Unit 9BC

Parametric & Polar

Calculator ActiveBC OnlyCED Unit 9

Parametric Equations

Curves defined by x = f(t) and y = g(t): derivatives, tangent lines, and arc length.

First Derivative

Slope of the Curve

This is the chain rule: divide out the parameter.

Second Derivative

Concavity of the Curve

Take the derivative of the first derivative with respect to

, then divide by

again.

⚠ Watch Out

Not d²y/dt² ÷ d²x/dt²

You cannot just take second derivatives of the top and bottom independently.

Equations of Tangent Lines

Point-Slope Form

Evaluate x, y, and slope at the same t-value.

1. Find the point: (f(t₀), g(t₀))

2. Find the slope: (dy/dt) / (dx/dt) at t = t₀

3. Write in point-slope form

Horizontal & Vertical Tangents

Horizontal Tangent

Vertical Tangent

⚠ Watch Out

Both zero = indeterminate

If dy/dt = 0 and dx/dt = 0 at the same t, the tangent is undefined, indicating a possible cusp or self-intersection.

Arc Length (Parametric)

Distance Along the Curve from t = a to t = b

This is the same as integrating speed when x(t) and y(t) describe motion.

→ Tip

Calculator: store the integrand, then use fnInt. Make sure you're in RADIAN mode.

AP® Calculus BC · Unit 9 Overview · Mr. Brantley