# Physical Interpretation of the Derivative

### Average Velocity

The average velocity is the ratio between distance travelled (Δd) and the time elapsed (Δt).

### Instantaneous Velocity

The instantaneous velocity is the limit of the velocity function when Δt approaches zero, that is to say, the derivative of space relative to time.

The relationship between the distance traveled in meters and the time in seconds is e(t) = 6t^{2}.

Calculate:

1 The average velocity between t = 1 and t = 4.

The average velocity is the incremental quotient in the interval [1, 4].

2 The instantaneous velocity at t = 1.

The instantaneous velocity is the derivative at t = 1.