Netherlands VWO 6 (eindexamenjaar) Mathematics B

Chapters: 2

1. Differential and integral calculus (part 2)

Integral calculus

Integrals – Integration undoes differentiation: if F′(x) = f(x), then ∫f(x) dx = F(x) + C. We integrate with standard formulas, substitution (replace an inside part by u), partial fractions (split a fraction) and by parts (∫u dv = uv − ∫v du). A definite integral ∫ₐᵇ f(x) dx is the signed area under the curve from a to b, and the fundamental theorem says it equals F(b) − F(a). Its properties make many hard integrals easy.

2. Geometry with coordinates (part 2)

Algebraic methods in plane geometry · Vectors and dot product · Applications

Straight Lines (Class 11): Slope, Forms of the Equation and Distance – The slope of a line is rise ÷ run = (y₂ − y₁)/(x₂ − x₁) = tan θ. Parallel lines have equal slopes; perpendicular lines have m₁m₂ = −1. The angle between two lines is given by tan θ = |(m₂ − m₁)/(1 + m₁m₂)|. A line can be written as y = b or x = a (parallel to an axis), y − y₁ = m(x − x₁) (point-slope), y = mx + c (slope-intercept), the two-point form, or x/a + y/b = 1 (intercept form). The distance of (x₁, y₁) from Ax + By + C = 0 is |Ax₁ + By₁ + C|/√(A² + B²).
Vector Algebra – A vector has a size (magnitude) and a direction. In 3D we write it as a = xî + yĵ + zk̂. Its length is |a| = √(x² + y² + z²). Its direction cosines are l = x/|a|, m = y/|a|, n = z/|a|, and l² + m² + n² = 1. Vectors are added head-to-tail (triangle law) or component by component. ka stretches a by k and flips it if k is negative. The point dividing AB in m : n has position vector (mb + na)/(m + n) inside and (mb − na)/(m − n) outside. Dot product a·b = |a||b|cosθ gives a number and tells the angle and the projection. Cross product a×b = |a||b|sinθ n̂ gives a vector at right angles to both; its length is the area of the parallelogram on a and b.
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.