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These are the statements describing what students need to learn:

[Higher Level only statements are in blue]

• introduction to the concept of a limit. Derivative interpreted as gradient function and as rate of change
• increasing and decreasing functions. Graphical interpretation of f'(x)>0, f'(x)=0,f'(x)<0
• derivative of f(x)=axⁿ is f'(x)=anx^n-1, The derivative of functions of the form f(x)=axⁿ+bx^n-1+… where all exponents are integers
• tangents and normals at a given point, and their equations
• introduction to integration as anti-differentiation of functions of the form f(x)=axⁿ+bx^n-1+.... Anti-differentiation with a boundary condition to determine the constant term. Definite integrals using technology. Area of a region enclosed by a curve y=f(x) and the x-axis, where f(x)>0
• derivative of xⁿ, sinx, cosx, e^x and lnx. Differentiation of a sum and a multiple of these functions. The chain rule for composite functions. The product and quotient rules
• the second derivative. Graphical behaviour of functions, including the relationship between the graphs of f,f' and f"
• local maximum and minimum points. Testing for maximum and minimum. Optimization. Points of inflexion with zero and non-zero gradients
• Kinematic problems involving displacement s, velocity v, acceleration a and total distance travelled.
• indefinite integral of xⁿ,sinx,cosx,1/x and e^x. The composites of any of these with the linear function ax+b. Integration by inspection (reverse chain rule) or by substitution for expressions of the form: ∫kg'(x)f(g(x))dx
• definite integrals, including analytical approach. Areas of a region enclosed by a curve y=f(x) and the x-axis, where f(x) can be positive or negative, without the use of technology. Areas between curves
• Informal understanding of continuity and differentiability of a function at a point.
  Understanding of limits (convergence and divergence).
  Definition of derivative from first principles.
  Higher derivatives.
• The evaluation of limits of the form lim f(x) ÷ g(x) as x approaches a and as x approaches ∞
  using l’Hôpital’s rule or the Maclaurin series.
  Repeated use of l’Hôpital’s rule.
• Implicit differentiation.
  Related rates of change.
  Optimisation problems.
  
• Derivatives of tanx, secx, cosecx, cotx, a^x, log_ax, arcsinx, arccosx, arctanx.
  Indefinite integrals of the derivatives of any of the above functions.
  The composites of any of these with a linear function.
  Use of partial fractions to rearrange the integrand.
• Integration by substitution.
  Integration by parts.
  Repeated integration by parts.
• Area of the region enclosed by a curve and the y-axis in a given interval.
  Volumes of revolution about the x-axis or y-axis.
• First order differential equations.
  Numerical solution of dy/dx=f(x,y) using Euler’s method.
  Variables separable.
  Homogeneous differential equation dy/dx=f(y/x)using the substitution y=vx.
  Solution of y'+P(x)y=Q(x), using the integrating factor.
• Maclaurin series to obtain expansions for e^x, sinx, cosx, ln(1+x), (1+x)^p, p ∈ Q
  Use of simple substitution, products, integration and differentiation to obtain other series.
  Maclaurin series developed from differential equations.

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