1. Find the derivative r'(t). 2. Evaluate r'(a).

1. Find the second derivative r''(t). 2. Evaluate r''(a).

1. Find $\frac{dx}{dt}$ and $\frac{dy}{dt}$. 2. Use the formula $L = \int_{a}^{b} \sqrt{(\frac{dx}{dt})^2 + (\frac{dy}{dt})^2} dt$.

1. Determine the limits of integration, a and b. 2. Identify the polar function r(θ). 3. Use the formula $A = \frac{1}{2} \int_{a}^{b} r^2 d\theta$.

1. Determine the limits of integration, a and b. 2. Identify the outer curve R(θ) and the inner curve r(θ). 3. Use the formula $A = \frac{1}{2} \int_{a}^{b} (R^2 - r^2) d\theta$.

1. Find $\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$. 2. Evaluate $\frac{dy}{dx}$ at t=t₀ to find the slope. 3. Find x(t₀) and y(t₀) to find the point. 4. Use point-slope form to write the equation of the tangent line.

1. Use the formulas x = r cos(θ) and y = r sin(θ). 2. Substitute the values of r and θ into the formulas. 3. Calculate x and y.

1. Find $\frac{dy}{dt}$. 2. Set $\frac{dy}{dt} = 0$ and solve for t. 3. Find the y-coordinate at that t value.

1. Find $\frac{dx}{dt}$. 2. Set $\frac{dx}{dt} = 0$ and solve for t. 3. Check if the sign of $\frac{dx}{dt}$ changes around that t value.

1. Find the velocity vector v(t) = <dx/dt, dy/dt>. 2. Calculate the magnitude of the velocity vector: speed = $\sqrt{(\frac{dx}{dt})^2 + (\frac{dy}{dt})^2}$

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>