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[Stack-cvs] stack-dev/sample_questions/diagnostictests FormulaSheet.tex, NONE, 1.1.2.1
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From: Chris S. <san...@us...> - 2010-09-01 11:50:07
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Update of /cvsroot/stack/stack-dev/sample_questions/diagnostictests In directory sfp-cvsdas-3.v30.ch3.sourceforge.com:/tmp/cvs-serv31389 Added Files: Tag: STACK2_2 FormulaSheet.tex Log Message: --- NEW FILE: FormulaSheet.tex --- \documentclass{article} \usepackage{amsmath,amssymb,latexsym,a4wide,times} \usepackage{epsfig} \usepackage{graphicx} \parindent=0pt \parskip=2mm \columnsep=1cm \newcommand{\R}{{\mathbb R}} \newcommand{\N}{{\mathbb N}} \newcommand{\Z}{{\mathbb Z}} \newcommand{\Q}{{\mathbb Q}} \newcommand{\C}{{\mathbb C}} \renewcommand{\d}{\mathrm{d}} \pagestyle{empty} \begin{document} \begin{center} {\Large\bf Facts and formulae} \end{center} Indices: \[ a^ma^n = a^{m+n},\quad \frac{a^m}{a^n} = a^{m-n},\quad (a^m)^n = a^{mn}\] \[ a^0 = 1,\quad a^{-m} = \frac{1}{a^m},\quad a^{\frac{1}{n}} = \sqrt[n]{a},\quad a^{\frac{m}{n}} = \left(\sqrt[n]{a}\right)^m.\] Logarithms: for any $b>0$, $b \neq 1$: $\log_b(a) = c$, means $a = b^c$. \[\log_b(a) + \log_b(b) = \log_b(ab),\quad \log_b(a) - \log_b(b) = \log_b\left(\frac{a}{b}\right),\quad n\log_b(a) = \log_b\left(a^n\right)\] \[\log_b(1) = 0,\quad \log_b(b) = 1,\quad \log_a(x) = \frac{\log_b(x)}{\log_b(a)}\] Natural logarithms use base $e\approx 2.718$, and are denoted $\log_e$ or alternatively $\ln$. Standard Trigonometric Values \[ \sin(45^\circ)={1\over \sqrt{2}}, \qquad \cos(45^\circ) = {1\over \sqrt{2}},\qquad \tan( 45^\circ)=1 \] \[ \sin (30^\circ)={1\over 2}, \qquad \cos (30^\circ)={\sqrt{3}\over 2},\qquad \tan (30^\circ)={1\over \sqrt{3}}\] \[ \sin (60^\circ)={\sqrt{3}\over 2}, \qquad \cos (60^\circ)={1\over 2},\qquad \tan (60^\circ)={ \sqrt{3}} \] Standard Trigonometric Identities \[\sin(a\pm b)\ = \ \sin(a)\cos(b)\ \pm\ \cos(a)\sin(b)\] \[\cos(a\ \pm\ b)\ = \ \cos(a)\cos(b)\ \mp \sin(a)\sin(b)\] \[\tan (a\ \pm\ b)\ = \ {\tan (a)\ \pm\ \tan (b)\over1\ \mp\ \tan (a)\tan (b)}\] \[2\sin(a)\cos(b)\ = \ \sin(a+b)\ +\ \sin(a-b)\] \[2\cos(a)\cos(b)\ = \ \cos(a-b)\ +\ \cos(a+b)\] \[2\sin(a)\sin(b) \ = \ \cos(a-b)\ -\ \cos(a+b)\] \[\sin^2(a)+\cos^2(a)\ = \ 1\] \[1+{\rm cot}^2(a)\ = \ {\rm cosec}^2(a),\quad \tan^2(a) +1 \ = \ \sec^2(a)\] \[\cos(2a)\ = \ \cos^2(a)-\sin^2(a)\ = \ 2\cos^2(a)-1\ = \ 1-2\sin^2(a)\] \[\sin(2a)\ = \ 2\sin(a)\cos(a)\] \[\sin^2(a) \ = \ {1-\cos (2a)\over 2}, \qquad \cos^2(a)\ = \ {1+\cos(2a)\over 2}\] Hyperbolic Functions \[\cosh(x) = \frac{e^x+e^{-x}}{2}, \qquad \sinh(x)=\frac{e^x-e^{-x}}{2}\] \[\tanh(x) = \frac{\sinh(x)}{\cosh(x)} = \frac{{e^x-e^{-x}}}{e^x+e^{-x}}\] \[{\rm sech}(x) ={1\over \cosh(x)}={2\over {\rm e}^x+{\rm e}^{-x}}, \qquad {\rm cosech}(x)= {1\over \sinh(x)}={2\over {\rm e}^x-{\rm e}^{-x}}\] \[{\rm coth}(x) ={\cosh(x)\over \sinh(x)} = {1\over {\rm tanh}(x)} ={{\rm e}^x+{\rm e}^{-x}\over {\rm e}^x-{\rm e}^{-x}}\] Hyperbolic Identities \[{\rm e}^x=\cosh(x)+\sinh(x), \quad {\rm e}^{-x}=\cosh(x)-\sinh(x)\] \[\cosh^2(x) -\sinh^2(x) = 1$$ $$1-{\rm tanh}^2(x)={\rm sech}^2(x)\] \[{\rm coth}^2(x)-1={\rm cosech}^2(x)$$ $$\sinh(x\pm y)=\sinh(x)\ \cosh(y)\ \pm\ \cosh(x)\ \sinh(y)\] \[\cosh(x\pm y)=\cosh(x)\ \cosh(y)\ \pm\ \sinh(x)\ \sinh(y)\] \[\sinh(2x)=2\,\sinh(x)\cosh(x)\] \[\cosh(2x)=\cosh^2(x)+\sinh^2(x)\] \[\cosh^2(x)={\cosh(2x)+1\over 2}\] \[\sinh^2(x)={\cosh(2x)-1\over 2}\] Inverse Hyperbolic Functions \[\cosh^{-1}(x)=\ln\left(x+\sqrt{x^2-1}\right) \quad \mbox{ for } x\geq 1\] \[ \sinh^{-1}(x)=\ln\left(x+\sqrt{x^2+1}\right)\] \[\tanh^{-1}(x) = \frac{1}{2}\ln\left({1+x\over 1-x}\right) \quad \mbox{ for } -1< x < 1\] Calculus rules The Product Rule: \[\frac{\mathrm{d}}{\mathrm{d}{x}} \big(f(x)g(x)\big) = f(x) \cdot \frac{\mathrm{d} g(x)}{\mathrm{d}{x}} + g(x)\cdot \frac{\mathrm{d} f(x)}{\mathrm{d}{x}}.\] The Quotient Rule: \[\frac{d}{dx}\left(\frac{f(x)}{g(x)}\right)=\frac{g(x)\cdot\frac{df(x)}{dx}\ \ - \ \ f(x)\cdot \frac{dg(x)}{dx}}{g(x)^2}. \] The Chain Rule \[\frac{df(g(x))}{dx} = \frac{dg(x)}{dx}\cdot\frac{df(u)}{du}.\] Integration by Substitution: \[\int f(u){{\rm d}u\over {\rm d}x}{\rm d}x=\int f(u){\rm d}u \quad\hbox{and}\quad \int_a^bf(u){{\rm d}u\over {\rm d}x}\,{\rm d}x = \int_{u(a)}^{u(b)}f(u){\rm d}u.\] Integration by Parts: \[\int_a^b u{{\rm d}v\over {\rm d}x}{\rm d}x=\left[uv\right]_a^b- \int_a^b{{\rm d}u\over {\rm d}x}v\,{\rm d}x.