France Terminale Complementary Mathematics
Chapters: 2
1. Study themes
Models defined by a function · Growth models · History of the logarithm · Computing areas · Wealth distribution and inequality · Bayesian inference · Repeated trials and sampling · Waiting times · Correlation and causation
- Functions: Input, Rule, Output – A function is a rule that gives exactly one output for each input. We write it as f(x). The output f(a) is called the image of a; an input that gives a certain output is a preimage. A function can be shown as words, a table, a formula or a graph. Its graph can go up (increasing), go down (decreasing) and have a highest point (maximum) or lowest point (minimum).
- Growth Models: Recursive Sequences and y' = ay + b – A growth model says how a quantity changes step by step: u(n+1) = a·u(n) + b. If b = 0 it is geometric, if a = 1 it is arithmetic. For 0 < a < 1 the sequence moves to the fixed point L = b/(1 − a). The continuous version is y' = ay + b with solution y = C·e^(ax) − b/a. Euler's method turns y' = f(y) back into steps, and the logistic model slows growth near a limit.
- Logarithms – A logarithm tells you the power you need. log_b x = y means b^y = x. So log₂ 8 = 3 because 2³ = 8. Logs turn multiplying into adding: log(ab) = log a + log b, log(a/b) = log a − log b, log(aⁿ) = n log a. Base 10 is the common log, base e is the natural log (ln). Any base can be changed: log_b x = log x ÷ log b.
- 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.
- Income Inequality – Income inequality means income is shared unevenly between people. Economists rank people from poorest to richest, split them into five groups of 20% (quintiles) and compare their shares. The Lorenz curve plots the cumulative share of income against the cumulative share of people; the further it bends from the straight line of equality, the more unequal the society. The Gini coefficient = A ÷ (A + B) turns this into one number between 0 (perfect equality) and 1 (one person has everything). Wealth (what you own) is usually more unequal than income (what you earn). Causes include differences in skills, education, inherited wealth, discrimination and technology. Governments reduce inequality with progressive taxes, benefits, minimum wages and public services such as free schooling and health care.
- Conditional Probability, Multiplication Rule and Independent Events – Conditional probability is the chance of A when we already know B has happened. We throw away every outcome outside B and count again: P(A|B) = P(A ∩ B) ÷ P(B). Turned around, this gives the multiplication rule P(A ∩ B) = P(B)·P(A|B). If knowing B does not change the chance of A, the events are independent and P(A ∩ B) = P(A)·P(B).
- Binomial Distribution – Repeat the same yes/no trial n times, independently, with the same chance of success p each time. The number of successes X follows the binomial distribution B(n, p): P(X = k) = ⁿCₖ pᵏ qⁿ⁻ᵏ with q = 1 − p. Its mean is np and its variance is npq.
- 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.
- Linear Regression and the Least Squares Line – Linear regression finds the straight line ŷ = a + bx that best follows paired data (x, y). A residual is the gap between a real point and the line: e = y − ŷ. The least squares line makes the sum of squared residuals as small as possible. Its slope is b = Sxy ÷ Sxx and it always passes through the mean point (x̄, ȳ). We use it to predict y from x, but only inside the data range, and a strong link does not prove that x causes y.
2. Content
Analysis toolkit · Probability distributions and bivariate statistics · Algorithms and programming
- 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.
- 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.
- Python Basics: Modes, Variables, Data Types and Operators – Python runs in interactive mode (one line at a time) or script mode (a saved .py file). Blocks are shown by indentation. Variables are names tied to values; every value has a data type, and some types are mutable. Operators build expressions that Python works out by precedence. input() reads text, int()/float()/str() convert types, and debugging fixes syntax, runtime and logical errors.
- Getting Started with Python – Python is a high-level, free and open-source, interpreted, portable, case-sensitive language that is easy to read. You can run it in interactive mode (type after the >>> prompt and see the result at once) or script mode (save code in a .py file and run it). Python's character set includes letters, digits, special symbols, whitespace and all Unicode characters. The smallest units of a program are tokens: keywords, identifiers, literals, operators and punctuators. A variable is a name that refers to a value; in an assignment the right side (r-value) is evaluated first and stored in the left side (l-value). Comments start with # and are ignored by Python.
- Python Revision for Class 12: Everything from Class 11 in One Place – Class 12 builds on Class 11 Python. You must know tokens (keywords, identifiers, literals, operators, punctuators), data types (int, float, bool, str, list, tuple, dict), which types are mutable (can change in place) and which are immutable, operator precedence, if-elif-else, for and while loops with break and continue, string slicing, and the main methods of lists, tuples and dictionaries.