Mathe-Abi ab 2004
Aufgabe: Löse die Gleichung $e^{2x}+e^{x}-2=0$
Lösung:
axiom
g:=%e**(2*x)+%e**(x)-2=0
\begin{equation}
\label{eq1}{{e^{\left(2 \ x \right)}}+{e^x}- 2}= 0\end{equation}
Type: Equation(Expression(Integer))
axiom
g:=g+2
\begin{equation}
\label{eq2}{{e^{\left(2 \ x \right)}}+{e^x}}= 2\end{equation}
Type: Equation(Expression(Integer))
axiom
f:=(2*x^2+2*x-1)/(x^2-1)
\begin{equation}
\label{eq3}{{2 \ {x^2}}+{2 \ x}- 1}\over{{x^2}- 1}\end{equation}
Type: Fraction(Polynomial(Integer))
axiom
differentiate(f,x)
\begin{equation}
\label{eq4}{-{2 \ {x^2}}-{2 \ x}- 2}\over{{x^4}-{2 \ {x^2}}+ 1}\end{equation}
Type: Fraction(Polynomial(Integer))
axiom
simplify(%)
\begin{equation}
\label{eq5}{-{2 \ {x^2}}-{2 \ x}- 2}\over{{x^4}-{2 \ {x^2}}+ 1}\end{equation}
Type: Expression(Integer)
axiom
f2:=3*(x^2-4*%e^(-2*x))
\begin{equation}
\label{eq6}-{{12}\ {e^{\left(-{2 \ x}\right)}}}+{3 \ {x^2}}\end{equation}
Type: Expression(Integer)
axiom
integrate(f2,x)
\begin{equation}
\label{eq7}{6 \ {e^{\left(-{2 \ x}\right)}}}+{x^3}\end{equation}
Type: Union(Expression(Integer),...)
Comparar $ea(n)$ con ordenes como $n^k$, $2^n$, $n!$, $n^n$ donde
$ea(n)=\frac{1}{n-\log_2{2^n-1}}$
Intento:
axiom
e:=1/(n-log(2^n-1))
\begin{equation}
\label{eq8}-{1 \over{{\log \left({{2^n}- 1}\right)}- n}}\end{equation}
Type: Expression(Integer)
axiom
simplify(e)
\begin{equation}
\label{eq9}-{1 \over{{\log \left({{2^n}- 1}\right)}- n}}\end{equation}
Type: Expression(Integer)
axiom
solve(n+2-log((2^(n+1)-1)^2/(2^n-1));n)
\begin{equation*}
\label{eq10}\left[{n = 0}\right]?\end{equation*}
Type: List(Equation(Fraction(Polynomial(Integer))))
axiom
solve([nw=h/cos(a)/(h/w+tan(a)),nh=nw*h/w],[w,h])
\begin{equation*}
\label{eq11}\begin{array}{@{}l}
\displaystyle
\left[ \left[{w ={{{{nw^2}\ {\cos \left({a}\right)}\ {\tan \left({a}\right)}}+{nh \ nw \ {\cos \left({a}\right)}}}\over nh}}, \: \right.
\
\
\displaystyle
\left.{h ={{nw \ {\cos \left({a}\right)}\ {\tan \left({a}\right)}}+{nh \ {\cos \left({a}\right)}}}}\right] \right]
\end{array}
\end{equation*}
Type: List(List(Equation(Expression(Integer))))
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