unit-9 – HighFiveAP

9.1

Parametric Functions (What + Direction)

Parametric equations describe a curve by making both coordinates depend on a parameter t: x = x(t) and y = y(t). Think of t as “time.”

Core Idea

A single point happens at a single time: (x(t), y(t)). As t increases, the curve is traced in a specific direction.

AP trap: The same curve can be traced in different directions depending on how t runs. When asked “direction of motion,” use increasing t.

Graph: Circle traced by a parameter (direction matters)

Curve Direction (t increasing)

Example: x=cos t, y=sin t, 0≤t≤2π. Same circle as x²+y²=1, but now you can see the motion.

AP-Style Quick Check

If x=t²-1 and y=t-2, then at t=2 the point is (3,0). For direction: compare points at t=2 and t=2.1.

9.2

Derivatives of Parametric Equations

In parametric form, slope means dy/dx, but you must use t first.

dy/dx = (dy/dt) / (dx/dt)

Horizontal vs. Vertical Tangents

Type	Condition
Horizontal	dy/dt = 0 AND dx/dt ≠ 0
Vertical	dx/dt = 0 AND dy/dt ≠ 0

Don’t lose the point: Always verify the “other derivative” is not zero. If dx/dt=0 AND dy/dt=0, it’s a “cusp/corner/undefined” situation — AP loves that.

Graph: Parametric curve with a horizontal tangent

The highlighted point corresponds to a t value where dy/dt=0. Use the tangent rules above, not “eyeballing.”

Mini Example (Typical MCQ)

If x=t² and y=t³-3t, then dx/dt=2t, dy/dt=3t²-3. Horizontal tangent: 3t²-3=0 → t=±1 (and check dx/dt≠0 at those t).

9.3

Second Derivative (Parametric)

You cannot just “differentiate twice” in t. You need the chain rule:

d²y/dx² = (d/dt(dy/dx)) / (dx/dt)

Concavity workflow (no chaos)

1) Find slope

dy/dx

2) Differentiate in t

d/dt(dy/dx)

3) Divide by dx/dt

÷ (dx/dt)

AP warning: concavity is about d²y/dx², not d²y/dt².

9.4

Arc Length (Parametric)

Arc length is “distance traveled along the curve” from t=a to t=b.

L = ∫_a^b √[(dx/dt)² + (dy/dt)²] dt

When it’s actually doable

When the inside simplifies to a perfect square (or classic trig identity).
Otherwise AP often asks setup + calculator evaluation.

AP-Style Setup

If x=t, y=t², 0≤t≤1, then dx/dt=1, dy/dt=2t so L=∫₀¹ √(1+4t²) dt.

9.5

Polar Functions (r = f(θ))

Polar uses an angle θ and a radius r. Convert to Cartesian when needed:

x = r cosθ, y = r sinθ

Two big AP facts

Negative r flips the point through the origin (same θ, opposite direction).
Symmetry tests save time: check θ→-θ, θ→π-θ, θ→θ+π.

Graphs: classic polar “families” (big + labeled)

Rose: r = 2cos(3θ)

Odd k ⇒ k petals. Here: 3 petals.

Cardioid: r = 1 - cosθ

A “heart” with a cusp. Great for area problems.

Graphing speed trick: Make a quick θ-table (0, π/2, π, 3π/2, 2π). Watch when r=0 (hits the origin).

9.6

Derivatives of Polar Functions

To find the slope of a polar curve, treat x(θ)=r cosθ and y(θ)=r sinθ, then use (dy/dθ)/(dx/dθ).

dy/dx = (r' sinθ + r cosθ) / (r' cosθ − r sinθ)

Tangent lines in polar

Horizontal tangent when numerator = 0 and denominator ≠ 0
Vertical tangent when denominator = 0 and numerator ≠ 0

Mini Example

If r=1-cosθ, then r'=sinθ. Plug into the slope formula and evaluate at the requested θ (AP loves θ=0, π/2, π).

9.7

Area in Polar Coordinates

Polar area is one of the highest-yield BC skills: it shows up in FRQs a lot.

A = ½ ∫_α^β [r(θ)]² dθ

Point-killer: You MUST square r. Also keep the ½. If the region is traced twice, fix your limits (don’t double-count).

Graph: “one loop” detection for area

Example: r = 2 + cosθ (limacon). For “one loop,” find where the curve starts repeating. Often: solve r=0 to locate origin hits.

AP-Style Setup (Area between curves)

If outer curve is r=f(θ) and inner is r=g(θ), then A = ½ ∫ (f(θ)² - g(θ)²) dθ.

9.8

Vector-Valued Functions (Motion)

A vector-valued function gives a position vector: r(t)=⟨x(t), y(t)⟩. BC uses this to model motion.

The Motion Triad

Position

r(t)

Velocity

v(t)=r'(t)

Acceleration

a(t)=r''(t)

Speed vs. Velocity

Speed is magnitude: |v(t)| = √[(x'(t))² + (y'(t))²]. Velocity is a vector (direction included).

Graph: Motion path + direction arrow

AP loves questions like: “Is the particle speeding up or slowing down?” (hint: compare velocity and acceleration directions).

Speeding up / slowing down (FRQ gold):
Speed increases when velocity and acceleration point in the same direction: v(t) · a(t) > 0. Speed decreases when v·a < 0.