Modulprüfung I1a / I1c 2005

1

a) a

f[x_]:= -2(6x^3 + 5x^2 - 6x^1 + x^0)/(-x^2+2x+1)

f'[x]//Expand

f'[x]//Simplify

f'[x]/.x->0

ArcTan[(f'[x]/.x->0)]

ArcTan[16] // N

ArcTan[(f'[x]/.x->0)]//N

Pi/ Degree//N

N[ArcTan[(f'[x]/.x->0)]]/(1 Degree)

Plot[Evaluate[f'[x]],{x,-3,3}];

Evaluate[f'[x]==0]

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

Solve[f'[x]==0,{x}]//N//Chop

a) c

f[x_]:= -2(6x^3 + 5x^2 - 6x^1 + x^0)/(x^2-2x+1)

f'[x]//Expand

f'[x]//Simplify

f'[x]/.x->0

ArcTan[(f'[x]/.x->0)]

ArcTan[8] // N

Pi/ Degree//N

1.44644/ Degree

Plot[Evaluate[f'[x]],{x,-3,3}];

Evaluate[f'[x]==0]

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

Solve[f'[x]==0,{x}]//N//Chop

a) x

f[x_]:= -2(6x^3 + 5x^2 - 8x^1 + x^0)/(x^2-2x+1)+x

f'[x]//Expand

f'[x]//Simplify

f'[x]/.x->0

ArcTan[(f'[x]/.x->0)]

ArcTan[8] // N

Pi/ Degree//N

1.44644/ Degree

Plot[Evaluate[f'[x]],{x,-3,3}];

Evaluate[f'[x]==0]

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

Solve[f'[x]==0,{x}]//N//Chop

b) a, c

f[x_]:= Sin[x] E^(x)-Cos[E^(-x)]-(x^3)/(x+1)

f'[x]//Expand

((f'[x]//Expand)[[1]]+(f'[x]//Expand)[[2]])//Together

(f'[x]/.x->1.0)//N

f''[x]

f''[x]//Simplify

((f''[x]//Expand)[[1]]+(f''[x]//Expand)[[2]]+(f''[x]//Expand)[[3]])//Together

Limit[Evaluate[f''[x]],x->Infinity]

b) x

f[x_]:= Sin[x] E^( x)-Cos[Log[x]]-(x^3)/(x+1)

f'[x]//Expand

((f'[x]//Expand)[[1]]+(f'[x]//Expand)[[2]])//Together

(f'[x]/.x->1.0)//N

f''[x]

f''[x]//Simplify

((f''[x]//Expand)[[1]]+(f''[x]//Expand)[[2]]+(f''[x]//Expand)[[3]])//Together

Limit[Evaluate[f''[x]],x->Infinity]

2

a

f[x_]:=-2 E^(-x^2)+1;
Plot[f[x],{x,-2,2}];

A[x_]:=f[x] 2x/2

Plot[A[x],{x,-2,2}];

A'[x]

Plot[A'[x],{x,-2,2}];

Solve[Evaluate[A'[x]==0],{x}]

Solve[Evaluate[A'[x]==0],{x}]//N

Plot[Evaluate[A'[x]],{x,-5,5}];

FindRoot[Evaluate[A'[x]==0],{x,0.5}]

u[1]=0.5;
u[n_]:=u[n-1]-A'[u[n-1]]/A''[u[n-1]];
Table[u[n],{n,1,5}]//MatrixForm

c

f[x_]:=2 E^(-x^2)-1;
A[x_]:=f[x] 2x/2

Plot[A[x],{x,-2,2}];

A'[x]

Plot[A'[x],{x,-2,2}];

Solve[Evaluate[A'[x]==0],{x}]

Solve[Evaluate[A'[x]==0],{x}]//N

Plot[Evaluate[A'[x]],{x,-5,5}];

FindRoot[Evaluate[A'[x]==0],{x,0.5}]

u[1]=0.5;
u[n_]:=u[n-1]-A'[u[n-1]]/A''[u[n-1]];
Table[u[n],{n,1,5}]//MatrixForm

x

f[x_]:=3 E^(-x^2)-2;
A[x_]:=f[x] 2x/2

Plot[A[x],{x,-2,2}];

A'[x]

Plot[A'[x],{x,-2,2}];

Solve[Evaluate[A'[x]==0],{x}]

Solve[Evaluate[A'[x]==0],{x}]//N

Plot[Evaluate[A'[x]],{x,-5,5}];

