polar derivative dy/dx

Convert to parametric with x = r(θ)cos θ and y = r(θ)sin θ, then use the parametric formula. Not dr/dθ.

Common polar curves

r = a is a circle, r = a ± b cos θ gives limaçons and cardioids, r = a cos(nθ) gives n petals for odd n and 2n for even n.

What examiners penalize here

• Keep the pattern straight: dy/dt = 0 → horizontal tangent, dx/dt = 0 → vertical tangent. Because the slope is a fraction (dy/dt)/(dx/dt), the numerator controls "flat" and the denominator controls "steep/vertical."
• The number-one polar-area error is writing (R − r)² instead of (R² − r²). Square each radius separately, then subtract. When in doubt, compute the outer area and inner area as two integrals and subtract the results.
• For planar motion, keep three vectors straight: position ⟨x, y⟩, velocity ⟨x′, y′⟩, acceleration ⟨x″, y″⟩. "Speed" and "total distance" both use the magnitude √((x′)² + (y′)²) — speed at an instant, distance as its integral over time.
• Parametric arc length and "total distance traveled" are the same integral, ∫√((dx/dt)² + (dy/dt)²) dt. If a free-response question asks for the distance a particle travels in the plane, this is the formula — not the magnitude of the displacement vector.

Practice Calculus BC

Our practice bank is drawn from across the whole course rather than filtered to one unit, which is closer to how the exam asks anyway — it will not tell you which unit a question is testing.

Questions about this unit

All 10 units of AP Calculus BC

1. Unit 1 · Limits & Continuity
2. Unit 2 · Differentiation: Definition & Fundamental Properties
3. Unit 3 · Differentiation: Composite, Implicit & Inverse Functions
4. Unit 4 · Contextual Applications of Differentiation
5. Unit 5 · Analytical Applications of Differentiation
6. Unit 6 · Integration & Accumulation of Change
7. Unit 7 · Differential Equations
8. Unit 8 · Applications of Integration
9. Unit 9 · Parametric, Polar & Vector-Valued Functions
10. Unit 10 · Infinite Sequences & Series