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).

A positive average rate of change indicates that (r) is increasing as (\theta) increases. A negative average rate of change indicates that (r) is decreasing as (\theta) increases.

It represents the instantaneous rate of change of the distance (r) from the origin with respect to the angle (\theta).

Examine the sign of the derivative (dr/d\theta). If (dr/d\theta > 0), the function is increasing; if (dr/d\theta < 0), it's decreasing.

Relative extrema (maxima or minima) occur where the function changes direction. A relative maximum is a point where (r) is locally largest, and a relative minimum is a point where (r) is locally smallest.

These points are critical points and potential locations of relative maxima or minima. They indicate where the function's rate of change is momentarily zero.

Polar coordinates use distance (r) and angle (\theta) to define a point, while Cartesian coordinates use horizontal distance (x) and vertical distance (y). They can be converted using trigonometric relationships.

As (\theta) increases, the point ((r, \theta)) moves around the origin. The value of (r) determines how far the point is from the origin at each angle (\theta).

When (r) is negative, the point is plotted in the opposite direction of the angle (\theta). It is a reflection through the origin.

Concavity can be determined by analyzing the second derivative (\frac{d^2y}{dx^2}). If (\frac{d^2y}{dx^2} > 0), the curve is concave up. If (\frac{d^2y}{dx^2} < 0), the curve is concave down.

Common symmetries include symmetry about the x-axis (polar axis), symmetry about the y-axis ((\theta = \frac{\pi}{2})), and symmetry about the origin (pole).

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

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.