PEPCVF - Polar Derivatives and Areas

Polar Derivatives and Areas

Derivatives, Slopes, and Tangent Lines of Polar Functions

To determine the slope of a tangent line to a polar curve, it is first necessary to find the derivative of a polar curve C given in polar coordinates by a function equation image indicator $LaTeX: r=f\left(\theta\right)$ $r=f\left(\theta\right)$ , where $LaTeX: \theta$ $\theta$ .

is the parameter. The parametric equations equation image indicator $LaTeX: x=r\cos\theta$ $x=r\cos\theta$ and $LaTeX: y=r\:\sin\theta$ $y=r\:\sin\theta$ become $LaTeX: x=f\left(\theta\right)\cos\theta\:\text{and}\:y=f\left(\theta\right)\sin\theta$ $x=f\left(\theta\right)\cos\theta\:\text{and}\:y=f\left(\theta\right)\sin\theta$ . If f is a differentiable function of equation image indicator , then x and y are also differentiable. When $LaTeX: \frac{dx}{d\theta}\ne0\:\text{at}\:\left(r,\theta\right),\frac{dy}{dx}$ $\frac{dx}{d\theta}\ne0\:\text{at}\:\left(r,\theta\right),\frac{dy}{dx}$ is calculated using the product rule and the method used to find the derivative of parametric equations: equation image indicator $LaTeX: \frac{dy}{dx}=\frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}}=\frac{r\:\cos\theta+\frac{dr}{d\theta}\sin\theta}{-r\:\sin\theta+\frac{dr}{d\theta}\cos\theta}$ $\frac{dy}{dx}=\frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}}=\frac{r\:\cos\theta+\frac{dr}{d\theta}\sin\theta}{-r\:\sin\theta+\frac{dr}{d\theta}\cos\theta}$ . An equivalent representation is equation image indicator $LaTeX: \frac{dy}{dx}=\frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}}=\frac{f'\left(\theta\right)\:\sin\theta+f\left(\theta\right)\cos\theta}{f'\left(\theta\right)\:\cos\theta-f\left(\theta\right)\sin\theta}$ $\frac{dy}{dx}=\frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}}=\frac{f'\left(\theta\right)\:\sin\theta+f\left(\theta\right)\cos\theta}{f'\left(\theta\right)\:\cos\theta-f\left(\theta\right)\sin\theta}$ . To explore finding polar derivatives using an interactive applet, CLICK HERE Links to an external site..

The slope of a polar curve at a point equation image indicator $LaTeX: \left(r,\:\theta\right)$ $\left(r,\:\theta\right)$ is given by $LaTeX: \frac{dy}{dx}\Biggr|_{\left(r,\:\theta\right)}=\frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}}=\frac{r\:\cos\theta+\frac{dt}{d\theta}\sin\theta}{-r\:\sin\theta+\frac{dt}{d\theta}\cos\theta}$ $\frac{dy}{dx}\Biggr|_{\left(r,\:\theta\right)}=\frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}}=\frac{r\:\cos\theta+\frac{dt}{d\theta}\sin\theta}{-r\:\sin\theta+\frac{dt}{d\theta}\cos\theta}$ . View the next presentation illustrating how to find slopes of tangent lines to polar curves.

With polar curves it is common for horizontal and vertical tangents to exist as shown below.

Area Bounded by a Single Curve

Finding the area of a polar region is similar to the method of finding the area of a region in the Cartesian coordinate system. Where the basic element of area in the rectangular system is a rectangle, the polar system uses sectors of a circle as the basic area element. Consider how the formula for area of a polar region is developed and applied.

The next three videos investigate how areas of polar regions are calculated given a variety of polar curves

Points of Intersection of Polar Graphs

When two polar curves intersect, finding their points of intersection is not as straightforward as with Cartesian (rectangular) curves. Recall from the previous lesson that a given point has infinitely many sets of polar coordinates, e.g., the pole. Just as with parametric equations, the path traced by the curve as the angle increases does not guarantee that the points of intersection are reached at the same time for a given angle value. The analogy of orbiting satellites discussed earlier is apparent with polar defined intersecting curves. Simultaneously solving two polar curves to determine their points of intersection may or may not reveal all solutions. Suggested ways to identify all intersection points are checking behavior at the pole and graphing the equations. An example is found below.

example of polar curves intersection points

Area Between Two Polar Curves

Finding the area bounded by two polar curves is conceptually similar to finding the area between two curves defined as functions, which was expressed as equation image indicator . A derivation of the formula for the area between two polar curves and example are featured below.

The next two presentations show several examples of how to find areas bounded by polar curves.

To investigate polar area using an interactive applet CLICK HERE Links to an external site..

Polar Derivatives and Areas Practice

Polar Derivatives and Areas: Even More Problems!

Complete problems from your textbook and/or online resources as needed to ensure your complete understanding of derivatives of polar functions and area of a polar region.

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