Državna Matura Math A flashcards

32 cards. Read the term and the definition here. The deck at the top of this page is optional.

Quadratic formula

For ax^2 + bx + c = 0 with a ≠ 0, the solutions are x = (-b ± √(b^2 - 4ac)) / (2a). The discriminant D = b^2 - 4ac determines the number of real roots.

Absolute value inequality

For a ≥ 0, |x| < a is equivalent to -a < x < a, and |x| > a is equivalent to x < -a or x > a. These describe intervals on the number line.

Logarithm product rule

For positive bases a ≠ 1 and positive x, y: log_a(xy) = log_a x + log_a y. This turns multiplication into addition and is used to solve exponential equations.

Binomial coefficient

C(n,k) = n! / (k!(n-k)!) counts the number of ways to choose k objects from n distinct objects without regard to order. It appears in the binomial expansion.

Vertex of parabola

The graph of f(x) = ax^2 + bx + c (a ≠ 0) has vertex at x = -b/(2a). The parabola opens upward if a > 0 and downward if a < 0.

Exponential function properties

f(x) = a^x with a > 0, a ≠ 1 has domain all real numbers, range (0, ∞), passes through (0,1), and is increasing if a > 1 or decreasing if 0 < a < 1.

Inverse function

A function f has an inverse f^{-1} exactly when it is one-to-one. The graph of f^{-1} is the reflection of the graph of f across the line y = x.

Graph transformation f(x - c)

Replacing x by x - c in f(x) shifts the graph horizontally by c units to the right if c > 0, or to the left if c < 0. The shape of the graph is unchanged.

Sine rule

In any triangle with sides a, b, c opposite angles A, B, C: a/sin A = b/sin B = c/sin C = 2R, where R is the circumradius of the triangle.

Cosine rule

In a triangle, c^2 = a^2 + b^2 - 2ab cos C, where C is the angle between sides a and b. It generalises the Pythagorean theorem to any triangle.

Pythagorean identity

For any angle θ, sin^2 θ + cos^2 θ = 1. Dividing by cos^2 θ gives 1 + tan^2 θ = sec^2 θ, valid where cos θ ≠ 0.

Sphere volume and surface

A sphere of radius r has volume V = (4/3)πr^3 and surface area S = 4πr^2. These formulas are used in solid geometry problems involving balls and domes.

Arithmetic sequence

A sequence in which each term after the first is obtained by adding a constant difference d: a_n = a_1 + (n-1)d. The sum of the first n terms is S_n = (n/2)(a_1 + a_n).

Geometric sequence

A sequence in which each term after the first is obtained by multiplying by a constant ratio q: a_n = a_1 q^{n-1}. The sum of the first n terms is S_n = a_1(1 - q^n)/(1 - q) for q ≠ 1.

Infinite geometric series

The sum a_1 + a_1 q + a_1 q^2 + ... converges if and only if |q| < 1, and then its sum is S = a_1/(1 - q). If |q| ≥ 1, the series diverges.

Sum of first n integers

Σ_{k=1}^n k = n(n+1)/2. This formula is frequently used in arithmetic series and in counting problems involving consecutive integers.

Distance between two points

For points A(x1,y1) and B(x2,y2), the distance is d = √((x2-x1)^2 + (y2-y1)^2). The midpoint has coordinates ((x1+x2)/2, (y1+y2)/2).

Perpendicular lines slopes

Two non-vertical lines are perpendicular if and only if the product of their slopes is -1: k1 k2 = -1. A horizontal line (slope 0) is perpendicular to a vertical line (undefined slope).

Circle equation

The circle with centre (p,q) and radius r has equation (x-p)^2 + (y-q)^2 = r^2. Expanded, it becomes x^2 + y^2 + Dx + Ey + F = 0.

Dot product of vectors

For u = (u1,u2) and v = (v1,v2), the dot product is u·v = u1 v1 + u2 v2 = |u||v| cos θ, where θ is the angle between the vectors. It is zero for perpendicular vectors.

Combinations formula

The number of ways to choose k objects from n distinct objects, order not important, is C(n,k) = n!/(k!(n-k)!). It is used in probability and binomial expansion.

Addition rule probability

For any events A and B, P(A ∪ B) = P(A) + P(B) - P(A ∩ B). If A and B are mutually exclusive, then P(A ∩ B) = 0 and the rule simplifies.

Conditional probability

P(A|B) = P(A ∩ B)/P(B), provided P(B) > 0. If A and B are independent, then P(A|B) = P(A) and P(A ∩ B) = P(A)P(B).

Standard deviation

The standard deviation is the square root of the variance. Population variance is σ^2 = Σ(x_i - μ)^2/n; sample variance uses n-1 in the denominator. It measures data spread.

Binomial probability

In n independent trials with success probability p, the probability of exactly k successes is C(n,k) p^k (1-p)^{n-k}. This is the binomial distribution.

Derivative definition

The derivative of f at x is f'(x) = lim_{h→0} (f(x+h) - f(x))/h, when this limit exists. It gives the slope of the tangent line and the instantaneous rate of change.

Chain rule

If y = f(g(x)), then dy/dx = f'(g(x)) · g'(x). The chain rule is used to differentiate composite functions such as sin(2x) or e^{x^2}.

First derivative test

At a critical point c where f'(c) = 0 or is undefined, if f' changes from positive to negative, f has a local maximum at c; from negative to positive gives a local minimum.

Power rule integration

For n ≠ -1, ∫ x^n dx = x^{n+1}/(n+1) + C. For n = -1, ∫ (1/x) dx = ln|x| + C. The constant C represents any antiderivative.

Fundamental Theorem of Calculus

If F is an antiderivative of f on [a,b], then ∫_a^b f(x) dx = F(b) - F(a). It connects definite integrals with antiderivatives and allows area computation.

Area between curves

If f(x) ≥ g(x) on [a,b], the area between their graphs from x = a to x = b is ∫_a^b (f(x) - g(x)) dx. The integrand is the vertical distance between the curves.

L'Hôpital's rule

For limits of the form 0/0 or ∞/∞, lim f(x)/g(x) = lim f'(x)/g'(x), provided the latter limit exists. It is used to evaluate indeterminate forms.