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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)=ax
^{n}is f'(x)=anx^{n-1}, The derivative of functions of the form f(x)=ax^{n}+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)=ax
^{n}+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 x
^{n}, 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 x
^{n},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_{a}x, 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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