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

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

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

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

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

-$\sqrt{3}/2$

-$\sqrt{3}$

-$\sqrt{3}/6$

-$\sqrt{3}/3$

Gradient max = $|-akI_0 \cdot \sin(k\theta)|$

Gradient max = $akI_0$

Gradient max = $\frac{aI_0}{k - a}$

Gradient max = $\frac{kI_0}{a}$

The number of petals halves.

It's impossible to determine without further information.

The number of petals remains unchanged.

The number of petals doubles.

-2

-1

0

2

$\frac{x\exp(\cos(\vartheta)) - \sin(\vartheta)}{\cos(\vartheta)\exp(\cos(\vartheta))}$

$\frac{-x}{1 - \sin(\vartheta)} + c$

$\frac{-x}{1 + e^{\cos(\vartheta)}} + \cos(\vartheta)$

$\frac{x\cos(\vartheta)}{1 + \sin(\vartheta)}$

Assuming the output should be treated the same regardless of the context of presentation

Ignoring the potential need to adjust for the presence of additional variables except time or space

By pulling through $\frac{dR}{d\Theta}$ times $\frac{dY}{dR}$ plus $\frac{dR}{d\Theta}$ times $\frac{dY}{dR}$

Applying standard rules without modification

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.

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.

Through explicit multiplication and substitution using polar-to-cartesian conversion formula for $r$

$\frac{1}{n} \cdot r^{n-1} \cdot \frac{d^{n+1}r}{d\theta^{n+1}} \cdot (\frac{dr}{d\theta})^{-n} \cdot (\frac{d\theta}{dr})^{-1}$

Using the relationship between rectangular and polar coordinates with $r=(x^2)+(y^2)$

Employment of simplification techniques that utilize basic properties of exponentiation and square roots