Ukraine 11 клас Mathematics (profile level)
Chapters: 8
1. Algebra: exponential and logarithmic functions (40 h)
Exponential function · Logarithms and logarithmic function · Exponential and logarithmic equations and inequalities · Derivatives of exponential and log functions
- Exponential Functions – An exponential function has the form y = a·bˣ, where a ≠ 0 is the starting value and b > 0, b ≠ 1 is the growth factor. If b > 1 it grows; if 0 < b < 1 it decays. Each step of 1 in x multiplies y by b (a constant ratio), unlike a linear function which adds a constant. The graph of y = bˣ passes through (0, 1), has domain all real numbers, range y > 0, and the x-axis as a horizontal asymptote. The general form y = a·b^(k(x − d)) + c stretches, reflects and shifts it.
- 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.
- 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.
2. Algebra: integral and its applications (30 h)
Antiderivative and indefinite integral · 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.
3. Algebra: combinatorics and probability (30 h)
Combinatorics · Probability theory
- Permutations and Combinations – Counting without listing is the heart of this chapter. The fundamental principle of counting says: if one job can be done in m ways and the next in n ways, both together can be done in m × n ways. n! (n factorial) is 1 × 2 × … × n, with 0! = 1. A permutation is an arrangement, where order matters: the number of ways to arrange r things out of n different things is ⁿPᵣ = n!/(n − r)!. A combination is a selection, where order does not matter: ⁿCᵣ = n!/(r!(n − r)!). Each selection of r things can be arranged in r! ways, so ⁿPᵣ = ⁿCᵣ × r!. Useful facts: ⁿCᵣ = ⁿCₙ₋ᵣ and ⁿCᵣ + ⁿCᵣ₋₁ = ⁿ⁺¹Cᵣ. When some objects repeat, divide by the factorial of each repeat count.
- 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).
4. Algebra: equations, inequalities and systems — review (30 h)
Methods for equations and inequalities · Systems of equations
- Inequalities: Rules, Intervals and Solving Them – An inequality says one amount is bigger or smaller than another, using <, >, ≤ or ≥. On a number line, the smaller number is on the left. You may add or subtract the same number on both sides, and multiply or divide by the same positive number, and the sign stays. If you multiply or divide by a negative number, the sign flips. The answer is usually a whole set of numbers, written as an interval such as (−∞, 4]. A quadratic inequality is solved from its roots and the shape of its graph. |x| < a means −a < x < a. Some inequalities are true for every number, like x² ≥ 0 and the AM–GM inequality.
- Simultaneous Equations – Simultaneous equations are two or more equations that must be true at the same time. Their solution is the set of values that fits every equation. On a graph, each solution is a point where the graphs meet. With one straight line and one curve, put the line into the curve (substitution) to get a quadratic; its discriminant tells you if there are 2, 1 or 0 meeting points. With three linear equations, eliminate one letter at a time.
5. Geometry: polyhedra (24 h)
Polyhedral angles and polyhedra · Sections and surface areas
- 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.
- Surface Area and Volume of Cuboid, Cylinder, Cone, Pyramid and Sphere – Surface area is the total area of the outside skin of a solid, like the paper needed to wrap it. Volume is the space inside, like the water it can hold. A cuboid's volume is the number of 1 cm cubes that fit in it. A cone holds one third of a cylinder with the same base and height, and a pyramid holds one third of the matching prism.
6. Geometry: solids of revolution (21 h)
Cylinder, cone, frustum · Sphere and ball
- 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.
7. Geometry: volumes of polyhedra (16 h)
Volume of polyhedra
- 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.
8. Geometry: volumes and surfaces of round solids (16 h)
Volumes of round solids · Surface areas of round solids
- 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.