Netherlands VWO 6 (eindexamenjaar) Mathematics D
Chapters: 3
1. Dynamical systems (part 2)
Continuous dynamical systems · Applications of dynamical systems
- Differential Equations – A differential equation connects a function y with its derivatives. Its order is the highest derivative present and its degree is the power of that derivative (when the equation is a polynomial in derivatives). A general solution has arbitrary constants; a condition like y(0) = 1 fixes them to give a particular solution. Class 12 solves first-order equations of three kinds: variables separable, homogeneous (put y = vx) and linear dy/dx + Py = Q (multiply by the integrating factor e^∫P dx).
- Mathematical Modelling: Using Maths to Describe the Real World – A mathematical model is an equation, graph or table that describes a real situation in a simple way. The modelling cycle: understand the real problem → choose variables and make assumptions → build a model (for example linear, quadratic or exponential) → solve and predict → check the answer against real data → improve the model or state its limits. No model is perfect; a good one is simple and close enough to be useful.
2. Complex numbers
Basic operations · Profile-specific deepening
- Complex Numbers and Quadratic Equations – Some equations like x² + 1 = 0 have no real answer, because no real number has a negative square. So we add a new number i with i² = −1. A complex number is z = a + ib: a is the real part and b is the imaginary part. We add, subtract and multiply complex numbers like algebra, and replace i² by −1. To divide, we multiply top and bottom by the conjugate. On the Argand plane, z = a + ib is the point (a, b). The conjugate a − ib is its mirror image in the real axis, and the modulus √(a² + b²) is its distance from the origin. Every quadratic ax² + bx + c = 0 with D = b² − 4ac < 0 has two complex roots that are conjugates of each other.
3. Mathematics in science
Mathematics in science
- Mathematical Modelling: Using Maths to Describe the Real World – A mathematical model is an equation, graph or table that describes a real situation in a simple way. The modelling cycle: understand the real problem → choose variables and make assumptions → build a model (for example linear, quadratic or exponential) → solve and predict → check the answer against real data → improve the model or state its limits. No model is perfect; a good one is simple and close enough to be useful.