An equation where x and y are defined in terms of a third variable, usually time (t).

A function that maps a real number to a vector in a vector space, often representing position, velocity, or acceleration.

A two-dimensional coordinate system where a point is located by its distance (r) from the origin and angle (θ) from the positive x-axis.

The distance along a curve defined by a function between two points.

A function of the form r = f(θ), where r is the distance from the origin and θ is the angle.

A two-dimensional coordinate system in which the position of a point is determined by the distance from the origin (r) and the angle (theta θ) between the positive x-axis and the line connecting the point to the origin, counterclockwise.

Time (t).

x and y.

The xy-plane, also known as ℝ^2.

Position, velocity, and acceleration of an object moving in space.

$\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$

$\frac{d^2y}{dx^2} = \frac{\frac{d}{dt}(\frac{dy}{dx})}{\frac{dx}{dt}}$

$L = \int_{a}^{b} \sqrt{(\frac{dx}{dt})^2 + (\frac{dy}{dt})^2} dt$

$A = \frac{1}{2} \int_{a}^{b} r^2 d\theta$

$A = \frac{1}{2} \int_{a}^{b} (R^2 - r^2) d\theta$

$x = r \cos(\theta), y = r \sin(\theta)$

$r = \sqrt{x^2 + y^2}$

$\frac{dy}{dx} = \frac{\frac{dr}{d\theta} \sin(\theta) + r \cos(\theta)}{\frac{dr}{d\theta} \cos(\theta) - r \sin(\theta)}$

r'(t) = <f'(t), g'(t)>

∫v(t) dt = <∫f(t) dt, ∫g(t) dt>

Velocity is the first derivative of the position vector with respect to time. Acceleration is the second derivative of the position vector with respect to time.

Integrate each component of the vector-valued function separately.

Find $\frac{dy}{dx}$ by calculating $\frac{dy/dt}{dx/dt}$. Evaluate at the given t-value.

The velocity vector of the particle at that point.

Find where $\frac{dy}{dt} = 0$ and $\frac{dx}{dt} \neq 0$.

Find where $\frac{dx}{dt} = 0$ and $\frac{dy}{dt} \neq 0$.

Use the formula $A = \frac{1}{2} \int_{a}^{b} r^2 d\theta$, where r is the polar function and a and b are the limits of integration.

Use the formula $A = \frac{1}{2} \int_{a}^{b} (R^2 - r^2) d\theta$, where R is the outer curve and r is the inner curve.

Velocity is the derivative of position, and acceleration is the derivative of velocity (or the second derivative of position).

Analyze the signs of $\frac{dx}{dt}$ and $\frac{dy}{dt}$. Positive $\frac{dx}{dt}$ means moving right, negative means moving left. Positive $\frac{dy}{dt}$ means moving up, negative means moving down.