Lösungen

1

Remove["Global`*"];

a

Series[E^(-x^2),{x,0,8}]

N[%]

b

Evaluate[Normal[Series[E^(-x^2),{x,0,8}]]]

N[%]

Integrate[Evaluate[Normal[Series[E^(-x^2),{x,0,8}]]],{x,-2,2}]

%//N

Integrate[E^(-x^2),{x,-2,2}]//N

Integrate[Evaluate[Normal[Series[E^(-x^2),{x,0,100}]]],{x,-2,2}]//N

c

Evaluate[Normal[Series[Cos[x^2]+E^(x^2),{x,0,8}]]]

N[%]

Integrate[Evaluate[Normal[Series[E^(-x^2),{x,0,8}]]],{x,-2,2}]

%//N

Integrate[E^(-x^2),{x,-2,2}]//N

Integrate[Evaluate[Normal[Series[E^(-x^2),{x,0,100}]]],{x,-2,2}]//N

d

Evaluate[Normal[Series[Sqrt[x],{x,1,6}]]]

N[%]

e

r=1

f

Evaluate[Normal[Series[Log[x]-Sin[x],{x,1,6}]]]

N[%]

g

1-1/(1-1/2)+1/(1-1/3)

N[%]

2

Remove["Global`*"];

a

p1 = {1, 2, 3};  p2 = {3, 4, 2};  p3 = {8, 8, 10};  
    p4 = {4, 0, -2};

Det[{p4 - p1, p3 - p1, p2 - p1}]/6

% // N

b

(Norm[Cross[p4 - p1, p3 - p1]] + 
          Norm[Cross[p4 - p1, p2 - p1]] + 
Norm[
            Cross[p3 - p1, p2 - p1]] + 
Norm[
            Cross[p3 - p4, p2 - p4]])/2

% // N

c

p1 = {1, 2, 3};  p2 = {3, 4, 2};  p3 = {8, 8, 10};  
    p4 = {4, 0, -2};

(p1 + p2 + p3)/3

% // N

p1 + (p2 - p1)/2 + (p3 - (p1 + (p2 - p1)/2))/
        3

% // N

p1 + (p2 - p1)/2 + (p3 - (p1 + (p2 - p1)/2))/
        3 + (p4 - (p1 + (p2 - p1)/
                  2 + (p3 - (p1 + (p2 - p1)/2))/3))/
        4

% // N

(p1+p2+p3+p4)/4

%//N

d

Det[{p4 - p1, p3 - p1, p2 - p1}]/
      Norm[Cross[p3 - p1, p2 - p1]]

% // N

3

Remove["Global`*"];

a

p1={1,2,3}; p2={3,4,2}; p3={8,8,10}; p4={4,0,-2};

a = {7, 10, 7};  b = {-2, -2, 1};

Det[{p3 - p1, p2 - p1, a}]

Gerade durchstösst Ebene

Det[{p3 - p1, p2 - p1, b}]

Es existiert eine Schnittgerade

b

a1 = {7,10}; b1 = {-2,-2};

i

a11={-10,7};

ii

ArcCos[a1.b1/(Norm[a1] Norm[b1])]

φ1=ArcCos[a1.b1/(Norm[a1] Norm[b1])] /Degree //N

φ = φ1 + 90 - 12

360-φ

iii

dreh[φ_]:={{Cos[φ],-Sin[φ]},{Sin[φ],Cos[φ]}};
dreh[u]//MatrixForm

dreh[90 Degree].{{1},{1}}

dreh[12 Degree].Transpose[{b1}]

%//N //MatrixForm

4

Remove["Global`*"];

a

A = {{1,2},{-1,4}};  B = {{3,3},{1,4}};
Print[A//MatrixForm];
Print[B//MatrixForm];

X = Inverse[Transpose[A]].A.Inverse[B]//MatrixForm

%//N//MatrixForm

b

Inverse[A]//MatrixForm

5

Remove["Global`*"];

a

Cm = {{1,2,3},{0,1,1},{2,1,0}}; 
Print[Cm//MatrixForm];

Det[Cm]

b

Nein, wegen a)

c

Dm = {{1,2,3,4},{0,1,2,2},{2,1,0,4},{2,1,1,0}}; 
Print[Dm//MatrixForm];

Det[Dm]

Zusatz

Remove["Global`*"];

solv=DSolve[{f''[t] + f[t]== 0, f[0]== 2,f'[0]==0},f,t]//Flatten

f=f/.solv

f[x]//Simplify

Plot[f[x],{x,0,10}];

Anhang

Remove["Global`*"];

p1={1,2,3};
p2={3,4,2};
p3=2(p1+p2)

p3={8,8,10}

p4=2(p2-p1)

p4={4,0,-2}

Volumen=1/6 Det[{p2-p1,p3-p1,p4-p1}]

a1=p3-p1

b1=p2-p1

Det[{{1,2,3},{0,1,1},{2,1,0}}]

Det[{{1,2,3,4},{0,1,2,-2},{2,1,0,4},{2,1,1,0}}]