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\begin{document}
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\bigskip

{\bf K\'epletgy\H ujtem\'eny Matematika EP1 vizsg\'ara}
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&&&\\
\mc{\bf Nevezetes sorozathat\'ar\'ert\'ekek}&\mc{\bf Nevezetes f\"uggv\'enyhat\'ar\'ert\'ekek}\\
$\lim_{n\to\infty}q^n=0$&ha $|q|<1$&$\lim_{x\to0}\frac{\sin x}x=1$&$\lim_{x\to0}\frac{\tg x}x=1$\\
$\lim_{n\to\infty}q^n n^k=0$&ha $|q|<1$ \'es $k\in\mathbb N$&$\lim_{x\to\infty}\left(1+\frac1x\right)^x=e$&$\lim_{x\to0}(1+x)^{1/x}=e$\\
$n^k\ll e^n\ll n!\ll n^n$&amint $n\to\infty$ ha $k\in\mathbb N$&$\lim_{x\to0}\frac{\log_a(1+x)}x=\frac1{\ln a}$&ha $a>0$ \'es $a\neq1$\\
\m{\qquad ahol $a_n\ll b_n$ azt jelenti, hogy $\lim_{n\to\infty}\frac{a_n}{b_n}=0$}&$\lim_{x\to0}\frac{a^x-1}x=\ln a$&ha $a>0$ \'es $a\neq1$\\
\m{$\lim_{n\to\infty}\left(1+\frac1n\right)^n=e$}&$\lim_{x\to0}\frac{(1+x)^\mu-1}x=\mu$&ahol $\mu\in\mathbb R$\\
\m{$\lim_{n\to\infty}\left(1+b_n\right)^{c_n}=e^{\lim_{n\to\infty}b_nc_n}$ ha $b_n\to0$, $c_n\to\infty$}&&\\
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\m{\(\sin^2x+\cos^2x=1\)}&\mc{\bf Differenci\'al\'asi szab\'alyok}\\
\m{\(\sin(x\pm y)=\sin x\cos y\pm\cos x\sin y\)}&\((cu)'=cu'\)&(\(c\) konstans)\\
\m{\(\cos(x\pm y)=\cos x\cos y\mp\sin x\sin y\)}&\m{\((u+v)'=u'+v'\)}\\
\m{\(\tg(x\pm y)=\frac{\tg x\pm\tg y}{1\mp\tg x\cdot\tg y}\)}&\m{\((uv)'=u'v+uv'\)}\\
\m{\(\sin2x=2\sin x\cos x\)}&\m{\(\left(\frac{u}{v}\right)'=\frac{u'v-uv'}{v^2}\)}\\
\m{\(\cos2x=\cos^2x-\sin^2x\)}&\m{\((f(g(x)))'=f'(g(x))g'(x)\)}\\
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\m{\(\tg2x=\frac{2\tg x}{1-\tg^2x}\)}&\mc{\bf Integr\'al\'asi szab\'alyok}\\
\(\sin^2x=\frac{1-\cos2x}{2},\)&\(\cos^2x=\frac{1+\cos2x}{2}\)&\(\int cf\dx=c\int f\dx\)&(\(c\) konstans)\\
\m{\(\sin x+\sin y=2\sin\frac{x+y}{2}\cos\frac{x-y}{2}\)}&\m{\(\int(f+g)\dx=\int f\dx+\int g\dx\)}\\
\m{\(\sin x-\sin y=2\cos\frac{x+y}{2}\sin\frac{x-y}{2}\)}&\m{\(\int f(ax+b)\dx=\frac1aF(ax+b)+c\)}\\
\m{\(\cos x+\cos y=2\cos\frac{x+y}{2}\cos\frac{x-y}{2}\)}&\m{\(\qquad\text{ahol }F\text{ az }f\text{ primit\'\i v f\"uggv\'enye}\)}\\
\m{\(\cos x-\cos y=-2\sin\frac{x+y}{2}\sin\frac{x-y}{2}\)}&\m{\(\int f(g(x))g'(x)\dx=F(g(x))+c\)}\\
\m{\(\sin x\cos y=\frac12[\sin(x+y)+\sin(x-y)]\)}&\m{\(\qquad\text{ahol }F\text{ az }f\text{ primit\'\i v f\"uggv\'enye}\)}\\
\m{\(\cos x\cos y=\frac12[\cos(x+y)+\cos(x-y)]\)}&\m{\(\int f^\alpha f'\dx=\frac{f^{\alpha+1}}{\alpha+1}+c\text{, ha }\alpha\ne-1\)}\\
\m{\(\sin x\sin y=-\frac12[\cos(x+y)-\cos(x-y)]\)}&\m{\(\int\frac{f'}{f}\dx=\ln|f|+c\)}\\
