5.12 Behaviors of Implicit Relations

Slopes on implicit curves

Use implicit differentiation to find dy/dx for a curve like x² + y² = 25. Horizontal tangents occur where dy/dx = 0 (numerator 0, denominator ≠ 0).

Vertical tangents

Vertical tangents occur where dy/dx is undefined (denominator 0). The implicit second derivative gives concavity of the curve.

Key terms

Implicit relations

Curves not solved for y (e.g. x² + y² = 25).

Find slope

Use implicit differentiation to get dy/dx.

Horizontal tangent

Where dy/dx = 0 (numerator 0).

Vertical tangent

Where dy/dx is undefined (denominator 0).

Concavity

The implicit second derivative gives it.

Try an AP-style question

Where does x² + y² = 16 have horizontal tangent lines?

1. (4, 0) and (−4, 0)
2. (0, 4) and (0, −4)
3. (0, 0) only
4. Nowhere