Russia 11 класс Algebra and Elements of Calculus (advanced)

Chapters: 4

1. Numbers and calculations

Integers · Complex numbers

Divisibility: Multiples, Divisors, GCD and LCM – a is divisible by b when a = b × k for a whole number k: b is a divisor (factor) of a and a is a multiple of b. Every whole number a can be written as a = b × q + r with 0 ≤ r < b (division with remainder); b divides a exactly when r = 0. Quick tests tell divisibility by 2, 3, 4, 5, 6, 8, 9, 10 and 11 from the digits. The GCD is the greatest common divisor, found by prime factors or by Euclid's algorithm; LCM is the least common multiple, and GCD × LCM = a × b. Two numbers are coprime when their GCD is 1.
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.

Systems and equivalence · Trigonometric inequalities · Exponential, logarithmic, irrational inequalities · Parameters

Coming soon

Graphs · 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.

Applications of derivative · Antiderivative and integral · Differential equations

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.
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).