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THE SECOND DERIVATIVE

      (Technical background information. Not part of the novel.)

      To understand the second derivative, it is helpful to know what a vector is in physics. A vector is a quantity with direction and magnitude (both of which could have a zero value). Remembering Newton’s laws of motion, a force must be applied to change direction or magnitude.
      A particle moving in three-dimensional space must be plotted as a function of time from some fixed reference point in order to be located at any instant in time. Differentiating the function describes the direction of change and the rate of change. If the direction is changed and/or the speed is changed, the second derivative describes the rate of change to the prior rate of change.
      Calculus is concerned with the calculation of instantaneous rates of change. The first derivative test tells us about the direction of a function. It identifies increasing and decreasing properties of functions by finding the slope of the tangent line to the graph of the function.
      The second derivative measures how the rate of change of a quantity is itself changing. The second-order derivative of a function is the derivative of the derivative of the function.

Real World Application

      A particle moving through space travels a finite distance during a finite period of time. Speed is the value we give to the term magnitude in the description of a vector above. If there is a constant rate of change to distance traveled over time, the slope of the line in the first derivative will stay the same. If speed is not constant, the second derivative identifies the rate of change in speed. If speed is increasing, we call this acceleration. In the real world, constant speed is rare. Objects in motion are typically accelerating or decelerating continuously.
      Just as a force must be applied to change magnitude, force is also required to change direction. The first derivative is the tangent line at any instant in time. If the particle is moving in a circle. The slope of the tangent line is changing and the second derivative measures the rate of change in direction. If the circle becomes a spiral, then there is a further change to the rate of change.


Predicting Cause And Effect

      If the image of a particle moving through space is replaced with the concept of an event occurring at a location and during a period of time, that event has the potential to influence subsequent events. Calculating the second derivative (rate of change) is necessary to determine how events interweave over time. Depending on when and where an event occurs, much like a vector, every event has an information cone of influence projecting through time that defines which subsequent events it can influence.

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