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 10–15% 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
• Second Derivatives and Concavity of Parametric Curves15 min · 3 objectivesCompute d²y/dx² for a parametric curve using the correct two-step quotient · Determine concavity of a parametric curve at a given parameter value · Locate horizontal and vertical tangents on a parametric curve
• 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
• 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
• 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
• Polar Slopes and Regions Between Two Curves15 min · 3 objectivesFind dy/dx for a polar curve by treating it as a parametric curve in θ · Choose correct limits of integration for a polar region, including a single petal · Set up the area bounded by two polar curves, using their intersection angles

Formulas in Unit 9

Every term in Unit 9

All 34 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.

Parametric second derivative

d²y/dx² = [d/dt(dy/dx)]/(dx/dt). NOT a ratio of second derivatives — divide by dx/dt a second time.

Why the second division appears

Converting any t-derivative into an x-derivative divides by dx/dt, and the second derivative requires that conversion twice.

Velocity vector

⟨x′(t), y′(t)⟩. A vector, with direction; not the same object as speed.

Acceleration vector

⟨x″(t), y″(t)⟩. Its relationship to velocity decides whether the particle is speeding up in the plane.

Speed in the plane

√((x′)² + (y′)²) — a non-negative scalar, never a vector and never negative.

Distance equals the integral of speed

∫√((x′)² + (y′)²)dt is both total distance and parametric arc length, because they are the same quantity.

Displacement versus distance

Displacement is the vector of the component integrals. For a closed loop it is zero while the distance is the perimeter.

Parametric horizontal tangent

dy/dt = 0 with dx/dt ≠ 0. The second condition is not a technicality.

Parametric vertical tangent

dx/dt = 0 with dy/dt ≠ 0. Both vanishing signals a singular point, usually a cusp.

Polar area is built from sectors

A = (1/2)∫r²dθ. The factor of one half comes from the sector area (1/2)r²θ, not from anything rectangular.

Polar area limits are angles

Getting the limits wrong is far more common than misapplying the formula. Find them before setting up.

Area between polar curves

(1/2)∫(R² − r²)dθ — the difference of the SQUARES, not the square of the difference.

Limits for one petal

Consecutive angles where r = 0, since the curve passes through the pole at the start and end of a petal.

Petal count for a rose

r = cos(nθ) or sin(nθ) has n petals when n is odd and 2n when n is even.

Petal width

Each petal of cos(nθ) or sin(nθ) spans an angular width of π/n. For n = 2 that is π/2, not π.

Tracing twice

Integrating over an interval that traces the curve twice doubles the area. A sketch catches this in seconds.

Polar intersections can miss the pole

Setting r₁ = r₂ finds most intersections, but two curves can reach the origin at different θ and still meet there.

Polar dy/dx

Convert with x = r cos θ and y = r sin θ, then dy/dx = (dy/dθ)/(dx/dθ). It is not dr/dθ.

r² appears, so sign of r does not matter for area

A curve and its negative enclose the same region, because the integrand squares r.

Vector-valued position from velocity

Integrate each component separately and apply the initial position componentwise. There is no single constant of integration.

Speeding up in the plane

Speed increases when the velocity and acceleration vectors point in broadly the same direction — the dot product is positive.

Parametric equations describe motion, not just a curve

Two different parameterizations can trace the same curve at different speeds, which is why distance depends on the parameterization and arc length of the curve does not.

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."
• Parametric second-derivative questions are usually multiple choice and usually include the ratio-of-second-derivatives distractor. Compute dy/dx first, write it down, then differentiate that written expression — the discipline prevents the shortcut.
• 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.
• 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.
• 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.
• Polar free-response questions almost always award a point for correct limits separate from the integrand. Even if the algebra stalls, write the fully set-up integral with its bounds.

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