Rates of Change

• Calculus

  Two main branches: differential calculus and integral calculus, interconnected through the Fundamental Theorem of Calculus
• Differential calculus

  • Study of rates of change and slopes of curves, concepts like limits, derivatives, and differentials
• Integral calculus

  • Study of areas and volumes of curves and solids, concepts like integrals and antiderivatives
• Average Rate of Change (AROC)

  The average rate of change of f(x) with respect to x as x changes from a to b, given by (f(b)-f(a))/(b-a)
• Instantaneous Rate of Change (IROC)

  The rate of change in a function at the instant that x = a
• Geometrically, AROC is the slope of the secant between two points, IROC is the slope of the tangent at a point
• As Q approaches P, the secants turn into the tangent at point P
• Tangent line

  A line that touches a curve at only one point but does not cross it
• The limiting position of a sequence of secants through points P and Q on the curve, where Q approaches P, gives the tangent line at P
• Slope of the tangent line

  Instantaneous rate of change at that point