Priprema za prijemni
iz matematike

$$\begin{gathered} a^m\cdot a^n=a^{m+n} \\[5pt] \dfrac{a^m}{a^n}=a^{m-n} \\[5pt] (a^m)^n=a^{mn} \\[5pt] (ab)^n=a^n b^n,\quad \Big(\tfrac{a}{b}\Big)^n=\dfrac{a^n}{b^n} \\[5pt] a^0=1,\quad a^{-n}=\dfrac{1}{a^n} \end{gathered}$$

$$\begin{gathered} \sqrt[n]{a}=a^{1/n},\quad \sqrt[n]{a^m}=a^{m/n} \\[5pt] \sqrt[n]{ab}=\sqrt[n]{a}\,\sqrt[n]{b} \\[5pt] \sqrt[n]{\dfrac{a}{b}}=\dfrac{\sqrt[n]{a}}{\sqrt[n]{b}} \end{gathered}$$

$$\begin{gathered} (a\pm b)^2=a^2\pm 2ab+b^2 \\[5pt] (a\pm b)^3=a^3\pm 3a^2b+3ab^2\pm b^3 \\[5pt] a^2-b^2=(a-b)(a+b) \\[5pt] a^3\pm b^3=(a\pm b)(a^2\mp ab+b^2) \end{gathered}$$

$$ax+b=0 \;\Rightarrow\; x=-\dfrac{b}{a}\quad (a\neq 0)$$

$$\begin{gathered} ax^2+bx+c=0 \\[5pt] x_{1,2}=\dfrac{-b\pm\sqrt{D}}{2a},\quad D=b^2-4ac \end{gathered}$$

$$\begin{gathered} x_1+x_2=-\dfrac{b}{a},\quad x_1 x_2=\dfrac{c}{a} \\[5pt] ax^2+bx+c=a(x-x_1)(x-x_2) \end{gathered}$$

$$\sqrt{f(x)}=g(x)\iff f(x)=g(x)^2,\; g(x)\ge 0$$

$$a^{f(x)}=a^{g(x)}\iff f(x)=g(x)\quad (a>0,\ a\neq 1)$$

$$\begin{gathered} \log_a b=c \iff a^c=b \\[5pt] \log_a a=1,\quad \log_a 1=0,\quad a^{\log_a x}=x \end{gathered}$$

$$\begin{gathered} \log_a(xy)=\log_a x+\log_a y \\[5pt] \log_a\dfrac{x}{y}=\log_a x-\log_a y \\[5pt] \log_a x^k=k\log_a x \\[5pt] \log_a x=\dfrac{\log_b x}{\log_b a} \end{gathered}$$

$$a>1:\ \text{raste};\qquad 0

Pri \(0

$$\begin{gathered} \sin^2\alpha+\cos^2\alpha=1 \\[5pt] \operatorname{tg}\alpha=\dfrac{\sin\alpha}{\cos\alpha},\quad \operatorname{ctg}\alpha=\dfrac{\cos\alpha}{\sin\alpha} \\[5pt] 1+\operatorname{tg}^2\alpha=\dfrac{1}{\cos^2\alpha} \end{gathered}$$

$$\begin{gathered} \sin(x\pm y)=\sin x\cos y\pm\cos x\sin y \\[5pt] \cos(x\pm y)=\cos x\cos y\mp\sin x\sin y \\[5pt] \operatorname{tg}(x\pm y)=\dfrac{\operatorname{tg}x\pm\operatorname{tg}y}{1\mp\operatorname{tg}x\,\operatorname{tg}y} \end{gathered}$$

$$\begin{gathered} \sin 2x=2\sin x\cos x \\[5pt] \cos 2x=\cos^2 x-\sin^2 x \\[5pt] \operatorname{tg}2x=\dfrac{2\operatorname{tg}x}{1-\operatorname{tg}^2 x} \\[5pt] \sin^2\tfrac{x}{2}=\dfrac{1-\cos x}{2},\quad \cos^2\tfrac{x}{2}=\dfrac{1+\cos x}{2} \end{gathered}$$

$$\begin{gathered} \sin x+\sin y=2\sin\tfrac{x+y}{2}\cos\tfrac{x-y}{2} \\[5pt] \sin x-\sin y=2\cos\tfrac{x+y}{2}\sin\tfrac{x-y}{2} \\[5pt] \cos x+\cos y=2\cos\tfrac{x+y}{2}\cos\tfrac{x-y}{2} \\[5pt] \cos x-\cos y=-2\sin\tfrac{x+y}{2}\sin\tfrac{x-y}{2} \end{gathered}$$

$$\begin{gathered} \sin x\cos y=\tfrac{1}{2}\big[\sin(x-y)+\sin(x+y)\big] \\[5pt] \sin x\sin y=\tfrac{1}{2}\big[\cos(x-y)-\cos(x+y)\big] \\[5pt] \cos x\cos y=\tfrac{1}{2}\big[\cos(x-y)+\cos(x+y)\big] \end{gathered}$$