FindRoot[Evaluate[A'[x]==0],{x,0.5}]

u[1]=0.5;
u[n_]:=u[n-1]-A'[u[n-1]]/A''[u[n-1]];
Table[u[n],{n,1,5}]//MatrixForm

3 a

Remove["Global`*"]

a, b

f[x_]:=(x+3)(x+2)(x-4);
Plot[f[x],{x,-4,5}];

f1[x_,a_]:=a (x+3)(x+2)(x-4);
f2[x_,b_]:=x^2+b;

solv=Flatten[Solve[{f1[x,a]==f2[x,b],Evaluate[D[f1[x,a],x]==D[f2[x,b],x]]},{a,b}]]

q=solv/.x->-2

%//N

s1={-2,f1[-2]}

S2

f1[x_]:=a (x+3)(x+2)(x-4)/.q;
f2[x_]:=x^2+b/.q;
Plot[{f1[x],f2[x]},{x,-4,5}];

solv2=Flatten[Solve[{f1[x]==f2[x]},{x}]]

Evaluate[f2'[x]]

x0=x/.solv2[[3]]

%//N

f1[x0]

%//N

s2={x0,f1[x0]}

f3'

m=Evaluate[f1'[x]]/.x->s2[[1]]

f1''[x]//Simplify

solv3=Flatten[Solve[Evaluate[f1''[x]==0],{x}]]

%//N

xW=x/.solv3

sW={xW,f1[xW]}

h[x_,a1_,b1_,c1_,d1_]:= a1 x^3+b1 x^2+c1 x+d1;
solv4=Solve[{
h[s1[[1]],a1,b1,c1,d1]==s1[[2]],
h[s2[[1]],a1,b1,c1,d1]==s2[[2]],
h[sW[[1]],a1,b1,c1,d1]==sW[[2]],
Evaluate[(D[h[x,a1,b1,c1,d1],x]/.x->s1[[1]])==m]},{a1,b1,c1,d1}
]//Flatten

h[x_]:= a1 x^3+b1 x^2+c1 x+d1/.solv4;

h'[x]/.x->s1[[2]]

%//N

Plot[{f1[x],f2[x],h[x]},{x,-5,6}];

Old

m=Evaluate[f2'[x]]/.solv2[[1]]

f1''[x]//Simplify

solv3=Flatten[Solve[Evaluate[f1''[x]==0],{x}]]

h[x_]:=f2[x0]+m(x-x0);
h[-1/3]

Plot[{f1[x],f2[x],h[x]},{x,-5,4}];

f1[-1/3]

f1[x0]

%//N

s1={}

w={-1/3,f1[-1/3]}

%//N

3 c

Remove["Global`*"]

a, b

f[x_]:=(x-3)(x-2)(x+4);
Plot[f[x],{x,-5,4}];

f1[x_,a_]:=a (x-3)(x-2)(x+4);
f2[x_,b_]:=x^2+b;

solv=Flatten[Solve[{f1[x,a]==f2[x,b],Evaluate[D[f1[x,a],x]==D[f2[x,b],x]]},{a,b}]]

q=solv/.x->2

%//N

s1={-2,f1[-2]}

S2

f1[x_]:=a (x-3)(x-2)(x+4)/.q;
f2[x_]:=x^2+b/.q;
Plot[{f1[x],f2[x]},{x,-5,4}];

solv2=Flatten[Solve[{f1[x]==f2[x]},{x}]]

Evaluate[f2'[x]]

x0=x/.solv2[[1]]

%//N

f1[x0]

%//N

s2={x0,f1[x0]}

f3'

m=Evaluate[f1'[x]]/.x->s2[[1]]

f1''[x]//Simplify

solv3=Flatten[Solve[Evaluate[f1''[x]==0],{x}]]

%//N

xW=x/.solv3

sW={xW,f1[xW]}

h[x_,a1_,b1_,c1_,d1_]:= a1 x^3+b1 x^2+c1 x+d1;
solv4=Solve[{
h[s1[[1]],a1,b1,c1,d1]==s1[[2]],
h[s2[[1]],a1,b1,c1,d1]==s2[[2]],
h[sW[[1]],a1,b1,c1,d1]==sW[[2]],
Evaluate[(D[h[x,a1,b1,c1,d1],x]/.x->s1[[1]])==m]},{a1,b1,c1,d1}
]//Flatten

h[x_]:= a1 x^3+b1 x^2+c1 x+d1/.solv4;

h'[x]/.x->s1[[2]]

