Italy Secondaria di secondo grado – classe 5ª (esame di Stato) Mathematics (liceo scientifico)
Chapters: 4
1. Axiomatic method
Axiomatic method and modelling
- The Axiomatic Method and Modelling – In the axiomatic method we start from a few statements called axioms, which we accept without proof, and from clear rules of logic. Everything else, the theorems, must be proved from them step by step. A good set of axioms is consistent (no contradiction) and each axiom is independent (not provable from the others). A model is a real or mathematical thing in which all axioms are true. The same method is used for arithmetic (Peano), geometry (Euclid, Hilbert) and probability (Kolmogorov).
2. Geometry
Analytic geometry in space
- Three Dimensional Geometry – A line in space is fixed by one point on it and its direction. Its direction cosines l, m, n satisfy l² + m² + n² = 1; any numbers in the same ratio are direction ratios, and through two points they are (x₂ − x₁, y₂ − y₁, z₂ − z₁). Vector equation: r = a + λb. Cartesian equation: (x − x₁)/a = (y − y₁)/b = (z − z₁)/c. The angle between two lines is the angle between their directions: cosθ = |b₁·b₂|/(|b₁||b₂|). Two lines in space are parallel, intersecting or skew. The shortest distance between skew lines is d = |(a₂ − a₁)·(b₁ × b₂)|/|b₁ × b₂|; for parallel lines d = |b × (a₂ − a₁)|/|b|; d = 0 means they meet.
3. Relations and functions
Limits · Differential calculus · Integral calculus · Differential equations and optimisation
- 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.
- 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.
- 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.
- 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).
4. Data and prediction
Probability distributions
- Probability Distributions of Discrete Random Variables – A random variable X turns each outcome of an experiment into a number. Its probability distribution lists every value x with its probability P(X = x); each P is between 0 and 1 and they add to 1. The mean E(X) = Σx·P(x) is the long-run average (balance point). The variance Var(X) = Σ(x − μ)²P(x) = E(X²) − μ² measures spread; σ = √Var. Special models: uniform, binomial B(n, p) with mean np and variance np(1 − p), and Poisson with mean = variance = λ.