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).

Polar: Uses distance (r) and angle (\theta). Cartesian: Uses horizontal (x) and vertical (y) distances.

Average: Change over an interval. Instantaneous: Derivative at a point.

Relative: Local maximum in an interval. Absolute: Overall maximum of the function.

Increasing: (r) increases as (\theta) increases. Decreasing: (r) decreases as (\theta) increases.

x-axis: (f(\theta) = f(-\theta)). y-axis: (f(\theta) = -f(-\theta)) or (f(\theta) = f(\pi - \theta)).

Cartesian: $\int f(x) dx$. Polar: $\frac{1}{2} \int r^2 d\theta$.

Cartesian: $\frac{dy}{dx}$. Polar: $\frac{dy/d\theta}{dx/d\theta}$.

(r = a\cos(\theta)): Circle on x-axis. (r = a\sin(\theta)): Circle on y-axis.

Cardioid: Heart-shaped, (r = a(1 \pm \cos(\theta))). Circle: Centered at origin, (r = a).

Cartesian: (y = mx + b). Polar: More complex, often involving (\theta = constant) for lines through the origin.

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

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.