Functions and algebra: Investigate and use instantaneous rate of change of a variable when interpreting models both in mathematical and real life situations

• Establish the derivatives of the following functions from first principles:
  • [latex]\scriptsize f\left( x \right)=b;~f\left( x \right)=ax+b;f\left( x \right)=a{{x}^{2}}+b;f\left( x \right)={{x}^{3}};f\left( x \right)=a{{x}^{3}};f\left( x \right)=\displaystyle \frac{1}{x};f\left( x \right)=\displaystyle \frac{a}{x}[/latex]
    Note: The binomial theorem does not form part of the curriculum.
• Find the derivatives of the functions in the form:
  • [latex]\scriptsize f\left( x \right)=a{{x}^{n}}[/latex]
  • [latex]\scriptsize f\left( x \right)=a\ln kx[/latex]
  • [latex]\scriptsize f\left( x \right)=a{{e}^{{kx}}}[/latex]
  • [latex]\scriptsize f\left( x \right)=\text{asin}kx[/latex]
  • [latex]\scriptsize f\left( x \right)=\text{acos}kx[/latex]
  • [latex]\scriptsize f\left( x \right)=\text{atan}kx[/latex]

Where:

• • [latex]\scriptsize f\left( x \right)=a{{x}^{n}}~~~~~~~~~~~{f}'\left( x \right)=na{{x}^{{n-1}}}~[/latex]
  • [latex]\scriptsize f\left( x \right)=a\ln kx~~~~{f}'\left( x \right)=\displaystyle \frac{k}{x}[/latex]
  • [latex]\scriptsize f\left( x \right)=a{{e}^{{kx}}}~~~~~~~~{f}'\left( x \right)=k{{e}^{{kx}}}[/latex]
  • [latex]\scriptsize f\left( x \right)=\text{asin}kx~~~{f}'\left( x \right)=ka\cos kx~~[/latex]
  • [latex]\scriptsize f\left( x \right)=\text{acos}kx~~~~{f}'\left( x \right)=-ka\sin kx~~[/latex]

Examples to include are:

• • [latex]\scriptsize 3{{x}^{2}};\displaystyle \frac{3}{{{{x}^{{-3}}}}};~-\displaystyle \frac{2}{{\sqrt[3]{{{{x}^{2}}}}}};2\ln 3x;\displaystyle \frac{1}{2}{{e}^{{-2x}}};2\text{sin}3x;\displaystyle \frac{1}{3}\cos \displaystyle \frac{x}{2};~-4\text{tan}x;\text{etc}.[/latex]
• Use the constant, sum and/or difference, product, quotient and chain rules for differentiation. Note: Combinations of rules in the same problem are excluded.
• Find the equation of the tangent to a graph at a specific point.
• Solve practical problems involving rates of change. Note: velocity and acceleration may be included.
• Draw graphs of cubic functions by determining:
  • y-intercept
  • roots (x-intercepts)
  • turning points using derivatives
• Determine/prove maximum and minimum turning points by making use of second order derivatives (Only: quadratic and cubic functions)
• Determine the point of inflection of cubic graphs by using second order derivatives.

By the end of this unit you will be able to:

• Define the derivative using limits.
• Calculate the derivative from first principles.

By the end of this unit you will be able to:

• Use various forms of notation to represent the derivative.
• Apply the power rule.
• Find the derivative of a constant.
• Find the derivative of a constant multiplied by a function.
• Find the derivative of a sum/difference.
• Find the derivative of a product.
• Find the derivative of a quotient.
• Apply the chain rule.

By the end of this unit you will be able to:

• Define a tangent line.
• Determine the gradient of the tangent at a point.
• Find the equation of a tangent.

By the end of this unit you will be able to:

• Solve practical problems involving rates of change.

By the end of this unit you will be able to:

• Determine the shape of a cubic function.
• Determine the x and y intercepts.
• Find the turning points of the graph.
• Find the maximum and minimum values of the graph.
• Find the point of inflection and discuss concavity using second derivatives.