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Paper 1

CALCULUS
Factorise third degree polynomials (including examples which require the factor theorem)

(a) Investigate and use instantaneous rate of change of a variable when interpreting models of situations:

  • demonstrating an intuitive understanding of the limit concept in the context of approximating the rate of change or gradient at a point:
  • establishing the derivatives of the following functions from first principles:
    f(x) = b
    f(x) = x
    f(x) = x2
    f(x) = x3
    f(x) = x-1
    and then generalise to the derivative of:
    f(x) = xn

(b) Use the following rules of differentiation:

(c) Determine the equations of tangents to graphs.

(d) Generate sketch graphs of cubic functions using differentiation to determine the stationary points (maxima, minima and points of inflection) and the factor theorem and other techniques to determine the intercepts with the x-axis.

(e) Solve practical problems involving optimisation and rates of change.

Clarification

  • Besides using the Factor Theorem, methods of synthetic division or long division could also be used to factorise cubic polynomials and solve cubic equations.
    NOTE:
    This knowledge is necessary for the sketching of cubic functions.
  • Calculate from first principles the derivatives of the following functions:
    f(x) = ax
    f(x) = ax + b
    f(x) = ax2
    f(x) = ax2 + b

    NOTE:

  • Differentiate by using the power rule:
    If f(x) = axn then
    f '(x)=an.xn-1

    Examples

    Differentiate

  • NOTE:
    The following notations can be used:
    · f '(x)
    · Dx
    · dy/dx
    · y'
  • Candidates are expected to be able to interpret cubic functions:
    - By determining the equation of a cubic function from a given graph.
    - Using the second derivative or any other means to determine a point of inflection where applicable.
    - Discuss the nature of stationary points including local maximum, local minimum and points of inflection.
    - Integration with transformation.
  • Candidates are expected to interpret the graph of the derivative of a function.