\] \begin{center} \begin{tabular}{ll} $f(x)$ & $f'(x)$\\ $k$, constant & $0$ \\ $x^n$, any constant $n$ & $nx^{n-1}$\\ $e^x$ & $e^x$\\ $\ln(x)=\log_{\rm e}(x)$ & $\frac{1}{x}$ \\ $\sin(x)$ & $\cos(x)$ \\ $\cos(x)$ & $-\sin(x)$ \\ $\tan(x) = \frac{\sin(x)}{\cos(x)}$ & $\sec^2(x)$ \\ $\mbox{cosec}(x)=\frac{1}{\sin(x)}$ & $-\mbox{cosec}(x)\cot(x)$ \\ $\sec(x)=\frac{1}{\cos(x)}$ & $\sec(x)\tan(x)$ \\ $\cot(x)=\frac{\cos(x)}{\sin(x)}$ & $-\mbox{cosec}^2(x)$ \\ $\cosh(x)$ & $\sinh(x)$ \\ $\sinh(x)$ & $\cosh(x)$ \\ $\tanh(x)$ & $\mbox{sech}^2(x)$ \\ $\mbox{sech}(x)$ & $-\mbox{sech}sech(x)\tanh(x)$ \\ $\mbox{cosech}(x)$ & $-\mbox{cosech}(x)\coth(x)$ \\ $coth(x)$ & $-\mbox{cosech}^2(x)$ \\ \end{tabular} \end{center} \[\frac{d}{dx}\left(\sin^{-1}(x)\right) = \frac{1}{\sqrt{1-x^2}},\quad \frac{d}{dx}\left(\cos^{-1}(x)\right) = \frac{-1}{\sqrt{1-x^2}},\quad \frac{d}{dx}\left(\tan^{-1}(x)\right) = \frac{1}{1+x^2} \] \[\frac{d}{dx}\left(\cosh^{-1}(x)\right) = \frac{1}{\sqrt{x^2-1}},\quad \frac{d}{dx}\left(\sinh^{-1}(x)\right) = \frac{1}{\sqrt{x^2+1}},\quad \frac{d}{dx}\left(\tanh^{-1}(x)\right) = \frac{1}{1-x^2}\] \begin{center} \begin{tabular}{lll} $f(x)$ & $\int f(x)\ dx$\\ $e^x$ & $e^x+c$ & \\ $\cos(x)$ & $\sin(x)+c$ & \\ $\sin(x)$ & $-\cos(x)+c$ & \\ $\tan(x)$ & $\ln(\sec(x))+c$ & $-\frac{\pi}{2} < x < \frac{\pi}{2}$\\ $\sec x$ & $\ln (\sec(x)+\tan(x))+c$ & $-{\pi\over 2}< x < {\pi\over 2}$\\ cosec$\, x$ & $\ln ($cosec$(x)-\cot(x))+c$ & $0 < x < \pi$\\ cot$\,x$ & $\ln(\sin(x))+c$ & $0< x< \pi$ \\ $\cosh(x)$ & $\sinh(x)+c$ & \\ $\sinh(x)$ & $\cosh(x) + c$ & \\ $\tanh(x)$ & $\ln(\cosh(x))+c$ & \\ coth$(x)$ & $\ln(\sinh(x))+c $ & $x>0$\\ ${1\over x^2+a^2}$ & ${1\over a}\tan^{-1}{x\over a}+c$ & $a>0$\\ [2pt] ${1\over x^2-a^2}$ & ${1\over 2a}\ln{x-a\over x+a}+c$ & $|x|>a>0$\\ [2pt] ${1\over a^2-x^2}$ & ${1\over 2a}\ln{a+x\over a-x}+c$ & $|x|<a$\\ [3pt] ${1\over \sqrt{x^2+a^2}}$ & $\sinh^{-1}\left(\frac{x}{a}\right) + c$ & $a>0$ \\ ${1\over \sqrt{x^2-a^2}}$ & $\cosh^{-1}\left(\frac{x}{a}\right) + c$ & $x\geq a > 0$ \\ ${1\over \sqrt{x^2+k}}$ & $\ln (x+\sqrt{x^2+k})+c$ & \\ ${1\over \sqrt{a^2-x^2}}$ & $\sin^{-1}\left(\frac{x}{a}\right)+c$ & $-a\leq x\leq a$ \end{tabular} \end{center} \vfill {\scriptsize C J Sangwin, \verb$C.J...@bh...$, \today. This formula sheet is released under Creative Commons Attribution-Share Alike.\\ \includegraphics[width=1.5cm]{88x31.png}} \end{document} |
