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

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

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

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

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

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

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

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

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

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

Velocity is the derivative of position. It represents the instantaneous rate of change of position.

Acceleration is the derivative of velocity. It represents the instantaneous rate of change of velocity.

The velocity and acceleration have the same sign (both positive or both negative).

The velocity and acceleration have opposite signs (one positive and one negative).

Displacement is the change in position, while total distance traveled considers the absolute value of the velocity, accounting for changes in direction.

The particle changes direction when the velocity changes sign (from positive to negative or vice versa).

It indicates a potential change in direction or a moment when the object is at rest.

The second derivative, $x''(t)$, gives the acceleration, which describes how the velocity is changing.

Average velocity is the change in position divided by the change in time over a given interval.

The initial condition, such as $x(0) = c$, allows you to solve for the constant of integration when finding the position function from the velocity function.

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.