randRange(2,5)
randRange(2,5)
randRange(2,5)
randRange(2,5)
[
["x", "\\sin(x)", "1", "\\cos(x)", "x\\cos(x)", "\\sin(x)",
"x\\sin(x) + \\cos(x)","0","-\\pi","\\pi"],
["x", "\\sin(x)", "1", "\\cos(x)", "x\\cos(x)", "\\sin(x)",
"x\\sin(x) + \\cos(x)","0","-\\frac{\\pi}{2}","\\frac{\\pi}{2}"],
["x^2", "-\\cos(x)", "2x", "\\sin(x)", "x^2\\sin(x)", "-2x\\cos(x)",
"-x^2\\cos(x) + 2x\\sin(x) + 2\\cos(x)","0","-\\pi","\\pi"],
["x^2", "-\\cos(x)", "2x", "\\sin(x)", "x^2\\sin(x)", "-2x\\cos(x)",
"-x^2\\cos(x) + 2x\\sin(x) + 2\\cos(x)",0,"-\\frac{\\pi}{2}","\\frac{\\pi}{2}"],
["x", "-\\cos(x)", "1", "\\sin(x)", "x\\sin(x)", "-\\cos(x)",
"-x\\cos(x) + \\sin(x)","2\\pi","-\\pi","\\pi"],
["x", "-\\cos(x)", "1", "\\sin(x)", "x\\sin(x)", "-\\cos(x)",
"-x\\cos(x) + \\sin(x)","2","-\\frac{\\pi}{2}","\\frac{\\pi}{2}"],
["x", "e^x", "1", "e^x", "xe^x", "e^x", "xe^x - e^x",B + "\\ln(" + B +") -" + (B-1),0,"\\ln("+ B +")"],
["\\ln(x)", "\\frac{1}{"+(n+1)+"}x^{"+(n+1)+"}", "\\frac{1}{x}",
"x^"+n+"", "x^"+n+"\\ln(x)", "\\frac{1}{"+(n+1)+"}x^{"+n+"}",
"\\frac{1}{"+(n+1)+"}x^{"+(n+1)+"}\\ln(x) - \\frac{1}{"+(n+1)*(n+1)+"}x^{"+(n+1)+"}",
"\\frac{" + (n*k+k-1) + "e^{" + (n*k+k) + "}+ 1}{" + (n+1)*(n+1) + "}",1, "e^{" + k + "}"],
["\\ln(x)", "\\frac{1}{"+(n+1)+"}x^{"+(n+1)+"}", "\\frac{1}{x}",
"x^"+n+"", "x^"+n+"\\ln(x)", "\\frac{1}{"+(n+1)+"}x^{"+n+"}",
"\\frac{1}{"+(n+1)+"}x^{"+(n+1)+"}\\ln(x) - \\frac{1}{"+(n+1)*(n+1)+"}x^{"+(n+1)+"}",
"\\frac{" + pow(l,n+1) + "(" + (n+1) + "\\ln(" + l + ")- 1) + 1}{" + (n+1)*(n+1) + "}",1,l],
["x^2", "\\sin(x)", "2x", "\\cos(x)", "x^2\\cos(x)", "2x\\sin(x)",
"x^2\\sin(x) + 2x\\cos(x) - 2\\sin(x)","-4 \\pi","-\\pi","\\pi"],
["x^2", "\\sin(x)", "2x", "\\cos(x)", "x^2\\cos(x)", "2x\\sin(x)",
"x^2\\sin(x) + 2x\\cos(x) - 2\\sin(x)","\\frac{\\pi^2}{2}-4","-\\frac{\\pi}{2}","\\frac{\\pi}{2}"],
["x^2", "e^x", "2x", "e^x", "x^2e^x", "2xe^x",
"x^2e^x - 2xe^x + 2 e^x",B + "\\ln(" + B +")(\\ln(" + B +") - 2 )+" + (2*B-2) ,0,"\\ln("+ B +")"]
]
randRange(0,functionBank.length-1)
functionBank[fNum]
Gegeben sei der euklidische Vektorraum
\left(C^0([f[8],f[9]]), \langle \ , \ \rangle \right) mit
\displaystyle \langle f, g \rangle = \int_{f[8]}^{f[9]} f(x)g(x) \; dx.
Berechnen Sie das Skalarprodukt für {\color{red}f}, \ {\color{blue}g} \; : [f[8],f[9]] \to \mathbb R
mit {\color{red}f(x) = f[0]} und {\color{blue}g(x) = f[3]}.
\langle f, g \rangle =
f[7]
Es ist
\displaystyle
\langle {\color{red}f}, \ {\color{blue}g} \rangle =
\int_{f[8]}^{f[9]} {\color{red}f(x)}{\color{blue}g(x)} \; dx =
\int_{f[8]}^{f[9]} {\color{red}f[0]}\ {\color{blue}f[3]} \; dx
Für das bestimmte Integral rechnen wir zunächst mit partieller Integration
\displaystyle
\int u(x)v'(x) \; dx = u(x)v(x) - \int u'(x)v(x)\; dx + C
.
Wir erhalten \displaystyle \int {\color{red}f[0]}\ {\color{blue}f[3]} \; dx = f[6] + C.
Mit dem Hauptsatz ist dann
\displaystyle \int_{f[8]}^{f[9]} {\color{red}f[0]}\ {\color{blue}f[3]} \; dx =
f[6]\biggl|_{f[8]}^{f[9]} = f[7].
Die beiden Vektoren sind orthogonal.
Alternativ folgt das auch direkt aus
der Symmetrieeigenschaft der ungeraden Funktion
x \mapsto {\color{red}f[0]}\ {\color{blue}f[3]}.
Mit dem Hauptsatz ist dann
\displaystyle \int_{f[8]}^{f[9]} {\color{red}f[0]}\ {\color{blue}f[3]} \; dx =
f[6]\biggl|_{f[8]}^{f[9]} = f[7].