Mathematics Advanced State Examination Formula Reference Guide

Algebraic Identities

• Cube of a sum and cube of a difference:   $(a \pm b)^3 = a^3 \pm 3a^2 b + 3ab^2 \pm b^3$

• Sum and difference of cubes:   $(a \pm b)(a^2 \mp ab + b^2) = a^3 \pm b^3$

Logarithms

• Basic logarithmic identity:   $a^{\log_a(b)} = b$

• Logarithm of a product:   $\log_a(b \cdot c) = \log_a(b) + \log_a(c)$

• Logarithm of a quotient:   $\log_a\left(\frac{b}{c}\right) = \log_a(b) - \log_a(c)$

• Power rule for argument:   $\log_a(b^k) = k \cdot \log_a(b)$

• Power rule for base:   $\log_{a^k}(b) = \frac{1}{k} \cdot \log_a(b)$

• Change of base formula:   $\log_a(b) = \frac{\log_c(b)}{\log_c(a)}$

Trigonometry

Double-Angle Formulas

• Sine of double angle:   $\sin(2\alpha) = 2\sin(\alpha)\cos(\alpha)$

• Cosine of double angle:   $\cos(2\alpha) = \cos^2(\alpha) - \sin^2(\alpha)$

• Tangent of double angle:   $\tan(2\alpha) = \frac{2\tan(\alpha)}{1 - \tan^2(\alpha)}$

Sum and Difference Angle Formulas

• Sine of sum/difference:   $\sin(\alpha \pm \beta) = \sin(\alpha)\cos(\beta) \pm \cos(\alpha)\sin(\beta)$

• Cosine of sum/difference:   $\cos(\alpha \pm \beta) = \cos(\alpha)\cos(\beta) \mp \sin(\alpha)\sin(\beta)$

• Tangent of sum/difference:   $\tan(\alpha \pm \beta) = \frac{\tan(\alpha) \pm \tan(\beta)}{1 \mp \tan(\alpha)\tan(\beta)}$

Table of Trigonometric Values for Standard Angles

• Angle $\alpha = 0^\circ$ ($0\text{ rad}$):   $\sin(0^\circ) = 0$$\cos(0^\circ) = 1$$\tan(0^\circ) = 0$

• Angle $\alpha = 30^\circ$ $\left(\frac{\pi}{6}\text{ rad}\right)$:$\sin(30^\circ) = \frac{1}{2}$$\cos(30^\circ) = \frac{\sqrt{3}}{2}$$\tan(30^\circ) = \frac{\sqrt{3}}{3}$

• Angle $\alpha = 45^\circ$ $\left(\frac{\pi}{4}\text{ rad}\right)$:$\sin(45^\circ) = \frac{\sqrt{2}}{2}$$\cos(45^\circ) = \frac{\sqrt{2}}{2}$$\tan(45^\circ) = 1$

• Angle $\alpha = 60^\circ$ $\left(\frac{\pi}{3}\text{ rad}\right)$:$\sin(60^\circ) = \frac{\sqrt{3}}{2}$$\cos(60^\circ) = \frac{1}{2}$$\tan(60^\circ) = \sqrt{3}$

• Angle $\alpha = 90^\circ$ $\left(\frac{\pi}{2}\text{ rad}\right)$:$\sin(90^\circ) = 1$$\cos(90^\circ) = 0$$\tan(90^\circ)$ is undefined ($-$)

Basic Trigonometric Equations

• Sine equation:   If $\sin(x) = a$, where $a \in [-1; 1]$, then:   $x = (-1)^k \arcsin(a) + \pi k, \quad k \in \mathbb{Z}$

• Cosine equation:   If $\cos(x) = a$, where $a \in [-1; 1]$, then:   $x = \pm \arccos(a) + 2\pi k, \quad k \in \mathbb{Z}$

• Tangent equation:   If $\tan(x) = a$, where $a \in \mathbb{R}$, then:   $x = \arctan(a) + \pi k, \quad k \in \mathbb{Z}$

Progressions

Arithmetic Progression

• $n$-th term formula:   $a_n = a_1 + d(n - 1)$

• Common difference:   $d = a_{n+1} - a_n$

• Sum of the first $n$ terms:   $S_n = \frac{a_1 + a_n}{2} \cdot n = \frac{2a_1 + d(n - 1)}{2} \cdot n$

• Variable definitions: $a_n$ is the $n$-th term, $d$ is the common difference, $n$ is the term position/index, and $S_n$ is the sum of the first $n$ terms.

