# Physical Applications of the Derivative

### Average Velocity

The average velocity is the ratio of distance travelled (Δe) and the time elapsed (Δt).

### Instantaneous Velocity

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

### Instantaneous Acceleration

The instantaneous acceleration is the derivative of velocity relative to time.

Therefore, the acceleration is the second derivative of distance relative to time.

The distance travelled by a vehicle is given by the function e(t) = 3t² − t + 1. The distance is measured in meters and time in seconds.

Find the equation of velocity.

v(t) = e′(t) = 6t − 1

Find the velocity at time t = 0.

v(0)= 6** · **0 − 1 = −1 m/s

Find the acceleration equation.

a(t) = v′(t) = e′′(t) = 6 m/s^{2}

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