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1

a

f[x_]:=100 x^100+10 x^10+5x^2-3x+1;
D[f[x],x]

b

D[f[x],x]/.x->1

c

f[x_]:=a x^a+10 x^10+5x^2-3x+1;
D[f[x],x]

D[f[x],x]/.x->a

d

f[x_]:=Cos[x] Cot[x]//Simplify;
D[f[x],x]

e

f[x_]:=Cos[x] Cot[x]//Simplify;
D[f[x],x]/.x->Pi/4

N[%]

f

f[x_]:=Cos[x]/x^2//Simplify;
D[f[x],x]

g

f[x_]:=Cos[x]/x^2//Simplify;
D[f[x],x]/.x->Pi

h

f[x_]:=E^(2x^2-4x+5)//Simplify;
D[f[x],x]

f[x_]:=E^(2x^2-4x+5)//Simplify;
D[f[x],x] /.x->1

Winkel?

2

a

f[x_]:=(x-3)(x-1)/(x-1)^2;
Plot[f[x],{x,-3,3}];

f[x]//Apart

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

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

Solve[Denominator[f[x]]==0,{x}]

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

Solve[Evaluate[D[f[x],{x,2}]==0],{x}]

b

f[x_]:=1+((x-2)^2)/(x-3)^2;
Plot[f[x],{x,-3,4}];

1+((x-2)^2)/(x-3)^2//Apart

1+((x-2)^2)/(x-3)^2//Together

f[x_]:=1+((x-2)^2)/(x-3)^2;
D[f[x],x]//Together

f[x_]:=1+((x-2)^2)/(x-3)^2;
Plot[f[x],{x,1,2.5}];

f[x_]:=1+((x-2)^2)/(x-3)^2;
D[f[x],{x,2}]//Together

f[x_]:=1+((x-2)^2)/(x-3)^2;
Plot[f[x],{x,-10,2.5}];

f[x_]:=1+((x-2)^2)/(x-3)^2;
Plot[{2,f[x]},{x,-20,2.6},PlotRange->{0,2.2}];

f[x]//Apart

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

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

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

Komplex

Solve[Denominator[Together[f[x]]]==0,{x}]

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

Solve[Evaluate[D[f[x],{x,2}]==0],{x}]

c

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

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

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

D[f[x],{x,1}]//Together

D[f[x],{x,2}]//Together

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

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

Solve[Evaluate[D[f[x],{x,2}]==0],{x}]

N[%]//Chop

NSolve[Evaluate[D[f[x],{x,2}]==0],{x}]

d

f[x_]:=Sin[(E^x)];
Plot[f[x],{x,0,2}];

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

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

D[f[x],{x,1}]//Together

D[f[x],{x,2}]//Together

D[f[x],{x,1}]//Together

D[f[x],{x,2}]//Together

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

FindRoot[f[x]==0,{x,1.2}]

FindRoot[f[x]==0,{x,1.8}]

{Log[Pi/2],Log[3Pi/2],Log[5Pi/2]}//N

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

N[%]

Solve[Evaluate[D[f[x],{x,2}]==0],{x}]

FindRoot[Evaluate[D[f[x],{x,2}]==0],{x,0}]

FindRoot[Evaluate[D[f[x],{x,2}]==0],{x,0.8}]

FindRoot[Evaluate[D[f[x],{x,2}]==0],{x,1.2}]

FindRoot[Evaluate[D[f[x],{x,2}]==0],{x,1.8}]

3

a Höhensatz r^2 = h (2R-h)

V[h_]:=r^2 Pi h/3 /. r->Sqrt[h(4-h)]; V[h]

Plot[V[h],{h,0,5}];

D[V[h],{h,1}]//Together

Solve[Evaluate[D[V[h],{h,1}]==0],{h}]

N[%]

b Proportion (h^2+r^2) : (h-R^2)=r^2 : R^2

Solve[(h^2+r^2)/(h-2)^2==r^2/2^2, {h}]

Plot[((4 (r^2))/(-4 + (r^2))), {r, 0, 10}];

V[r_]:=r^2 Pi 4 r^2/(r^2-4) ; V[r]

Plot[V[r], {r, 0, 10}];

D[V[r],{r,1}]//Together

Solve[Evaluate[D[V[r],{r,1}]==0],{r}]

N[%]

c

Remove[x, a, b, c, d]

f[x_,a_,b_,c_,d_]:=(a x^2+b x+ c)/(x + d);
f[x,a,b,c,d]

f[x,a,b,c,d]//Apart

Evaluate[D[f[x,a,b,c,d],x]==0/.x->4]

Evaluate[D[f[x,a,b,c,d],x]==0]

solv=Solve[{Evaluate[D[f[x,a,b,c,d],x]==0/.x->4],8+d==0, a==0.5, b-a d==-3},{a,b,c,d}]//Flatten

f[x_]:=f[x,a,b,c,d]/.solv;
f[0]