Physical Applications of the Derivative
The average velocity is the ratio of distance travelled (Δe) and the time elapsed (Δt).
The instantaneous velocity is the average speed limit when Δt approaches zero, that is to say, the derivative of space relative to time.
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/s2