\(\sh x=\frac{\e{x}-\e{-x}}{2},\)&\(\ch x=\frac{\e{x}+\e{-x}}{2}\)&\m{\(\int uv'\dx=uv-\int u'v\dx\)}\\
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\m{\(\ch^2x-\sh^2x=1\)}&\mc{\bf Nevezetes helyettes\'\i t\'esek}\\
\m{\(\sh2x=2\sh x\ch x\)}&\(R(\e{x})\)&\(\e{x}=t\)\\
\m{\(\ch2x=\ch^2x+\sh^2x\)}&\(R(\sqrt{ax+b})\)&\(\sqrt{ax+b}=t\)\\
\(\ch^2x=\frac{\ch2x+1}{2},\)&\(\sh^2x=\frac{\ch2x-1}{2}\)&\(R\left(\frac{\sqrt{ax+b}}{\sqrt{cx+d}}\right)\)&\(\frac{\sqrt{ax+b}}{\sqrt{cx+d}}=t\)\\
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\m{}&\(R(\sin x,\cos x)\)&\(\sin x,\cos x,\tg x,\tg\frac{x}{2}\!\!=\!t\)\\
\mc{\bf Deriv\'altak}&\(R(x,\,\sqrt{a^2-x^2})\)&\(x=a\sin t,\ x=a\cos t\)\\
\m{\((\sh x)'=\ch x\)}&\(R(x,\,\sqrt{a^2+x^2})\)&\(x=a\sh t\)\\
\m{\((\ch x)'=\sh x\)}&\(R(x,\,\sqrt{x^2-a^2})\)&\(x=a\ch t\)\\
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\m{\((\log_ax)'=\frac{1}{x\ln a}\)}&\mc{\bf Integr\'alok}\\
\m{\((x^\alpha)'=\alpha x^{\alpha-1}\)}&\(\int x^\alpha\dx=\frac{x^{\alpha+1}}{\alpha+1}+c\)&\((\alpha\ne-1)\)\\
\m{\((\e{x})'=\e{x}\)}&\m{\(\int\e{ax}\dx=\frac1a\,\e{ax}+c\)}\\
\m{\((a^x)'=a^x\ln(a)\)}&\m{\(\int a^x\dx=\frac{a^x}{\ln a}+c\)}\\
\m{\((\sin x)'=\cos x\)}&\m{\(\int\cos x\dx=\sin x+c\)}\\
\m{\((\cos x)'=-\sin x\)}&\m{\(\int\sin x\dx=-\cos x+c\)}\\
\m{\((\tg x)'=\frac{1}{\cos^2x}\)}&\m{\(\int\frac{1}{\cos^2x}\dx=\tg x+c\)}\\
\m{\((\ctg x)'=-\frac{1}{\sin^2x}\)}&\m{\(\int\frac{1}{\sin^2x}\dx=-\ctg x+c\)}\\
\m{\((\ln x)'=\frac1x\)}&\m{\(\int\frac1x\dx=\ln|x|+c\)}\\
\m{\((\as x)'=\frac{1}{\sqrt{1-x^2}}\)}&\m{\(\int\frac{\dx}{\sqrt{a^2-x^2}}=\as\frac{x}{a}+c\)}\\
\m{\((\arctg x)'=\frac{1}{1+x^2}\)}&\m{\(\int\frac{\dx}{x^2+a^2}=\frac1a\arctg\frac{x}{a}+c\)}\\
\m{\((\arsh x)'=\frac{1}{\sqrt{1+x^2}}\)}&\m{\(\int\frac{\dx}{\sqrt{x^2+a^2}}=\arsh\frac{x}{a}+c\)}\\
\m{\((\arch x)'=\frac{1}{\sqrt{x^2-1}}\)}&\m{\(\int\frac{\dx}{\sqrt{x^2-a^2}}=\arch\frac{x}{a}+c\)}\\
\m{\((\arth x)'=\frac{1}{1-x^2}\)}&\(\int\frac{\dx}{a^2-x^2}=\frac1a\arth\frac{x}{a}+c,\)&ha \(\left|\frac{x}{a}\right|<1\)\\
\m{\((\arcth x)'=\frac{1}{1-x^2}\)}&\(\int\frac{\dx}{a^2-x^2}=\frac1a\arcth\frac{x}{a}+c,\)&ha \(\left|\frac{x}{a}\right|>1\)\\
\m{\((\ac x)'=-\frac{1}{\sqrt{1-x^2}}\)}&\m{\(\int\tg x\dx=-\ln|\cos x|+c\)}\\
\m{\((\arcctg x)'=-\frac{1}{1+x^2}\)}&\m{\(\int\ctg x\dx=\ln|\sin x|+c\)}
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\begin{center}
{\bf Taylor-polinom}
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Legyen $f$ az $x_0$ pontban $n$-szer differenci\'alhat\'o val\'os f\"uggv\'eny.
