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

Limits for $\frac{dy}{dt}$ should be zero at each junction and match exactly.

The limits as $t \to c^-$ for $\frac{dx}{dt}$ should equal each other while limits for $\frac{dy}{dt}$ may differ.

Both $(\lim_{t \to c^-} \frac{dx}{dt}) = (\lim_{t \to c^+} \frac{dx}{dt})$ and $(\lim_{t \to c^-} \frac{dy}{dt}) = (\lim_{t \to c^+} \frac{dy}{dt})$, where c is any junction between pieces.

Only continuity of the parametric equations is required; differentiability depends solely on the physical context of the problem.

Attempting partial fractions results in non-elementary functions.

Applying direct comparison tests simplifies evaluation limits at infinity.

Equating this integral solving differential equations involves separable variables.

By observing similarities between this form and integrals evaluated using residues in complex analysis.

The tangent line to the curve is horizontal or vertical depending on the $r$ value

The radius $r$ is maximum or minimum value

The curve forms a closed loop with no tangent line

The curve passes through the origin at that point

To evaluate limits of the function.

To calculate the area under the curve.

To find the slope of the tangent line.

To determine concavity.

$\frac{1}{e^{4}}$

$\frac{1}{e^{2}}$

$\frac{-1}{e^{4}}$

$\frac{e^{4}}{4}$

$t^3 - 6t + 3 = 0$

$t^3 + 6t - 3 = 0$

$3t^4 + 6t^2 - 1 = 0$

$t^4 - 6t^2 + 3 = 0$

2

0

4

-2

$\frac{dx}{dt} = 3e^{3t}$

$\frac{dx}{dt} = e^{4t}$

$\frac{dx}{dt} = e^{3t + t}$

$\frac{dx}{dt} = e^{6t}$

$\frac{d^2y}{dt^2}$

$\frac{d^2x}{dt^2}$

$\frac{d^2y}{dx^2}$

$\frac{dy}{dt} \cdot \frac{dx}{dt}$

Function $f(x)$ must have vertical asymptotes within interval [$a$, $b$] for limits to define an area properly.

The derivative of function $f(x)$, denoted by $f'(x)$, must exist for all points between [$a$, $b$].

Function $f(x)$ must be integrable on the interval [$a$, $b$], with no infinite discontinuities.

The second derivative, denoted by $f''(x)$, must change signs across any critical points within [$a$, $b$].