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.

Parametric: x and y are functions of t | Cartesian: y is a function of x

Vector-Valued: Integrate each component separately, result is a vector | Scalar: Integrate a single function, result is a scalar

Cartesian: Integrate difference of functions wrt x, A = ∫(top - bottom) dx | Polar: Integrate difference of squared polar functions wrt θ, A = 1/2 ∫(R² - r²) dθ

Velocity: Vector quantity with magnitude and direction | Speed: Scalar quantity, magnitude of velocity

Position Vector: Represents location at a given time | Velocity Vector: Represents rate of change of position at a given time

Polar: Defined by radius and angle (r, θ) | Cartesian: Defined by horizontal and vertical distance (x, y)

Parametric: Use chain rule, $\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$ | Cartesian: Direct differentiation, $\frac{dy}{dx}$

Parametric: Integrate with respect to t | Cartesian: Integrate with respect to x

Vector-valued: Output is a vector | Scalar: Output is a single number

Cartesian: Integrate $\sqrt{1 + (dy/dx)^2}$ dx | Parametric: Integrate $\sqrt{(dx/dt)^2 + (dy/dt)^2}$ dt

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 = $\sqrt{(\frac{dx}{dt})^2 + (\frac{dy}{dt})^2}$