Ekkor az $f$-hez tartoz\'o $x_0$-ban fel\'\i rt $n$-edfok\'u Taylor-polinom
$T(x)=f(x_0)+f'(x_0)(x-x_0)+\frac12f''(x_0)(x-x_0)^2+\dots+\frac1{n!}f^{(n)}(x_0)(x-x_0)^n$.\\
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{\bf Integr\'al\'as alkalmaz\'asai}
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G\"orbe alatti ter\"ulet: $T=\int_a^b f(x)\dx=\int_{t_1}^{t_2} y(t)\frac{\dx(t)}{\di t}\di t$\\
Szektorter\"ulet: $T=\frac12\int_{t_1}^{t_2}\left|y(t)\frac{\dx(t)}{\di t}-x(t)\frac{\di y(t)}{\di t}\right|\di t=\frac12\int_{\varphi_1}^{\varphi_2}r^2(\varphi)\di\varphi$\\
S\'\i kg\"orbe \'\i vhossza: $s=\int_a^b\sqrt{1+(f'(x))^2}\dx=\int_{t_1}^{t_2}\sqrt{\big(\frac{\dx(t)}{\di t}\big)^2+\big(\frac{\di y(t)}{\di t}\big)^2}\di t
=\int_{\varphi_1}^{\varphi_2}\sqrt{r^2(\varphi)+\big(\frac{\di r(\varphi)}{\di\varphi}\big)^2}\di\varphi$\\
Forg\'astest t\'erfogata: $V=\pi\int_a^b f^2(x)\dx=\pi\int_{t_1}^{t_2} y^2(t)\frac{\dx(t)}{\di t}\di t$\\
Forg\'astest felsz\'\i ne: $A=2\pi\int_a^b f(x)\sqrt{1+(f'(x))^2}\dx=2\pi\int_{t_1}^{t_2} y(t)\sqrt{\big(\frac{\dx(t)}{\di t}\big)^2+\big(\frac{\di y(t)}{\di t}\big)^2}\di t$\\
S\'\i kidom t\"omegk\"oz\'eppontj\'anak koordin\'at\'ai: $x_{\text{tkp}}=\frac{\int_a^b xf(x)\dx}{\int_a^b f(x)\dx}$, $y_{\text{tkp}}=\frac{\frac12\int_a^b f^2(x)\dx}{\int_a^b f(x)\dx}$\\
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{\bf Vektort\'er} ($V$ vektort\'er, ha minden $\ul u,\ul v,\ul w\in V$ \'es $\lambda,\mu\in\mathbb R$ eset\'en teljes\"ulnek az al\'abbiak)
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$\ul u+\ul v\in V$&$\lambda\ul u\in V$\\
$\ul u+\ul v=\ul v+\ul u$&$(\lambda\mu)\ul u=\lambda(\mu\ul u)$\\
$(\ul u+\ul v)+\ul w=\ul u+(\ul v+\ul w)$&$1\in\mathbb R$-re $1\ul u=\ul u$\\
l\'etezik $\ul 0\in V$, hogy $\ul u+\ul 0=\ul u$&$(\lambda+\mu)\ul u=\lambda\ul u+\mu\ul u$\\
minden $\ul u\in V$-hez l\'etezik $-\ul u\in V$, hogy $\ul u+(-\ul u)=\ul 0$&$\lambda(\ul u+\ul v)=\lambda\ul u+\lambda\ul v$\\
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{\bf Determin\'ans}
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$\left|\begin{matrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{matrix}\right|=a_{11}a_{22}-a_{12}a_{21}$&