Geometric Progression

• $n$-th term formula:   $b_n = b_1 \cdot q^{n-1}$

• Common ratio ($q \neq 0$):   $q = \frac{b_{n+1}}{b_n}$

• Sum of the first $n$ terms:   $S_n = \frac{b_1(1 - q^n)}{1 - q} = \frac{b_1 - b_n q}{1 - q}$

• Sum of an infinitely decreasing geometric progression ($|q| < 1$):   $S = \frac{b_1}{1 - q}$

• Variable definitions: $b_n$ is the $n$-th term, $q$ is the common ratio ($q \neq 0$), $n$ is the term index, $S_n$ is the sum of the first $n$ terms, and $S$ is the sum of the infinitely decreasing geometric progression.

Vectors

• Magnitude (length) of vector $\mathbf{a} = (x_1; y_1)$:   $|\mathbf{a}| = \sqrt{x_1^2 + y_1^2}$

• Dot product of vectors $\mathbf{a} = (x_1; y_1)$ and $\mathbf{b} = (x_2; y_2)$:   $\mathbf{a} \cdot \mathbf{b} = x_1 x_2 + y_1 y_2 = |\mathbf{a}| \cdot |\mathbf{b}| \cdot \cos(\alpha)$

• Variable definitions: $|\mathbf{a}|$ is vector magnitude, $(x_1; y_1)$ and $(x_2; y_2)$ are vector coordinates, and $\alpha$ is the angle between the two vectors.

Triangle Geometry

• Law of Cosines:   $a^2 = b^2 + c^2 - 2bc \cdot \cos(A)$

• Law of Sines:   $\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)} = 2R$

• Area of a Triangle Formulas:

  • Using two sides and the included angle:     $S = \frac{1}{2} ab \cdot \sin(C)$
  • Heron's formula:     $S = \sqrt{p(p - a)(p - b)(p - c)}$
  • Using inradius $r$:     $S = r \cdot p$
  • Using circumradius $R$:     $S = \frac{abc}{4R}$

• Variable definitions: $a, b, c$ are the side lengths; $A, B, C$ (or $\angle A, \angle B, \angle C$) are the opposite angles; $p = \frac{a + b + c}{2}$ is the semi-perimeter; $r$ is the radius of the inscribed circle (inradius); $R$ is the radius of the circumscribed circle (circumradius).

Solid Geometry (Stereometry)

Cylinder

• Lateral surface area:   $S_{\text{lat}} = 2\pi R H$

• Volume:   $V = \pi R^2 H$

• Variable definitions: $R$ is the base radius, $H$ is the height.

Cone

• Lateral surface area:   $S_{\text{lat}} = \pi R l$

• Volume:   $V = \frac{1}{3} \pi R^2 H$

• Variable definitions: $R$ is the base radius, $l$ is the slant height (generator), $H$ is the height.

Frustum of a Cone

• Lateral surface area:   $S_{\text{lat}} = \pi(R + r)l$

• Volume:   $V = \frac{1}{3} \pi H (R^2 + Rr + r^2)$

• Variable definitions: $R$ and $r$ are base radii, $l$ is the slant height, $H$ is the height.

Sphere

• Total surface area:   $S_{\text{total}} = 4\pi R^2$

• Volume:   $V = \frac{4}{3} \pi R^3$

• Variable definitions: $R$ is the sphere radius.

Spherical Segment (Cap)

• Curved surface area:   $S_{\text{curved}} = 2\pi R H$

• Volume:   $V = \frac{1}{3} \pi H^2 (3R - H)$

• Variable definitions: $R$ is the sphere radius, $H$ is the segment height.

Pyramid Volume

• Volume of a pyramid:   $V = \frac{1}{3} S H$

• Variable definitions: $S$ is the base area, $H$ is the height.

Frustum of a Pyramid Volume

• Volume of a frustum of a pyramid:   $V = \frac{1}{3} H (S_1 + \sqrt{S_1 S_2} + S_2)$

• Variable definitions: $S_1$ and $S_2$ are base areas, $H$ is the height.