%//N

Plot[{f1[x],f2[x],h[x]},{x,-5,4}];

Old

m=Evaluate[f2'[x]]/.solv2[[1]]

f1''[x]//Simplify

%//N

solv3=Flatten[Solve[Evaluate[f1''[x]==0],{x}]]

%//N

h[x_]:=f2[x0]+m(x-x0);
h[1/3]

%//N

Plot[{f1[x],f2[x],h[x]},{x,-5,4}];

f1[1/3]

%//N

s1={}

w={1/3,f1[1/3]}

%//N

3 x

Remove["Global`*"]

a, b

f[x_]:=(x-6)(x-2)(x+4);
Plot[f[x],{x,-5,4}];

f1[x_,a_]:=a (x-3)(x-2)(x+4);
f2[x_,b_]:=x^2+b;

solv=Flatten[Solve[{f1[x,a]==f2[x,b],Evaluate[D[f1[x,a],x]==D[f2[x,b],x]]},{a,b}]]

q=solv/.x->2

%//N

s1={-2,f1[-2]}

S2

f1[x_]:=a (x-3)(x-2)(x+4)/.q;
f2[x_]:=x^2+b/.q;
Plot[{f1[x],f2[x]},{x,-5,4}];

solv2=Flatten[Solve[{f1[x]==f2[x]},{x}]]

Evaluate[f2'[x]]

x0=x/.solv2[[1]]

%//N

f1[x0]

%//N

s2={x0,f1[x0]}

f3'

m=Evaluate[f1'[x]]/.x->s2[[1]]

f1''[x]//Simplify

solv3=Flatten[Solve[Evaluate[f1''[x]==0],{x}]]

%//N

xW=x/.solv3

sW={xW,f1[xW]}

h[x_,a1_,b1_,c1_,d1_]:= a1 x^3+b1 x^2+c1 x+d1;
solv4=Solve[{
h[s1[[1]],a1,b1,c1,d1]==s1[[2]],
h[s2[[1]],a1,b1,c1,d1]==s2[[2]],
h[sW[[1]],a1,b1,c1,d1]==sW[[2]],
Evaluate[(D[h[x,a1,b1,c1,d1],x]/.x->s1[[1]])==m]},{a1,b1,c1,d1}
]//Flatten

h[x_]:= a1 x^3+b1 x^2+c1 x+d1/.solv4;

h'[x]/.x->s1[[2]]

%//N

Plot[{f1[x],f2[x],h[x]},{x,-5,4}];

4 Hyperbel

p[1]={3,6};
p[2]={-2,6};
p[3]={-3,-5};
p[4]={3,-4};

f[x_,y_,a_,c_,d_,e_]:=a x^2+c y^2+d x+e y + 1;

tab1=Table[f[p[k][[1]],p[k][[2]],a,c,d,e]==0,{k,1,4}]

solv1=Flatten[Solve[tab1,{a,c,d,e}]]

f[x_,y_,a_,c_,d_,e_]:=a x^2+c y^2+d x+e y + 1/.solv;

f[x_,y_]:=f[x,y,a,c,d,e];
f[x,y]

solv2=Flatten[Solve[f[x,y]==0,{y}]]

graf1[x_]:=y/.solv2[[1]];  pl1= Plot[Evaluate[graf1[x]],{x,-3,4}];

graf2[x_]:=y/.solv2[[2]];  pl2= Plot[Evaluate[graf2[x]],{x,-3,4}];

Show[pl1,pl2];

4  Ellipse a

p[1]={3,-6};
p[2]={-3,-6};
p[3]={-3,5};
p[4]={0,6};
p[5]={3,4};

f[x_,y_,a_,b_,c_,d_,e_]:=a x^2+ b x y+c y^2+d x+e y+ 1;

tab1=Table[f[p[k][[1]],p[k][[2]],a,b,c,d,e]==0,{k,1,5}]

solv1=Flatten[Solve[tab1,{a,b,c,d,e}]]

f[x_,y_,a_,b_,c_,d_,e_]:=a x^2+b x y+c y^2+d x+e y + 1/.solv1;

f[x_,y_]:=f[x,y,a,b,c,d,e];
f[x,y]

solv2=Flatten[Solve[f[x,y]==0,{y}]]

graf1[x_]:=y/.solv2[[1]];  pl1= Plot[Evaluate[graf1[x]],{x,-5,5}];

graf2[x_]:=y/.solv2[[2]];  pl2= Plot[Evaluate[graf2[x]],{x,-5,5}];

Show[pl1,pl2];

D[graf2[x],x]

Solve[Evaluate[D[graf2[x],x]]==0,{x}]

Solve[Evaluate[D[graf2[x],x]]==0,{x}]//N

Frage: x max > -0.5 ??  !!

