Section 9.7

Tangents in Polar

To find the slope of a polar curve , we treat it as a parametric curve where and .

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The Formula & The Visual

Slope is still "Rise over Run"

Even though we are in polar land, the tangent line is a Cartesian concept. We still want . To find this, we treat as a set of parametric equations:


Then we use the Chain Rule (specifically the parametric form ) and the Product Rule:

Visualizing the Tangent

Consider the cardioid . At , the tangent line grazes the curve. This line has a slope calculated by our formula.

  • Horizontal Tangent: Slope = 0 (Top/Bottom peaks)
  • Vertical Tangent: Slope is undefined (Sides)

Vertical tangent at

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Worked Example

Horizontal Tangents

Find horizontal tangents for .

1. Set dy/dtheta = 0
.
.
2. Solve
.
.
3. Exclude pole?
At , (cusp). Technically vertical tangent at cusp.
So horizontal tangents at .
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Level Up Examples

A. Tangent at the Pole

Find the slope of at the pole.

The Shortcut: When a curve passes through the pole ($r=0$), the formula simplifies greatly. Since $r=0$, terms with $r$ vanish:

1. Find when : .
2. Slopes are simply and .

B. Vertical Tangents

Find vertical tangents for .

Set denominator :





(Check numerator at $\pi$: $y'=0$ too? Limit needed. It's a cusp.)

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Practice Quiz

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