Differential Calculus

Differentiation Rules

• Product rule:   $(f(x) \cdot g(x))' = f'(x) \cdot g(x) + f(x) \cdot g'(x)$

• Quotient rule:   $\left(\frac{f(x)}{g(x)}\right)' = \frac{f'(x) \cdot g(x) - f(x) \cdot g'(x)}{(g(x))^2}$

• Composite function rule (Chain rule):   $(f(g(x)))' = f'(g(x)) \cdot g'(x)$

Derivatives of Basic Functions

• Trigonometric derivatives:   $(\sin(x))' = \cos(x)$$(\cos(x))' = -\sin(x)$$(\tan(x))' = \frac{1}{\cos^2(x)}$

• Exponential and logarithmic derivatives:   $(a^x)' = a^x \cdot \ln(a)$$(\log_a(x))' = \frac{1}{x \cdot \ln(a)}$

Tangent Line Equation

• Equation of the tangent line to the function graph $y = f(x)$ at point $(x_0; f(x_0))$:   $y = f'(x_0)(x - x_0) + f(x_0)$

• Slope (direction coefficient) of the tangent line:   $k = f'(x_0)$

Integral Calculus

Indefinite Integrals

• Power function:   $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1)$

• Reciprocal function:   $\int \frac{1}{x} \, dx = \ln|x| + C$

• Exponential function:   $\int a^x \, dx = \frac{a^x}{\ln(a)} + C$

• Trigonometric functions:   $\int \sin(x) \, dx = -\cos(x) + C$$\int \cos(x) \, dx = \sin(x) + C$$\int \frac{1}{\cos^2(x)} \, dx = \tan(x) + C$

• Constant definition: $C$ represents a real constant ($C \in \mathbb{R}$).

Volume of Solid of Revolution

• Volume generated by revolving $f(x)$ around the x-axis from $a$ to $b$:   $V = \pi \int_a^b (f(x))^2 \, dx$

Combinatorics, Probability, and Statistics

Combinatorics Formulas

• Number of combinations of $n$ elements taken $k$ at a time:   $C_n^k = \frac{n!}{k!(n - k)!}$

• Number of arrangements (permutations of $k$ from $n$):   $A_n^k = \frac{n!}{(n - k)!}$

Discrete Random Variables

• Random variable $X$ values: $x_1, x_2, \dots, x_n$

• Corresponding probabilities: $p_1, p_2, \dots, p_n$

• Expected Value (Mathematical Expectation / Mean) $\mathrm{E}X$:   $\mathrm{E}X = x_1 p_1 + x_2 p_2 + \dots + x_n p_n$

• Variance $\mathrm{D}X$:   $\mathrm{D}X = (x_1 - \mathrm{E}X)^2 p_1 + (x_2 - \mathrm{E}X)^2 p_2 + \dots + (x_n - \mathrm{E}X)^2 p_n$

Binomial Experiments (Bernoulli Trials)

• Binomial probability distribution formula:   $P(X = k) = P_n(k) = C_n^k p^k q^{n - k}$

• Variable definitions: $X$ is the random variable, $n$ is the total number of trials, $k$ is the number of successes, $p$ is the probability of success in a single trial, and $q = 1 - p$ is the probability of failure.

Newton's Binomial Formula

• Binomial expansion formula:   $(a + b)^n = a^n + C_n^1 a^{n-1}b + C_n^2 a^{n-2}b^2 + \dots + C_n^k a^{n-k}b^k + \dots + b^n$

Algebrinės tapatybės

• Kubo suma ir skirtumas: $(a \pm b)^3 = a^3 \pm 3a^2 b + 3ab^2 \pm b^3$
  • 1 pavyzdys: $(x + 2)^3 = x^3 + 3 \cdot x^2 \cdot 2 + 3 \cdot x \cdot 2^2 + 2^3 = x^3 + 6x^2 + 12x + 8$
  • 2 pavyzdys: $(2x - 3)^3 = (2x)^3 - 3 \cdot (2x)^2 \cdot 3 + 3 \cdot (2x) \cdot 3^2 - 3^3 = 8x^3 - 36x^2 + 54x - 27$
• Kubų suma ir skirtumas: $(a \pm b)(a^2 \mp ab + b^2) = a^3 \pm b^3$
  • 1 pavyzdys: $(x + 3)(x^2 - 3x + 9) = x^3 + 3^3 = x^3 + 27$
  • 2 pavyzdys: $(2x - 1)(4x^2 + 2x + 1) = (2x)^3 - 1^3 = 8x^3 - 1$