$$\begin{gathered} \sin x=\dfrac{2\operatorname{tg}\frac{x}{2}}{1+\operatorname{tg}^2\frac{x}{2}} \\[5pt] \cos x=\dfrac{1-\operatorname{tg}^2\frac{x}{2}}{1+\operatorname{tg}^2\frac{x}{2}} \end{gathered}$$

$$\begin{gathered} \sin x=a\Rightarrow x=(-1)^k\arcsin a+k\pi \\[5pt] \cos x=a\Rightarrow x=\pm\arccos a+2k\pi \\[5pt] \operatorname{tg}x=a\Rightarrow x=\operatorname{arctg}a+k\pi \end{gathered}$$

$$\vec a=(a_x,a_y,a_z),\quad |\vec a|=\sqrt{a_x^2+a_y^2+a_z^2}$$

$$\begin{gathered} \vec a\cdot\vec b=|\vec a|\,|\vec b|\cos\varphi \\[5pt] \vec a\cdot\vec b=a_xb_x+a_yb_y+a_zb_z \\[5pt] \cos\varphi=\dfrac{\vec a\cdot\vec b}{|\vec a|\,|\vec b|},\quad \vec a\perp\vec b\iff \vec a\cdot\vec b=0 \end{gathered}$$

$$|\vec a\times\vec b|=|\vec a|\,|\vec b|\sin\varphi$$

$$\begin{gathered} d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2} \\[5pt] S=\Big(\dfrac{x_1+x_2}{2},\ \dfrac{y_1+y_2}{2}\Big) \end{gathered}$$

$$\begin{gathered} y=kx+n,\quad y-y_1=k(x-x_1) \\[5pt] k=\operatorname{tg}\alpha=\dfrac{y_2-y_1}{x_2-x_1} \end{gathered}$$

$$\begin{gathered} d=\dfrac{|Ax_0+By_0+C|}{\sqrt{A^2+B^2}} \\[5pt] (x-p)^2+(y-q)^2=r^2 \end{gathered}$$

$$\begin{gathered} a^2+b^2=c^2\ (\text{pravougli}) \\[5pt] P=\dfrac{a\,h_a}{2}=\dfrac{1}{2}ab\sin\gamma \\[5pt] P=\sqrt{s(s-a)(s-b)(s-c)} \\[5pt] s=\dfrac{a+b+c}{2} \end{gathered}$$

$$\begin{gathered} \dfrac{a}{\sin\alpha}=\dfrac{b}{\sin\beta}=\dfrac{c}{\sin\gamma}=2R \\[5pt] c^2=a^2+b^2-2ab\cos\gamma \end{gathered}$$

$$\begin{gathered} P_{\text{trapez}}=\dfrac{a+b}{2}\,h \\[5pt] P_{\square}=a^2,\quad P_{\text{pravoug.}}=ab \\[5pt] O=2r\pi,\quad P_{\text{krug}}=r^2\pi \end{gathered}$$

$$\begin{gathered} V_{\text{prizma}}=B\,H,\quad V_{\text{kvadar}}=abc \\[5pt] V_{\text{kocka}}=a^3,\quad P_{\text{kocka}}=6a^2 \end{gathered}$$

$$\begin{gathered} V_{\text{piramida}}=\dfrac{1}{3}B\,H \\[5pt] V_{\text{kupa}}=\dfrac{1}{3}r^2\pi H,\quad M_{\text{kupa}}=r\pi s \end{gathered}$$

$$\begin{gathered} V_{\text{valjak}}=r^2\pi H,\quad P_{\text{valjak}}=2r\pi(r+H) \\[5pt] V_{\text{lopta}}=\dfrac{4}{3}r^3\pi,\quad P_{\text{lopta}}=4r^2\pi \end{gathered}$$

$$\begin{gathered} a_n=a_1+(n-1)d \\[5pt] S_n=\dfrac{n}{2}(a_1+a_n)=\dfrac{n}{2}\big(2a_1+(n-1)d\big) \end{gathered}$$

$$\begin{gathered} a_n=a_1\,q^{\,n-1} \\[5pt] S_n=a_1\dfrac{q^n-1}{q-1}\quad (q\neq 1) \end{gathered}$$

$$S=\dfrac{a_1}{1-q},\quad |q|<1$$

$$\begin{gathered} P_n=n!,\quad V_n^k=\dfrac{n!}{(n-k)!} \\[5pt] \binom{n}{k}=\dfrac{n!}{k!\,(n-k)!} \end{gathered}$$

$$\text{1) baza }P(1);\quad \text{2) }P(n)\Rightarrow P(n+1)$$

$$\begin{gathered} (a+b)^n=\sum_{k=0}^{n}\binom{n}{k}a^{n-k}b^k \\[5pt] T_{k+1}=\binom{n}{k}a^{n-k}b^k \end{gathered}$$

