$\frac{\Delta r}{\Delta \theta} = \frac{r(\theta_2) - r(\theta_1)}{\theta_2 - \theta_1}$

x = r \cos(\theta), y = r \sin(\theta)

r = \sqrt{x^2 + y^2}, \theta = \arctan(\frac{y}{x})

$\frac{dr}{d\theta} = f'(\theta)$

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

$A = \frac{1}{2} \int_{a}^{b} [f(\theta)]^2 d\theta$

$L = \int_{a}^{b} \sqrt{r^2 + (\frac{dr}{d\theta})^2} d\theta$

r = a, where 'a' is the radius of the circle.

$\theta = c$, where (c) is a constant angle.

Solve the equation (r_1(\theta) = r_2(\theta)) for (\theta). Also, check if the pole (origin) is a point on either curve.

It forms a circle centered at the origin with radius (r).

It implies that (f(\theta) = f(-\theta)), meaning the function is even.

Look for points where the curve reaches its farthest or closest distance from the origin locally.

It has a heart-like shape, typically represented by equations of the form (r = a(1 \pm \cos(\theta))) or (r = a(1 \pm \sin(\theta))).

It has a figure-eight shape, represented by equations of the form (r^2 = a^2 \cos(2\theta)) or (r^2 = a^2 \sin(2\theta)).

The period determines how often the graph repeats itself as (\theta) increases. For example, (r = \sin(2\theta)) completes a full graph in (\pi) radians.

It means that there exists some value of (\theta) for which (r = 0).

The graph is symmetric about the y-axis if replacing (\theta) with (-\theta) changes the sign of (r), or if replacing (\theta) with (\pi - \theta) leaves the equation unchanged.

It represents the instantaneous rate of change of (y) with respect to (x) at that point, indicating the direction of the curve.

The constant term (a) affects the size and position of the graph. If (a = b), it forms a cardioid. If (a < b), it forms a looped curve.

A coordinate system where a point is located by its distance (r) from the origin and an angle (\theta) from the polar axis.

A function defined in polar coordinates, typically in the form (r = f(\theta)), where (r) is the distance from the origin and (\theta) is the angle.

A point where the distance (r) from the origin is locally the greatest, changing from increasing to decreasing as (\theta) increases.

A point where the distance (r) from the origin is locally the smallest, changing from decreasing to increasing as (\theta) increases.

The reference line ((\theta = 0)) from which the angle (\theta) is measured in polar coordinates, analogous to the positive x-axis in Cartesian coordinates.

The change in (r) with respect to (\theta) over an interval, calculated as (\frac{\Delta r}{\Delta \theta} = \frac{r(\theta_2) - r(\theta_1)}{\theta_2 - \theta_1}).

The distance (r) from the origin is increasing as (\theta) increases.

The distance (r) from the origin is decreasing as (\theta) increases.

Values of (\theta) where the derivative of (r) with respect to (\theta) (i.e., (r'(\theta))) is either zero or undefined. These points are candidates for relative extrema.

It indicates that the point is at the origin, regardless of the value of (\theta).