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Diffstat (limited to 'info/mathtrip/src/calc.tex')
-rw-r--r-- | info/mathtrip/src/calc.tex | 68 |
1 files changed, 68 insertions, 0 deletions
diff --git a/info/mathtrip/src/calc.tex b/info/mathtrip/src/calc.tex new file mode 100644 index 0000000000..60c9c2e557 --- /dev/null +++ b/info/mathtrip/src/calc.tex @@ -0,0 +1,68 @@ +%This command provides the mathematic (calculus) of the second +%column of the page 6. +% +%This command has one parameter: +% 1) The width of the math text +\newcommand\TSixCalculusOne[1]{% + \def\dudx{\unskip\,\frac{du}{dx}} + \parbox[t]{#1}{% + \TSixColTwoFontSize + \deflength{\VSpace}{1.15\TSixSkipFormulae} %Useful to add more space for the derivatives. + \TSixTitle{Derivatives:} + \AdjustSpace{1ex plus .5ex minus .5ex} + \begin{DisplayFormulae}{1}{\SpaceBeforeFormula}{\VSpace}{\BigChar}{\StyleBold} + \Fm{\frac{d(cu)}{dx} = c \dudx} + \Fm{\frac{d(u+v)}{dx} = \dudx + \frac{dv}{dx}} + \Fm{\frac{d(uv)}{dx} = u \frac{dv}{dx} + v \dudx} + \Fm{\frac{d(u^n)}{dx} = nu^{n-1}\dudx} + \Fm{\frac{d(u/v)}{dx} = \frac{v\big(\dudx\big) - u\big(\frac{dv}{dx}\big)}{v^2}} + \Fm{\frac{d(e^{cu})}{dx} = ce^{cu}\dudx} + \Fm{\frac{d(c^u)}{dx} = (\ln c) c^u\dudx} + \Fm{\frac{d(\ln u)}{dx} = \frac{1}{u} \dudx} + \Fm{\frac{d(\sin u)}{dx} = \cos u \dudx} + \Fm{\frac{d(\cos u)}{dx} = - \sin u \dudx} + \Fm{\frac{d(\tan u)}{dx} = \sec^2 u \dudx} + \Fm{\frac{d(\cot u)}{dx} = \csc^2 u \dudx} + \Fm{\frac{d(\sec u)}{dx} = \tan u \,\sec u \dudx} + \Fm{\frac{d(\csc u)}{dx} = - \cot u \,\csc u \dudx} + \Fm{\frac{d(\arcsin u)}{dx} = \frac{1}{\sqrt{1 - u^2}}\dudx} + \Fm{\frac{d(\arccos u)}{dx} = \frac{-1}{\sqrt{1 - u^2}}\dudx} + \Fm{\frac{d(\arctan u)}{dx} = \frac{1}{1 + u^2}\dudx} + \Fm{\frac{d(\arccot u)}{dx} = \frac{-1}{1 + u^2}\dudx} + \Fm{\frac{d(\arcsec u)}{dx} = \frac{1}{u \sqrt{1 - u^2}}\dudx} + \Fm{\frac{d(\arccsc u)}{dx} = \frac{-1}{u \sqrt{1 - u^2}}\dudx} + \Fm{\frac{d(\sinh u)}{dx} = \cosh u \dudx} + \Fm{\frac{d(\cosh u)}{dx} = \sinh u \dudx} + \Fm{\frac{d(\tanh u)}{dx} = \sech^2 u \dudx} + \Fm{\frac{d(\coth u)}{dx} = -\csch^2 u \dudx} + \Fm{\frac{d(\sech u)}{dx} = -\sech u \,\tanh u\dudx} + \Fm{\frac{d(\csch u)}{dx} = -\csch u \,\coth u\dudx} + \Fm{\frac{d(\arcsinh u)}{dx} = \frac{1}{\sqrt{1 + u^2}}\dudx} + \Fm{\frac{d(\arccosh u)}{dx} = \frac{1}{\sqrt{u^2 - 1}}\dudx} + \Fm{\frac{d(\arctanh u)}{dx} = \frac{1}{1 - u^2}\dudx} + \Fm{\frac{d(\arccoth u)}{dx} = \frac{1}{u^2 - 1}\dudx} + \Fm{\frac{d(\arcsech u)}{dx} = \frac{-1}{u \sqrt{1 - u^2}}\dudx} + \Fm{\frac{d(\arccsch u)}{dx} = \frac{-1}{\vert u \vert \sqrt{1 + u^2}}\dudx.} + \end{DisplayFormulae} + + \TSixTitle{Integrals:} + \begin{DisplayFormulae}{1}{\SpaceBeforeFormula}{\TSixSkipFormulae}{\BigChar}{\StyleBold} + \Fm{\int c u \dx = c \int u \dx} + \Fm{\int (u + v) \dx = \int u \dx + \int v \dx} + \Fm{\int x^n \dx = \frac{1}{n+1}x^{n+1} + \MathRemark{n \neq -1}} + \Fm{\int \frac{1}{x} dx = \ln \ValAbs{x}} + \Fm{\int e^x \dx = e^x} + \Fm{\int \frac{dx}{1 + x^2} = \arctan x} + \Fm{\int u \frac{dv}{dx} dx = uv - \int v {du \over dx} dx} + \Fm{\int \sin \xdx = -\cos x} + \Fm{\int \cos \xdx = \sin x} + \Fm{\int \tan \xdx = -\ln \ValAbs{\cos x}} + \Fm{\int \cot \xdx = \ln \ValAbs{\cos x}} + \Fm{\int \sec \xdx = \ln \ValAbs{\sec x + \tan x}} + \Fm{\int \csc \xdx = \ln \ValAbs{\csc x + \cot x}} + \Fm{\int \arcsin \tfrac{x}{a} dx = \arcsin \tfrac{x}{a} + \sqrt{a^2 - x^2} + \MathRemark{a > 0}} + \end{DisplayFormulae} + } +} |