$$a:b=c:d\iff a\,d=b\,c$$

$$\begin{gathered} p\%\ \text{od}\ x=\dfrac{p}{100}\cdot x \\[5pt] \text{uvećanje: } x\Big(1+\dfrac{p}{100}\Big) \\[5pt] \text{umanjenje: } x\Big(1-\dfrac{p}{100}\Big) \end{gathered}$$

$$\begin{gathered} \lim_{n\to\infty}\Big(1+\dfrac{1}{n}\Big)^n=e \\[5pt] \lim_{x\to 0}\dfrac{\sin x}{x}=1 \end{gathered}$$

$$f'(x)=\lim_{h\to0}\dfrac{f(x+h)-f(x)}{h}$$

$$\begin{gathered} (c)'=0,\quad (x)'=1,\quad (x^\alpha)'=\alpha x^{\alpha-1} \\[5pt] (a^x)'=a^x\ln a,\quad (e^x)'=e^x \\[5pt] (\log_a x)'=\dfrac{1}{x\ln a},\quad (\ln|x|)'=\dfrac{1}{x} \\[5pt] (\sin x)'=\cos x,\quad (\cos x)'=-\sin x \\[5pt] (\operatorname{tg}x)'=\dfrac{1}{\cos^2 x},\quad (\operatorname{ctg}x)'=-\dfrac{1}{\sin^2 x} \\[5pt] (\arcsin x)'=\dfrac{1}{\sqrt{1-x^2}},\quad (\arccos x)'=-\dfrac{1}{\sqrt{1-x^2}} \\[5pt] (\operatorname{arctg}x)'=\dfrac{1}{1+x^2},\quad (\operatorname{arcctg}x)'=-\dfrac{1}{1+x^2} \end{gathered}$$

$$\begin{gathered} (fg)'=f'g+fg' \\[5pt] \Big(\dfrac{f}{g}\Big)'=\dfrac{f'g-fg'}{g^2} \\[5pt] \big(f(g(x))\big)'=f'(g)\cdot g' \end{gathered}$$

$$\begin{gathered} \int dx=x+C,\quad \int x^n\,dx=\dfrac{x^{n+1}}{n+1}+C\ (n\neq-1) \\[5pt] \int\dfrac{dx}{x}=\ln|x|+C \\[5pt] \int e^x\,dx=e^x+C,\quad \int a^x\,dx=\dfrac{a^x}{\ln a}+C \\[5pt] \int\sin x\,dx=-\cos x+C,\quad \int\cos x\,dx=\sin x+C \\[5pt] \int\dfrac{dx}{\cos^2 x}=\operatorname{tg}x+C,\quad \int\dfrac{dx}{\sin^2 x}=-\operatorname{ctg}x+C \end{gathered}$$

$$\begin{gathered} \int\dfrac{dx}{x^2+a^2}=\dfrac{1}{a}\operatorname{arctg}\dfrac{x}{a}+C \\[5pt] \int\dfrac{dx}{x^2-a^2}=\dfrac{1}{2a}\ln\Big|\dfrac{x-a}{x+a}\Big|+C \\[5pt] \int\dfrac{dx}{\sqrt{x^2\pm a^2}}=\ln\big|x+\sqrt{x^2\pm a^2}\big|+C \\[5pt] \int\dfrac{dx}{\sqrt{a^2-x^2}}=\arcsin\dfrac{x}{a}+C \end{gathered}$$

$$\begin{gathered} \int\sqrt{a^2-x^2}\,dx=\dfrac{x}{2}\sqrt{a^2-x^2}+\dfrac{a^2}{2}\arcsin\dfrac{x}{a}+C \\[5pt] \int\sqrt{x^2+A}\,dx=\dfrac{x}{2}\sqrt{x^2+A}+\dfrac{A}{2}\ln\big|x+\sqrt{x^2+A}\big|+C \end{gathered}$$

$$\int_a^b f(x)\,dx=F(b)-F(a),\quad F'(x)=f(x)$$

$$\begin{gathered} \text{Površina: } P=\int_a^b |f(x)|\,dx \\[5pt] \text{Dužina luka: } \ell=\int_a^b\sqrt{1+\big(f'(x)\big)^2}\,dx \\[5pt] \text{Zapremina obrtnog tela: } V=\pi\int_a^b f^2(x)\,dx \\[5pt] \text{Omotač obrtnog tela: } P=2\pi\int_a^b |f(x)|\sqrt{1+\big(f'(x)\big)^2}\,dx \end{gathered}$$

$$\begin{gathered} z=a+bi,\quad i^2=-1 \\[5pt] \bar z=a-bi,\quad |z|=\sqrt{a^2+b^2} \end{gathered}$$

$$\begin{gathered} z=r(\cos\varphi+i\sin\varphi) \\[5pt] z^n=r^n(\cos n\varphi+i\sin n\varphi) \end{gathered}$$

$$\dfrac{1}{z}=\dfrac{\bar z}{|z|^2}$$