MCV4U – Grade 12 Calculus & Vectors – Rates of Change

Grade 12 – Calculus and Vectors


Rates of Change Test


Rates of Change and the slope of a curve

  • Average rate of change: refers to the rate of change of a function over an interval. It corresponds to the slope of the secant connecting the two endpoints of the interval
  • Instantaneous rate of change: refers to the rate of change at a specific point. It corresponds to the slope of a tangent passing through a single point, or tangent point, on the graph of a function.
    • An estimate of the instantaneous rate of change can be obtained by calculating the average rate of change over the smallest interval for which data are available.
    • It can also be determined by sketching a tangent on a graph


Rates of Change using Equations

  • For a given function y = f(x), the instantaneous rate of change at x = a is estimated by calculating the slope of a secant over a very small interval, a <= x <= a + h such that h is a very small number
  • The expression f(a+h) – f(a) / h, where h not = 0, is called the difference quotient
    • It is used to calculate the slope of the secant between (a, f(a)) and (a + h, f(a + h)). It will let you have an estimate for the slope of the tangent as h approaches 0.



  • A sequence is a function, f(n) = tn , whose domain is the set of natural numbers N
  • limit as x approaches a of f(x) exists if the following are true:
    • limit as x approaches a- of f(x) exists (left hand limit exists)
    • limit as x approaches a+ of f(x) exists (right hand limit exists)
    • right side limit and left side limit are equal
  • If they aren’t equal and they criteria aren’t met, then the limit does not exists
  • Limits can be a real number of infinity
  • A function is continues at a value if these are true:
    • f(a) exists
    • limit as x approaches a exists
    • The limit as x approaches a and f(a) are the same


Limits and Continuity

  • The limit of a function at x = a may exist even though the function is discontinuous at x = a.
  • In short, the graph of a discontinuous function cannot be drawn without lifting your pencil
  • There are 3 kinds if discontinuity: Jump, Removable, and Infinite
  • When direct substitution of x = a results in a limit of an indeterminate , 0/0, determine an equivalent function that represents f(x).
  • Discontinuity may be removed by factoring, rationalizing the numerator or denominator, expanding, and simplifying.


Introduction to derivatives

  • The derivatives of y = f(x) is a new function y = f’(x), which represents the slope of the tangent, or instantaneous rate of change, at any point on the curve of y = f(x).
  • The derivative function is defined by the first principles definition for the derivative, f’(x) = lim h -> 0   f(x+h) – f(x) / h, if the limit exists
  • Different notations for the derivative of y = f(x) are y = f’(x), y’, dy/dx, and d/dx f(x).
  • If the derivative does not exist on a point on the curve, the function is non-differentiable at that x-value. This can occur at points where the function is discontinuous or in cases where the function has an abrupt change, which is represented by a cusp or corner on the graph.