Logaritmai

• Pagrindinė logaritmo tapatybė: $a^{\log_a(b)} = b$
  • 1 pavyzdys: $2^{\log_2(8)} = 8$
  • 2 pavyzdys: $5^{\log_5(x + 1)} = x + 1$
• Sandaugos logaritmas: $\log<em>a(b \cdot c) = \log</em>a(b) + \log_a(c)$
  • 1 pavyzdys: $\log<em>2(4 \cdot 8) = \log</em>2(4) + \log_2(8) = 2 + 3 = 5$
  • 2 pavyzdys: $\log<em>3(9x) = \log</em>3(9) + \log<em>3(x) = 2 + \log</em>3(x)$
• Dalmens logaritmas: $\log<em>a\left(\frac{b}{c}\right) = \log</em>a(b) - \log_a(c)$
  • 1 pavyzdys: $\log<em>5\left(\frac{125}{5}\right) = \log</em>5(125) - \log_5(5) = 3 - 1 = 2$
• Argumento rodiklio taisyklė: $\log<em>a(b^k) = k \cdot \log</em>a(b)$
  • 1 pavyzdys: $\log<em>2(8^4) = 4 \cdot \log</em>2(8) = 4 \cdot 3 = 12$
• Pagrindo rodiklio taisyklė: $\log<em>{a^k}(b) = \frac{1}{k} \cdot \log</em>a(b)$
  • 1 pavyzdys: $\log<em>8(2) = \log</em>{2^3}(2) = \frac{1}{3} \cdot \log_2(2) = \frac{1}{3}$
• Pagrindo keitimo formulė: $\log<em>a(b) = \frac{\log</em>c(b)}{\log_c(a)}$
  • 1 pavyzdys: $\log<em>4(8) = \frac{\log</em>2(8)}{\log_2(4)} = \frac{3}{2} = 1{,}5$

Trigonometrija

Dvigubojo kampo formulės

• Dvigubo kampo sinusas: $\sin(2\alpha) = 2\sin(\alpha)\cos(\alpha)$
  • 1 pavyzdys: Jei $\sin(\alpha) = \frac{3}{5}$ ir $\cos(\alpha) = \frac{4}{5}$, tai $\sin(2\alpha) = 2 \cdot \frac{3}{5} \cdot \frac{4}{5} = \frac{24}{25}$
• Dvigubo kampo kosinusas: $\cos(2\alpha) = \cos^2(\alpha) - \sin^2(\alpha)$
  • 1 pavyzdys: Jei $\cos(\alpha) = \frac{4}{5}$ ir $\sin(\alpha) = \frac{3}{5}$, tai $\cos(2\alpha) = \left(\frac{4}{5}\right)^2 - \left(\frac{3}{5}\right)^2 = \frac{16}{25} - \frac{9}{25} = \frac{7}{25}$
• Dvigubo kampo tangentas: $\tan(2\alpha) = \frac{2\tan(\alpha)}{1 - \tan^2(\alpha)}$
  • 1 pavyzdys: Jei $\tan(\alpha) = 3$, tai $\tan(2\alpha) = \frac{2 \cdot 3}{1 - 3^2} = \frac{6}{-8} = -\frac{3}{4}$

Kampų sumos ir skirtumo formulės

• Kampų sumos ir skirtumo sinusas: $\sin(\alpha \pm \beta) = \sin(\alpha)\cos(\beta) \pm \cos(\alpha)\sin(\beta)$
  • 1 pavyzdys: $\sin(75^\circ) = \sin(45^\circ + 30^\circ) = \sin(45^\circ)\cos(30^\circ) + \cos(45^\circ)\sin(30^\circ) = \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2} = \frac{\sqrt{6} + \sqrt{2}}{4}$
• Kampų sumos ir skirtumo kosinusas: $\cos(\alpha \pm \beta) = \cos(\alpha)\cos(\beta) \mp \sin(\alpha)\sin(\beta)$
  • 1 pavyzdys: $\cos(15^\circ) = \cos(45^\circ - 30^\circ) = \cos(45^\circ)\cos(30^\circ) + \sin(45^\circ)\sin(30^\circ) = \frac{\sqrt{6} + \sqrt{2}}{4}$
• Kampų sumos ir skirtumo tangentas: $\tan(\alpha \pm \beta) = \frac{\tan(\alpha) \pm \tan(\beta)}{1 \mp \tan(\alpha)\tan(\beta)}$
  • 1 pavyzdys: $\tan(75^\circ) = \tan(45^\circ + 30^\circ) = \frac{1 + \frac{\sqrt{3}}{3}}{1 - 1 \cdot \frac{\sqrt{3}}{3}} = 2 + \sqrt{3}$

