1. Take the derivative of $x(t)$ with respect to $t$. 2. $v(t) = x'(t)$.

1. Take the derivative of $v(t)$ with respect to $t$. 2. $a(t) = v'(t)$.

1. Find the velocity function $v(t)$. 2. Solve for when $v(t) > 0$.

1. Find the velocity function $v(t)$. 2. Solve for when $v(t) < 0$.

1. Find $v(t)$ and $a(t)$. 2. Determine when $v(t)$ and $a(t)$ have the same sign.

1. Find $v(t)$ and $a(t)$. 2. Determine when $v(t)$ and $a(t)$ have opposite signs.

1. Find the velocity function $v(t)$. 2. Evaluate $\int_{a}^{b} v(t) dt$.

1. Find the velocity function $v(t)$. 2. Evaluate $\int_{a}^{b} |v(t)| dt$.

1. Integrate $v(t)$ to find $x(t) = \int v(t) dt + C$. 2. Use the initial condition $x(0)$ to solve for $C$.

1. Find the second derivative $a(t) = x''(t)$. 2. Evaluate $a(c)$.

$v(t) = x'(t) = \frac{dx}{dt}$

$a(t) = v'(t) = \frac{dv}{dt}$

$a(t) = x''(t) = \frac{d^2x}{dt^2}$

$\frac{x(b) - x(a)}{b - a}$

$v(c) = x'(c)$

$a(c) = v'(c) = x''(c)$

$x(t) = \int v(t) dt$

$v(t) = \int a(t) dt$

$\int_{a}^{b} v(t) dt$

$\int_{a}^{b} |v(t)| dt$

Velocity: a vector quantity with magnitude and direction. Speed: a scalar quantity representing the magnitude of velocity.

Displacement: change in position (can be negative). Total Distance: total length traveled (always non-negative).

Average Velocity: Velocity over an interval. Instantaneous Velocity: Velocity at a specific time.

Positive acceleration: velocity increasing in the positive direction. Negative acceleration: velocity increasing in the negative direction.

Speeding up: velocity and acceleration have the same sign. Slowing down: velocity and acceleration have opposite signs.

Velocity: rate of change of position. Acceleration: rate of change of velocity.