4  Ellipse c

p[1]={3,6};
p[2]={-3,6};
p[3]={-3,-5};
p[4]={0,-6};
p[5]={3,-4};

f[x_,y_,a_,b_,c_,d_,e_]:=a x^2+ b x y+c y^2+d x+e y+ 1;

tab1=Table[f[p[k][[1]],p[k][[2]],a,b,c,d,e]==0,{k,1,5}]

solv1=Flatten[Solve[tab1,{a,b,c,d,e}]]

f[x_,y_,a_,b_,c_,d_,e_]:=a x^2+b x y+c y^2+d x+e y + 1/.solv1;

f[x_,y_]:=f[x,y,a,b,c,d,e];
f[x,y]

solv2=Flatten[Solve[f[x,y]==0,{y}]]

graf1[x_]:=y/.solv2[[1]];  pl1= Plot[Evaluate[graf1[x]],{x,-5,5}];

graf2[x_]:=y/.solv2[[2]];  pl2= Plot[Evaluate[graf2[x]],{x,-5,5}];

Show[pl1,pl2];

D[graf1[x],x]

Solve[Evaluate[D[graf1[x],x]]==0,{x}]

Solve[Evaluate[D[graf1[x],x]]==0,{x}]//N

Frage: x max > -0.5 ??  Ja!!

4  Ellipse x

p[1]={3,8};
p[2]={-3,8};
p[3]={-3,-5};
p[4]={0,-6};
p[5]={3,-4};

f[x_,y_,a_,b_,c_,d_,e_]:=a x^2+ b x y+c y^2+d x+e y+ 1;

tab1=Table[f[p[k][[1]],p[k][[2]],a,b,c,d,e]==0,{k,1,5}]

solv1=Flatten[Solve[tab1,{a,b,c,d,e}]]

f[x_,y_,a_,b_,c_,d_,e_]:=a x^2+b x y+c y^2+d x+e y + 1/.solv1;

f[x_,y_]:=f[x,y,a,b,c,d,e];
f[x,y]

solv2=Flatten[Solve[f[x,y]==0,{y}]]

graf1[x_]:=y/.solv2[[1]];  pl1= Plot[Evaluate[graf1[x]],{x,-5,5}];

graf2[x_]:=y/.solv2[[2]];  pl2= Plot[Evaluate[graf2[x]],{x,-5,5}];

Show[pl1,pl2];

D[graf1[x],x]

Solve[Evaluate[D[graf1[x],x]]==0,{x}]

Solve[Evaluate[D[graf1[x],x]]==0,{x}]//N

Frage: x max > -0.5 ??

5 a

a

Remove[a,b,c]

a[n_]:=n^4;
b[n_]:=a[n+1]-a[n];
c[n_]:=b[n+1]-b[n];

{b[n],c[n]}//Simplify

Table[c[n],{n,1,10}]

c[9000]

c[9000]//N

b

Solve[(y-4)/8==4/(x-8),{y}]

Solve[(y-4)/8==4/(x-8),{x}]

la[x_]:=Sqrt[x^2+(4 x/(x-8))^2]

Plot[la[x],{x,11,15}];

la'[x]//Simplify

Solve[Evaluate[la'[x]==0],{x}]

Solve[Evaluate[la'[x]==0],{x}]//N

5 c

a

Remove[a,b,c]

a[n_]:=n^4;
b[n_]:=a[n+1]-a[n];
c[n_]:=b[n+1]-b[n];

{b[n],c[n]}//Simplify

Table[c[n],{n,1,10}]

c[10000]

c[10000]//N

b

Solve[(y-3)/6==3/(x-6),{y}]

Solve[(y-3)/6==3/(x-6),{x}]

la[x_]:=Sqrt[x^2+(3x/(x-6))^2]

Plot[la[x],{x,7,12}];

la'[x]//Simplify

Solve[Evaluate[la'[x]==0],{x}]

Solve[Evaluate[la'[x]==0],{x}]//N

5 x

a

Remove[a,b,c]

a[n_]:=n^4;
b[n_]:=a[n]-a[n+1];
c[n_]:= -2(b[n+1]-b[n]);

{b[n],c[n]}//Simplify

Table[c[n],{n,1,10}]

c[20000]

c[20000]//N

b