Pagrindinių kampų trigonometrinių reikšmių lentelė

• Kampas $\alpha = 0^\circ$ ($0\text{ rad}$): $\sin(0^\circ) = 0$, $\cos(0^\circ) = 1$, $\tan(0^\circ) = 0$
• Kampas $\alpha = 30^\circ$ $\left(\frac{\pi}{6}\text{ rad}\right)$: $\sin(30^\circ) = \frac{1}{2}$, $\cos(30^\circ) = \frac{\sqrt{3}}{2}$, $\tan(30^\circ) = \frac{\sqrt{3}}{3}$
• Kampas $\alpha = 45^\circ$ $\left(\frac{\pi}{4}\text{ rad}\right)$: $\sin(45^\circ) = \frac{\sqrt{2}}{2}$, $\cos(45^\circ) = \frac{\sqrt{2}}{2}$, $\tan(45^\circ) = 1$
• Kampas $\alpha = 60^\circ$ $\left(\frac{\pi}{3}\text{ rad}\right)$: $\sin(60^\circ) = \frac{\sqrt{3}}{2}$, $\cos(60^\circ) = \frac{1}{2}$, $\tan(60^\circ) = \sqrt{3}$
• Kampas $\alpha = 90^\circ$ $\left(\frac{\pi}{2}\text{ rad}\right)$: $\sin(90^\circ) = 1$, $\cos(90^\circ) = 0$, $\tan(90^\circ)$ neapibrėžtas

Pagrindinės trigonometrinės lygtys

• Sinuso lygtis: Jei $\sin(x) = a$, kai $a \in [-1; 1]$, tai: $x = (-1)^k \arcsin(a) + \pi k, \quad k \in \mathbb{Z}$
  • 1 pavyzdys: $\sin(x) = \frac{1}{2} \Rightarrow x = (-1)^k \frac{\pi}{6} + \pi k, \quad k \in \mathbb{Z}$
• Kosinuso lygtis: Jei $\cos(x) = a$, kai $a \in [-1; 1]$, tai: $x = \pm \arccos(a) + 2\pi k, \quad k \in \mathbb{Z}$
  • 1 pavyzdys: $\cos(x) = \frac{\sqrt{2}}{2} \Rightarrow x = \pm \frac{\pi}{4} + 2\pi k, \quad k \in \mathbb{Z}$
• Tangento lygtis: Jei $\tan(x) = a$, kai $a \in \mathbb{R}$, tai: $x = \arctan(a) + \pi k, \quad k \in \mathbb{Z}$
  • 1 pavyzdys: $\tan(x) = 1 \Rightarrow x = \frac{\pi}{4} + \pi k, \quad k \in \mathbb{Z}$

Progresijos

Aritmetinė progresija

• $n$–ojo nario formulė: $a<em>n = a</em>1 + d(n - 1)$
  • 1 pavyzdys: Jei $a<em>1 = 3$ ir $d = 4$, tai $a</em>5 = 3 + 4(5 - 1) = 19$
• Skirtumas: $d = a<em>{n+1} - a</em>n$
  • 1 pavyzdys: Jei $a<em>1 = 2$ ir $a</em>2 = 7$, tai $d = 7 - 2 = 5$
• Pirmųjų $n$ narių suma: $S<em>n = \frac{a</em>1 + a<em>n}{2} \cdot n = \frac{2a</em>1 + d(n - 1)}{2} \cdot n$
  • 1 pavyzdys: Kai $a<em>1 = 2$ ir $a</em>{10} = 20$: $S_{10} = \frac{2 + 20}{2} \cdot 10 = 110$

