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)=anxn-1, The derivative of functions of the form f(x)=axn+bxn-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+bxn-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, ex 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 ex. 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, ax, logax, 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 ex, 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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