Parametric Equations, Polar Coordinates, and Vector–Valued Functions (BC Only)

The derivative of $r(\theta)$ with respect to $\theta$ is zero at $\theta = \alpha$.

The graph has an asymptote at $\theta = \alpha$.

The graph approaches and touches a single, well-defined point as $\theta$ approaches $\alpha$ from both sides.

The graph creates a loop or cusp at $\theta = \alpha$.

$\int_{0}^{\pi} \left( \frac{1}{2} \right) (2\sin(\theta))^2 d\theta$

$\int_{0}^{\pi/2} 2\sin(\theta) d\theta$

$\int_{0}^{\pi/2} \left( \frac{1}{2} \right) (2\sin(\theta))^2 d\theta$

$\int_{0}^{2\pi} \left( \frac{1}{2} \right) (r)^2 d\theta$

$2\cos(2*\frac{\pi}{4})$

$-2\sin(2*\frac{\pi}{4})$

$\cos(2*\frac{\pi}{4})$

$2\sin(2*\frac{\pi}{4})$

The curvature of the curve

The rate of change of the curvature

The rate of change of the distance from the origin

The slope of the tangent line

To convert polar coordinates to Cartesian coordinates.

To calculate the rate of change of r.

To find the slope of the tangent line.

To find the derivative of a composite function.

Verifying there's an integer “K” such that the succession difference between items above “K” stays within ε/2.

Determining there’s an integer "M" such that when "m" exceeds "M", each term remains constant or decreases.

Showing there’s a number "N" so every sum over "N" has its absolute value less than any small positive number ε chosen.

Confirming there's some large enough integer “J” where adding any numbers past “J” results in differences under ε squared.

$\cos(\frac{\pi}{6})\sin(\frac{\pi}{6})-\sin(\frac{\pi}{6}) \times \sin^{2}(\frac{\pi}{6})$

$\sin(\frac{\pi}{6})-\cos(\frac{\pi}{6})\sin(\frac{\pi}{6})$

$\sin(\frac{\pi}{6})\cos(\frac{\pi}{6}) \times \cos^{2}(\frac{\pi}{6})$

$\cos(\frac{\pi}{6}) \times \sin(\frac{\pi}{6}) \times \sin^{2}(\frac{\pi}{6})$

$A = \frac{1}{2} r^2 \theta$

$A = \pi r^2 \theta$

$A = r^2 \theta$

$A = \frac{1}{2} \theta \pi r$

There is either a local maximum or minimum radius at angle $\beta$ on the polar curve.

The curve intersects itself at angle $\beta$.

The polar curve has no defined radius for angle $\beta$.

There is an inflection point where concavity changes on the curve at angle $\beta$.

Terms increase in absolute value without bound as n increases.

Each term is greater than the sum of all subsequent terms combined.

Terms alternate between positive integers only throughout the entire series.

Terms decrease in absolute value, approaching zero asymptotically.