Kinematics and Calculus

Summary

• The kinematic quantities of displacement, velocity, and acceleration can be described by all sorts of functions.
• The function describing one quantity can be transformed into functions describing the other two quantities.
• The procedure for doing so is either…
• differentiation (finding the derivative) or…
  • The derivative of displacement with time is velocity.
  • The derivative of velocity with time is acceleration.
• integration (finding the integral).
  • The integral of acceleration over time is velocity.
  • The integral of velocity over time is displacement.

• The techniques of calculus can also be used to analyze functions — including those that describe motion.
  • The first derivative of a function…
    • is the instantaneous rate of change of the function.
    • determines the slope of a line tangent to a graph of the function.
    • equals zero at a local extrema (maximum or minimum) or a saddle point of the function.
  • The second derivative of a function…
    • is used to determine the direction of concavity of the graph of a function.
      • The graph of a function is concave up if its second derivative is positive.
      • The graph of a function is concave down if its second derivative is negative.
      • An inflection point (the transition between two different concavities) occurs where the second derivative is zero.
    • is used to distinguish extrema.
      • An extremum is a local maximum if the second derivative of the function at that point is negative.
      • An extremum is a local minimum if the second derivative of the function at that point is positive.
      • A saddle point occurs at a location where both the first and second derivatives of a function are zero.