Ukraine 11 клас Mathematics (advanced, in-depth from grade 8)

Chapters: 10

1. Algebra: exponential and logarithmic functions (36 h)

Exponential and logarithmic functions · Equations, inequalities and systems

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.
Exponential Equations and Inequalities – An exponential equation has the unknown in the power, like 2^x = 8. Solve it by making the bases equal, by substitution when it hides a quadratic, or by taking logarithms. For inequalities, keep the sign when the base is bigger than 1 and flip it when the base is between 0 and 1.

Antiderivatives · Definite integral

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.

Binomial theorem · Probability

Binomial Theorem for Positive Integers – A binomial is a two-term expression like a + b. The binomial theorem tells us how to expand (a + b)ⁿ for any positive integer n without multiplying again and again: (a + b)ⁿ = ⁿC₀aⁿ + ⁿC₁aⁿ⁻¹b + ⁿC₂aⁿ⁻²b² + … + ⁿCₙbⁿ. There are n + 1 terms. The power of a goes down by 1 and the power of b goes up by 1 in each term, and the two powers always add up to n. The coefficients ⁿCᵣ are the numbers in row n of Pascal's triangle, where each number is the sum of the two above it. The (r + 1)th term is Tᵣ₊₁ = ⁿCᵣ aⁿ⁻ʳ bʳ. The pattern was known long ago in India (Pingala's Meru Prastara) and in Persia and China, and Pascal later studied the triangle in detail. The theorem is proved by mathematical induction using ⁿCᵣ₋₁ + ⁿCᵣ = ⁿ⁺¹Cᵣ.
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).

Complex numbers · Polynomials

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.
Introduction to Polynomials – An algebraic expression is made of terms like 3x², −5x and 7. It is a polynomial when every power of the variable is a whole number (0, 1, 2, …). The degree is the biggest power. Degree 1 polynomials, y = ax + b, are called linear. They model things that grow or shrink by the same amount each step. a is the slope (change per step) and b is the y-intercept (starting value).

Review of the course

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.

Polyhedra · Trihedral angle

Polyhedra: Prisms, Pyramids and the Platonic Solids – A polyhedron is a closed solid whose surface is made only of flat polygons (faces). Faces meet along edges, and edges meet at vertices. Prisms have two equal parallel bases joined by parallelograms; pyramids have one base and triangles meeting at an apex; a frustum is a pyramid with its top cut off by a plane parallel to the base. For every convex polyhedron, Euler's formula holds: V − E + F = 2. There are exactly five regular (Platonic) polyhedra.
Solid Geometry: Points, Lines and Planes in Space – Solid geometry studies figures in three dimensions. Three points not on one line fix a plane. Two lines in space can be parallel, intersecting or skew (not in one plane). A line can lie in a plane, cut it, or be parallel to it; it is perpendicular to a plane if it is perpendicular to two intersecting lines of that plane. Angles in space are found by projecting onto a plane and using right triangles.

Special tetrahedra

Polyhedra: Prisms, Pyramids and the Platonic Solids – A polyhedron is a closed solid whose surface is made only of flat polygons (faces). Faces meet along edges, and edges meet at vertices. Prisms have two equal parallel bases joined by parallelograms; pyramids have one base and triangles meeting at an apex; a frustum is a pyramid with its top cut off by a plane parallel to the base. For every convex polyhedron, Euler's formula holds: V − E + F = 2. There are exactly five regular (Platonic) polyhedra.

Round solids · Combinations of solids

Solids of Revolution – A solid of revolution is the 3D shape you get when a flat shape turns a full 360° around a straight line (the axis). A rectangle makes a cylinder, a right triangle makes a cone, a half-circle makes a sphere and a trapezium makes a frustum. Their volumes are V = πr²h, V = ⅓πr²h, V = ⁴⁄₃πr³ and V = ⅓πh(R² + Rr + r²). In calculus, any curve y = f(x) turned about the x-axis gives V = π∫y² dx.

Volumes

Volume of Solids: Prism, Pyramid and Frustum – Volume is the space a solid fills, counted in unit cubes. A prism or parallelepiped has V = base area × height. A pyramid has V = ⅓ × base area × height. A frustum (a pyramid with its top cut off) has V = h/3 × (B₁ + B₂ + √(B₁B₂)). All of these come from one idea: add up thin slices.

Volumes and surfaces

Surface Areas and Volumes: Combined Solids – Many real objects are two simple solids stuck together, like a cone on a cylinder. The surface area is only the outside skin you can touch, so the hidden joint is left out. The volume is the space inside, so you simply add the volumes of the parts.