Netherlands VWO 5 Mathematics B
Chapters: 3
1. Functions, graphs and equations (part 2)
Inverse functions · Equations and inequalities · Asymptotes and limits
- Inverse Functions – An inverse function undoes what a function does. If f sends a to b, then f⁻¹ sends b back to a. To find it, write y = f(x), swap x and y, and solve for y. The graph of f⁻¹ is the mirror image of the graph of f in the line y = x. Only one-to-one functions have an inverse; if a function is not one-to-one, limit its domain first (for example x² with x ≥ 0 has inverse √x).
- Quadratic Equations – A quadratic equation has x² as its highest power: ax² + bx + c = 0 with a ≠ 0. It has at most two roots. Find them by factorising (split the middle term, then set each bracket to zero) or by the formula x = (−b ± √(b² − 4ac)) / 2a. The discriminant D = b² − 4ac tells the nature of roots: D > 0 two different real roots, D = 0 two equal roots, D < 0 no real roots.
- Limits (Class 11): What a Function Gets Close To – A limit tells us which number f(x) gets close to when x gets close to a point. We walk towards the point from the left and from the right. If both sides reach the same number, that number is the limit. You will learn limits of polynomial, rational, trigonometric, exponential and log functions, with the standard results you must know.
2. Differential and integral calculus (part 1)
Derivative functions · Differentiation techniques
- Application of Derivatives – The derivative measures how fast one quantity changes compared with another. Its sign tells whether a function goes up (f′ > 0, increasing) or down (f′ < 0, decreasing). Where f′ = 0 the tangent is flat: these critical points may be a local maximum or minimum, checked by the first derivative test (sign change) or the second derivative test (sign of f″). This lets us solve real problems like the biggest box or the cheapest tank.
- Continuity and Differentiability – A function is continuous at a point when its graph has no break there: the left limit, the right limit and the value are all equal. The derivative is the slope of the tangent line. With the chain rule, implicit differentiation, the rules for eˣ, ln x and inverse trig functions, logarithmic differentiation and parametric forms, you can differentiate almost any Class 12 function. The second derivative tells how the slope itself changes, that is, how the curve bends.
3. Trigonometric functions
Trigonometric functions
- Trigonometric Functions: Radians, Unit Circle, Graphs and Identities – An angle of one radian cuts an arc equal to the radius, so π radians = 180°. On a unit circle, the point at angle x is P = (cos x, sin x), which gives sin²x + cos²x = 1 and extends sine and cosine to every real number. The signs follow 'All, Sin, Tan, Cos' in quadrants I to IV; sin x and cos x repeat every 2π and stay between −1 and 1. Compound-angle formulas such as cos(A + B) = cos A cos B − sin A sin B lead to tan(A + B), cot(A + B), sum-to-product, double-angle and triple-angle identities.