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.

Values of x where the gradient of a curve is zero. Solution of f ′(x) = 0. Local maximum and minimum points.

Optimisation problems in context.

Approximating areas using the trapezoidal rule.

The derivatives of sin x, cos x, tan x, ex, ln x, xn where n ∈ ℚ. The chain rule, product rule and quotient rules. Related rates of change.

The second derivative. Use of second derivative test to distinguish between a maximum and a minimum point.

Definite and indefinite integration of xn where n ∈ ℚ, including n = … 1 , sin x, cos x, Integration by inspection, or substitution of the form ∫ f (g(x))g′(x)dx.

Area of the region enclosed by a curve and the x or y-axes in a given interval. Volumes of revolution about the x- axis or y- axis.

Kinematic problems involving displacement s, velocity v and acceleration a.

Setting up a model/differential equation from a context. Solving by separation of variables.

Slope fields and their diagrams.

Euler’s method for finding the approximate solution to first order differential equations. Numerical solution of Numerical solution of the coupled system dx = f 1(x, y, t) and dt

Phase portrait for the solutions of coupled differential equations of the form: dx = ax + by dt dy = cx + dy. dt Qualitative analysis of future paths for distinct, real, complex and imaginary eigenvalues. Sketching trajectories and using phase portraits to identify key features such as equilibrium points, stable populations and saddle points.

d2 x Solutions of = f (x, , t) by Euler’s method. dt2