$\left|\begin{matrix}a_{11}&a_{12}&\dots&a_{1n}\\a_{21}&a_{22}&\dots&a_{2n}\\\vdots&\vdots&\ddots&\vdots\\a_{n1}&a_{n2}&\dots&a_{nn}\end{matrix}\right|=\sum_{j=1}^n(-1)^{i+j}a_{ij}|A_{ij}|$\\
\m{ahol $|A_{ij}|$ az $a_{ij}$ elem sor\'anak \'es oszlop\'anak elhagy\'as\'aval kapott $(n-1)\times(n-1)$-es r\'eszm\'atrix determin\'ansa.}\\
\m{Az $\ul a=\left[\begin{matrix}a_1\\a_2\\a_3\end{matrix}\right]$ \'es $\ul b=\left[\begin{matrix}b_1\\b_2\\b_3\end{matrix}\right]$ $\mathbb R^3$-beli vektorok vektori\'alis szorzata
$\ul a\times\ul b=\left|\begin{matrix}\ul i&\ul j&\ul k\\a_1&a_2&a_3\\b_1&b_2&b_3\end{matrix}\right|=\left[\begin{matrix}a_2b_3-a_3b_2\\-a_1b_3+a_3b_1\\a_1b_2-a_2b_1\end{matrix}\right]$.}\\
\m{Az $\ul a=\left[\begin{matrix}a_1\\a_2\\a_3\end{matrix}\right]$, $\ul b=\left[\begin{matrix}b_1\\b_2\\b_3\end{matrix}\right]$ \'es $\ul c=\left[\begin{matrix}c_1\\c_2\\c_3\end{matrix}\right]$
$\mathbb R^3$-beli vektorok vegyesszorzata $\ul a\,\ul b\,\ul c=(\ul a\times\ul b)\ul c=\left|\begin{matrix}a_1&a_2&a_3\\b_1&b_2&b_3\\c_1&c_2&c_3\end{matrix}\right|$.}\\
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{\bf M\'atrix inverze}
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$A^{-1}=\displaystyle\frac{\adj(A)}{\det(A)}$\qquad
ahol $\adj(A)=\left[\begin{matrix}|A_{11}|&-|A_{12}|&\dots&(-1)^{1+n}|A_{1n}|\\-|A_{21}|&|A_{22}|&\dots&(-1)^{2+n}|A_{2n}|\\
\vdots&\vdots&\ddots&\vdots\\(-1)^{n+1}|A_{n1}|&(-1)^{n+2}|A_{n2}|&\dots&(-1)^{2n}|A_{nn}|\end{matrix}\right]^T$\\
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\begin{center}
{\bf Koordin\'atageometria}
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A $\ul v$ vektor felbont\'asa az $\ul u$ vektorral p\'arhuzamos \'es arra mer\H oleges komponensekre:\\
$\ul v=\ul v_p+\ul v_m$, ahol $\ul v_p||\ul u$ \'es $\ul v_m\bot\ul u$, akkor $\ul v_p=\displaystyle\frac{\langle \ul v,\ul u\rangle}{\|\ul u\|^2}\ul u$ \'es
$\ul v_m=\displaystyle\ul v-\frac{\langle \ul v,\ul u\rangle}{\|\ul u\|^2}\ul u$.\\
Az $(x_1,y_1,z_1)$ \'es $(x_2,y_2,z_2)$ pontok t\'avols\'aga $\mathbb R^3$-ben $\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}$.\\
Az $(x_1,y_1,z_1)$ pont \'es az $n_1(x-x_0)+n_2(y-y_0)+n_3(z-z_0)=0$ egyenlet\H u s\'\i k el\H ojeles t\'avols\'aga $\displaystyle\frac{n_1(x_1-x_0)+n_2(y_1-y_0)+n_3(z_1-z_0)}{\sqrt{n_1^2+n_2^2+n_3^2}}$.
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