==== 1. Mai 2017 bis 5. Mai 2017 ====
=== Freitag 5. Mai 2017 ===
Keine Miniaufgabe
==== 8. Mai 2017 bis 12. Mai 2017 ====
=== Dienstag 9. Mai 2017 ===
function generate(jQuery, idex, idsol) {
var randi = function(m) {
return Math.floor(Math.random()*m);
};
var dispVec = function(a) {
return "\\begin{pmatrix}"+a[0]+"\\\\ "+a[1]+"\\\\ "+a[2]+" \\end{pmatrix}";
};
var dispPoint = function(a) {
return "\\left("+a[0]+", "+a[1]+", "+a[2]+" \\right)";
};
var vecDiff = function(a,b) {
return [a[0]-b[0], a[1]-b[1], a[2]-b[2]];
};
var vecSum = function(a,b) {
return [a[0]+b[0], a[1]+b[1], a[2]+b[2]];
};
var lensq = function(a) {
return a[0]*a[0]+a[1]*a[1]+a[2]*a[2];
};
var quadrFactor = function(w) {
var f = 1;
for (var i=2; i*i<=w; i++) {
while (w % (i*i) == 0) {
w/=i*i;
f*=i;
}
}
return f;
}
var normalform = function(w) {
var r = "\\sqrt{"+w+"}";
var f = quadrFactor(w);
w = w/(f*f);
if (f!=1) {
if (w!=1) {
r+="="+f+"\\sqrt{"+w+"}";
} else {
r+="="+f;
}
}
return r;
};
var nicePoints = function(maxComp) {
var vecs;
var pts;
for (var k=0; k<2000; k++) {
vecs = [];
for (var i=0; i<2; i++) {
vecs[i]=[];
for (var j=0; j<3; j++) {
vecs[i].push((randi(maxComp)+1));
}
if (i==0) {
vecs[i][0] = -vecs[i][0];
} else {
vecs[i][1] = -vecs[i][1];
vecs[i][2] = -vecs[i][2];
}
}
var pts = [[randi(maxComp)+1, -randi(maxComp)-1, randi(maxComp)+1]];
pts.push(vecSum(pts[0],vecs[0]));
pts.push(vecSum(pts[1],vecs[1]));
var v3 = vecSum(vecs[0], vecs[1]);
if (quadrFactor(lensq(vecs[0]))>1 &&
quadrFactor(lensq(vecs[1]))>1 &&
quadrFactor(lensq(v3))>1 &&
v3[0]*v3[1]*v3[2]!=0 &&
quadrFactor(lensq(vecs[0]))*quadrFactor(lensq(vecs[0]))!=lensq(vecs[0]) &&
quadrFactor(lensq(vecs[1]))*quadrFactor(lensq(vecs[1]))!=lensq(vecs[1]) &&
quadrFactor(lensq(v3))*quadrFactor(lensq(v3))!=lensq(v3)) {
return pts;
}
}
console.log("Sorry...");
return pts;
};
var vecs=nicePoints(7);
var pnames = ["A","B", "C"];
var ex = "Gegeben sind drei Punkte: ";
for (var i=0; i<3; i++) {
ex+="$"+pnames[i]+"="+dispPoint(vecs[i])+"$"+(i==2 ? "." : ", ");
}
ex += "
Berechnen Sie die Komponenten und die Länge (Wurzeln in Normalform) von
\
1. $\\vec{AB}$
\
2. $\\vec{AC}$
\
3. $\\vec{BC}$";
jQuery(idex).append(ex);
var sol = ""
var n = 1;
for (var i=0; i<2; i++) {
for (var j=i+1; j<3; j++) {
var diff = vecDiff(vecs[j],vecs[i])
sol+=n+". $"+dispVec(diff)+"$";
sol+=", Länge $"+normalform(lensq(diff))+"$
";
n++;
}
}
jQuery(idsol).append(sol);
}
jQuery = jQuery ? jQuery : $,1
jQuery(function() {generate(jQuery, "#exos","#sol");});
=== Donnerstag 11. Mai 2017 ===
Vereinfachen Sie:
- $$\left(x^{\frac{2}{5}}\right)^{-\frac{35}{6}}\cdot x^{\frac{7}{2}}$$
- $$\left(x^{-\frac{4}{3}}\right)^{-\frac{23}{10}}: x^{\frac{2}{5}}$$
- $$\left(x^{\frac{3}{2}}: x^{\frac{7}{3}}\right)^{-\frac{2}{5}}$$
- $$x^{\frac{7}{6}}$$
- $$x^{\frac{8}{3}}$$
- $$x^{\frac{1}{3}}$$