Lösungen - Solutions

1

Remove["Global`*"]

a

E^
  (Sin[
      Cos[
          Cot[x]
          ]
       ]/Log[x Log[x^5-Log[x]]]) Tan[Cos[-3Sin[x]+1/x^4]]//TeXForm

D[E^
  (Sin[
      Cos[
          Cot[x]
          ]
       ]/Log[x Log[x^5-Log[x]]]) Tan[Cos[-3Sin[x]+1/x^4]],x]

b

x^x^x^x^x^x//TeXForm

D[x^x^x^x^x^x,x]

((((x^x)^x)^x)^x)^x//TeXForm

D[((((x^x)^x)^x)^x)^x,x]

x^(x^(x^(x^(x^x))))//TeXForm

D[x^(x^(x^(x^(x^x)))),x]

c

g[x_]:=x^x

?Nest

Nest[g,x,3]

Nest[g,x,5]//TeXForm

Nest[g,x,5]

D[Nest[g,x,5],x]

2

Remove["Global`*"]

a

a2=0; a3=0;

For[i=1,i<10,
i++,
c1[i]=(2.59-2.08 a2-1.44 a3)/6.25; 
a1=c1[i];
c2[i]=(-5.22-1.78 a1-0.44 a3)/4.61;
a2=c2[i];
c3[i]=(3.61+1.36 a1+0.95 a2)/3.75;
a3=c3[i];Print[{a1,a2,a3}]]

b

x2[0]=0; x3[0]=0;
x1[n_]:=(2.59-2.08 x2[n-1]-1.44 x3[n-1])/6.25;
x2[n_]:=(-5.22-1.78 x1[n]-0.44 x3[n-1])/4.61;
x3[n_]:=(3.61+1.36 x1[n]+0.95 x2[n])/3.75;

Table[{x1[n],x2[n],x3[n]},{n,1,2}]

Table[{x1[n],x2[n],x3[n]},{n,1,8}]

c

Solve[{y1==(2.59-2.08 y2-1.44 y3)/6.25,
y2==(-5.22-1.78 y1 -0.44 y3 )/4.61,
y3==(3.61+1.36 y1 +0.95 y2 )/3.75},{y1,y2,y3}]

d

a2=0; a3=0;

For[i=1,i<30,
i++,
c1[i]=(2.59-2 a2-3 a3)/1; 
a1=c1[i];
c2[i]=(5.22-2 a1-4 a3)/3;
a2=c2[i];
c3[i]=(-3.61-1 a1+1 a2)/1;
a3=c3[i];Print[{a1,a2,a3}]]

3

Remove["Global`*"]

f[x_,a3_,a2_,a1_,a0_]:= a3 x^3 + a2 x^2 + a1 x + a0

D[f[x ,a3 ,a2 ,a1 ,a0 ],x]

f1[x_,a3_,a2_,a1_,a0_]:= 3 a3 x^2 + 2 a2 x + a1

solv= Solve[{f[-2,a3,a2,a1,a0]==2,f[1/2,a3,a2,a1,a0]==0,
f[2,a3,a2,a1,a0]==0,f1[2,a3,a2,a1,a0]==2},{a3,a2,a1,a0}]//
Flatten

f[x_,a3_,a2_,a1_,a0_]:= a3 x^3 + a2 x^2 + a1 x + a0 /. solv

f[x,a3,a2,a1,a0]

p1=Plot[f[x,a3,a2,a1,a0],{x,-5,3}];

Remove["Global`*"]

f[x_,a3_,a2_,a1_,a0_]:= a3 x^3 + a2 x^2 + a1 x + a0

D[f[x ,a3 ,a2 ,a1 ,a0 ],x]

f1[x_,a3_,a2_,a1_,a0_]:= 3 a3 x^2 + 2 a2 x + a1

solv= Solve[{f[-2,a3,a2,a1,a0]==2,f[-1/2,a3,a2,a1,a0]==0,
f[2,a3,a2,a1,a0]==0,f1[2,a3,a2,a1,a0]==2},{a3,a2,a1,a0}]//
Flatten

f[x_,a3_,a2_,a1_,a0_]:= a3 x^3 + a2 x^2 + a1 x + a0 /. solv

f[x,a3,a2,a1,a0]

p1=Plot[f[x,a3,a2,a1,a0],{x,-5,3}];

D[f[x,a3,a2,a1,a0],{x,2}]

Solve[Evaluate[D[f[x,a3,a2,a1,a0],{x,2}]]==0,{x}]

p2=Show[Graphics[{PointSize[0.03],Point[{-47/42,f[-47/42,a3,a2,a1,a0]}]}]]

Show[p1,p2]

4

Remove["Global`*"]

f[x_]:=(297-2x)(210-2x)x;

f[x]//Expand

D[f[x],x]//Simplify

Solve[Evaluate[D[f[x],x]]==0,{x}]

Solve[Evaluate[D[f[x],x]]==0,{x}]//N

Plot[f[x],{x,-5,140}]

g[x_]:=(297-2x)(210-2x)x + 4 (x/3)^3; g[x]//Expand

D[g[x],x]//Simplify

Solve[Evaluate[D[g[x],x]]==0,{x}]

%//N

Plot[g[x],{x,-5,150}];

5

Remove["Global`*"]

f[a_,x_]:=a(x-1)(x+1);

f[a,x]//Expand

t1[x_,b_]:=(Evaluate[D[f[a,y],y]/.y->1] x + b); t1[1,b1]

solv=Solve[t1[1,b]==0,{b}]//Flatten

t2[u_]:=t1[u,b]/.solv;
t2[v]

A= (1-(-1)) t2[0]/2

6

f1[x_]:=x^(1/5);  D[f1[x],x]

f1[x_]:=ArcSin[x];  D[f1[x],x]

7

f[x_]:=1/2 (x+1)(x-1)(x-2)//Expand;  f[x]

f[-2]

f[3]

ms=(f[3]-f[-2])/(3-(-2))

D[f[x],x]

solv=Solve[Evaluate[D[f[x],x]]==ms,{x}]

%//N

x1=x/.solv[[1]]

x2=x/.solv[[2]]

s[x_]:=ms x + bs; s[-2]==f[-2]

solv=Solve[s[-2]==f[-2],{bs}]//Flatten

s[x_]:=ms x + bs/. solv

Plot[{f[x],s[x]},{x,-2,3},PlotRange->All,Epilog->{{PointSize[0.03],Point[{x1,f[x1]}],Point[{x2,f[x2]}]}}]