Differentiation Recap

In the last lesson, you learned about the derivative. This section is just here to remind you of what you learned.

Understanding the Derivative

You saw a few ways to understand the derivative:

1. The "Rate of Change" Interpretation

If f(t) gives the value of a function at any t, then \dot{f}(t_0) gives the instantaneous rate of change of f(t) at the value t=t_0.

2. The Graphical Interpretation

The slope of the line tangent to f(t) at t=t_0 is \dot{f}(t_0)

3. The Formal Definition

The formal mathematical definition is the following:

The derivative of a function f(t) is the function \dot{f}(t) (or \frac{df}{dt}), and is defined as:

Derivatives and Motion

Position, velocity, and acceleration are all useful quantities when describing a vehicle's motion and these quantities are related through the derivative.

• velocity is the derivative of position

  • v(t)=\dot{x}(t)

• acceleration is the derivative of velocity and the second derivative of position.

  • a(t) = \dot{v}(t) = \ddot{x}(t)

Coding the Derivative

def get_derivative_from_data(position_data, time_data):
    """
    Calculates a list of speeds from position_data and 
    time_data.

    Arguments:
      position_data - a list of values corresponding to 
        vehicle position

      time_data     - a list of values (equal in length to
        position_data) which give timestamps for each 
        position measurement

    Returns:
      speeds        - a list of values (which is shorter 
        by ONE than the input lists) of speeds.
    """
    # 1. Check to make sure the input lists have same length
    if len(position_data) != len(time_data):
        raise(ValueError, "Data sets must have same length")

    # 2. Prepare empty list of speeds
    speeds = []

    # 3. Get first values for position and time
    previous_position = position_data[0]
    previous_time     = time_data[0]

    # 4. Begin loop through all data EXCEPT first entry
    for i in range(1, len(position_data)):

        # 5. get position and time data for this timestamp
        position = position_data[i]
        time     = time_data[i]

        # 6. Calculate delta_x and delta_t
        delta_x = position - previous_position
        delta_t = time - previous_time

        # 7. Speed is slope. Calculate it and append to list
        speed = delta_x / delta_t
        speeds.append(speed)

        # 8. Update values for next iteration of the loop.
        previous_position = position
        previous_time     = time

    return speeds