AP Calculus BC glossary

Slope of a polar curve

Also called: Polar slope

The slope of a polar curve is dy over dx, not dr over d theta. Using x equals r cos theta and y equals r sin theta, dy over dx equals dr over d theta times sin theta plus r cos theta, all divided by dr over d theta times cos theta minus r sin theta.

The curve is really parametric in $\theta$ : $x = r\cos\theta$ and $y = r\sin\theta$ with $r = f(\theta)$ . Differentiating each with the product rule gives $\frac{dx}{d\theta}$ and $\frac{dy}{d\theta}$ , and the slope is their ratio.

\frac{dy}{dx} = \frac{\frac{dr}{d\theta}\sin\theta + r\cos\theta}{\frac{dr}{d\theta}\cos\theta - r\sin\theta}

The mistake

Reporting $\frac{dr}{d\theta}$ as the slope. That is the rate at which the radius changes, not the slope of the curve in the $xy$ -plane. Convert to $x$ and $y$ first, then form $\frac{dy}{dx}$ .

Appears in: Unit 9: Parametric, Polar, and Vector (BC)