Geometrinė progresija

• $n$–ojo nario formulė: $b<em>n = b</em>1 \cdot q^{n-1}$
  • 1 pavyzdys: Jei $b<em>1 = 2$ ir $q = 3$, tai $b</em>4 = 2 \cdot 3^3 = 54$
• Vardiklis ($q \neq 0$): $q = \frac{b<em>{n+1}}{b</em>n}$
  • 1 pavyzdys: Jei $b<em>1 = 5$ ir $b</em>2 = 10$, tai $q = \frac{10}{5} = 2$
• Pirmųjų $n$ narių suma: $S<em>n = \frac{b</em>1(1 - q^n)}{1 - q} = \frac{b<em>1 - b</em>n q}{1 - q}$
  • 1 pavyzdys: Jei $b<em>1 = 3$, $q = 2$, $n = 4$: $S</em>4 = \frac{3(1 - 2^4)}{1 - 2} = 45$
• Nykstamosios geometrinės progresijos suma ($|q| < 1$): $S = \frac{b_1}{1 - q}$
  • 1 pavyzdys: Jei $b_1 = 6$ ir $q = 0{,}5$, tai $S = \frac{6}{1 - 0{,}5} = 12$

Vektoriai

• Vektoriaus $\mathbf{a} = (x<em>1; y</em>1)$ ilgis: $|\mathbf{a}| = \sqrt{x<em>1^2 + y</em>1^2}$
  • 1 pavyzdys: Jei $\mathbf{a} = (3; 4)$, tai $|\mathbf{a}| = \sqrt{3^2 + 4^2} = 5$
• Skaliarinė sandauga: $\mathbf{a} \cdot \mathbf{b} = x<em>1 x</em>2 + y<em>1 y</em>2 = |\mathbf{a}| \cdot |\mathbf{b}| \cdot \cos(\alpha)$
  • 1 pavyzdys: Jei $\mathbf{a} = (1; 2)$ ir $\mathbf{b} = (3; 4)$, tai $\mathbf{a} \cdot \mathbf{b} = 1 \cdot 3 + 2 \cdot 4 = 11$

Trikampio geometrija

• Kosinusų teorema: $a^2 = b^2 + c^2 - 2bc \cdot \cos(A)$
  • 1 pavyzdys: Jei $b = 3$, $c = 4$ ir $\angle A = 60^\circ$, tai $a^2 = 3^2 + 4^2 - 2 \cdot 3 \cdot 4 \cdot \frac{1}{2} = 13 \Rightarrow a = \sqrt{13}$
• Sinusų teorema: $\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)} = 2R$
  • 1 pavyzdys: Jei $a = 6$ ir $\angle A = 30^\circ$, tai $2R = \frac{6}{\sin(30^\circ)} = 12 \Rightarrow R = 6$
• Trikampio ploto formulės:
  • Pagal dvi kraštines ir kampą: $S = \frac{1}{2} ab \cdot \sin(C)$
  • Pavyzdys: Jei $a = 4$, $b = 5$, $\angle C = 30^\circ$, tai $S = \frac{1}{2} \cdot 4 \cdot 5 \cdot \frac{1}{2} = 5$
  • Herono formulė: $S = \sqrt{p(p - a)(p - b)(p - c)}$, kur $p = \frac{a + b + c}{2}$
  • Pavyzdys: Jei $a = 3$, $b = 4$, $c = 5$, tai $p = 6$, $S = \sqrt{6(3)(2)(1)} = 6$
  • Naudojant įbrėžtinio apskritimo spindulį $r$: $S = r \cdot p$
  • Naudojant apibrėžtinio apskritimo spindulį $R$: $S = \frac{abc}{4R}$

Erdvės geometrija (Stereometrija)

Ritinys

• Šoninio paviršiaus plotas: $S_{\text{šon}} = 2\pi R H$
• Tūris: $V = \pi R^2 H$
  • 1 pavyzdys: Jei $R = 3$ ir $H = 5$, tai $V = \pi \cdot 3^2 \cdot 5 = 45\pi$

Kūgis

• Šoninio paviršiaus plotas: $S_{\text{šon}} = \pi R l$
• Tūris: $V = \frac{1}{3} \pi R^2 H$
  • 1 pavyzdys: Jei $R = 3$ ir $H = 4$, tai $V = \frac{1}{3} \pi \cdot 3^2 \cdot 4 = 12\pi$

