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, n ∈ ℤ 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 + ...., where n ∈ ℤ, 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 (n ∈ ℚ), 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 (n ∈ ℚ), sinx, cosx, 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 f (x + h) − f (x) f ′(x) = lim h→0 h Higher derivatives.

The evaluation of limits of the form lim f (x) lim using l’Hôpital’s rule or the Maclaurin x→∞ g(x) series. Repeated use of l’Hôpital’s rule.

Implicit differentiation. Related rates of change. Optimisation problems.

Derivatives of tanx, secx, cosecx, cotx, ax, loga 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 using Euler’s method. Variables separable. Homogeneous differential equation 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.