Nupjautinis kūgis

• Šoninio paviršiaus plotas: $S_{\text{šon}} = \pi(R + r)l$
• Tūris: $V = \frac{1}{3} \pi H (R^2 + Rr + r^2)$

Rutulys

• Paviršiaus plotas: $S_{\text{pav}} = 4\pi R^2$
• Tūris: $V = \frac{4}{3} \pi R^3$
  • 1 pavyzdys: Jei $R = 3$, tai $V = \frac{4}{3} \pi \cdot 3^3 = 36\pi$

Rutulio nuopjova

• Paviršiaus plotas: $S_{\text{nuopj}} = 2\pi R H$
• Tūris: $V = \frac{1}{3} \pi H^2 (3R - H)$

Piramidė

• Tūris: $V = \frac{1}{3} S H$
• Nupjautinės piramidės tūris: $V = \frac{1}{3} H (S<em>1 + \sqrt{S</em>1 S<em>2} + S</em>2)$

Diferencialinis skaičiavimas

Išvestinių taisyklės

• Sandaugos išvestinė: $(f(x) \cdot g(x))' = f'(x) \cdot g(x) + f(x) \cdot g'(x)$
  • 1 pavyzdys: $(x \cdot \sin(x))' = \sin(x) + x\cos(x)$
• Dalmens išvestinė: \left(\frac{f(x)}{g(x)}\right)' = \frac{f'(x) \cdot g(x) - f(x)
  \cdot g'(x)}{(g(x))^2}
• Sudėtinės funkcijos išvestinė: $(f(g(x)))' = f'(g(x)) \cdot g'(x)$
  • 1 pavyzdys: $(\sin(3x))' = 3\cos(3x)$

Pagrindinių funkcijų išvestinės

• Trigonometrinės:
  $(\sin(x))' = \cos(x)$
  $(\cos(x))' = -\sin(x)$
  $(\tan(x))' = \frac{1}{\cos^2(x)}$
• Rodiklinės ir logaritminės:
  $(a^x)' = a^x \cdot \ln(a)$
  $(\log_a(x))' = \frac{1}{x \cdot \ln(a)}$

Liestinės lygtis

• Liestinės lygtis taške $(x<em>0; f(x</em>0))$: $y = f'(x<em>0)(x - x</em>0) + f(x_0)$
• Liestinės krypties koeficientas: $k = f'(x_0)$

Integralinė matematika

Neapibrėžtiniai integralai

• Laipsninė funkcija: $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1)$
  • 1 pavyzdys: $\int x^2 \, dx = \frac{x^3}{3} + C$
• Racionalioji funkcija: $\int \frac{1}{x} \, dx = \ln|x| + C$
• Rodiklinė funkcija: $\int a^x \, dx = \frac{a^x}{\ln(a)} + C$
• Trigonometrinės funkcijos:
  $\int \sin(x) \, dx = -\cos(x) + C$
  $\int \cos(x) \, dx = \sin(x) + C$

Sukinio tūris

• Tūris sukant apie x ašį nuo $a$ iki $b$: $V = \pi \int_a^b (f(x))^2 \, dx$

Kombinatorika, tikimybės ir statistika

Kombinatorikos formulės

• Deriniai: $C_n^k = \frac{n!}{k!(n - k)!}$
  • 1 pavyzdys: $C_5^2 = \frac{5!}{2! \cdot 3!} = 10$
• Gretiniai: $A_n^k = \frac{n!}{(n - k)!}$
  • 1 pavyzdys: $A_5^2 = \frac{5!}{3!} = 20$

Matematinė viltis ir dispersija

• Matematinė viltis: $\mathrm{E}X = x<em>1 p</em>1 + x<em>2 p</em>2 + \dots + x<em>n p</em>n$
• Dispersija: $\mathrm{D}X = (x<em>1 - \mathrm{E}X)^2 p</em>1 + (x<em>2 - \mathrm{E}X)^2 p</em>2 + \dots + (x<em>n - \mathrm{E}X)^2 p</em>n$

Binominiai bandymai

• Bernulio formulė: $P(X = k) = P<em>n(k) = C</em>n^k p^k q^{n - k}$