Lösungen

1

a

f[x_]:=2+Cos[x]-Sin[x]; a = 0; b= 2 Pi;

Plot[{2+Cos[x]-Sin[x],Cos[x],Sin[x]},{x,0,2 Pi},PlotRange->{-1,3}];

Integrate[f[x],x]

Integrate[f[x],{x,a,b}]

N[%]

b

f[x_]:=x^4/4+x^3/3-3 x^2+2x-4; a = 2; b= -2;

Integrate[f[x],x]

Integrate[f[x],{x,a,b}]

N[%]

c

f[x_,t_]:=(t^2+2 t) E^x; a = 0; b= -1;

g[t_]:=Evaluate[Integrate[f[x,t],x]]; g[t]

g[t_]:=Evaluate[Integrate[f[x,t],{x,a,b}]]; g[t]

D[g[t],{t}]

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

d

Remove[a]

f[x_]:= a Cos[2 x Pi-Pi/2]; u = 0; b = Pi;

Integrate[f[x],x]

Integrate[f[x],{x,u,b}]

N[%]

e

Remove[u]

f[x_]:= x^3 Log[u x]; a = 1; b= E;

Integrate[f[x],x]

Integrate[f[x],{x,a,b}]

N[%]

%//Expand

f

f[x_]:=3x/x/(x-1)/(x+1); a = 2; b= 4;

Integrate[f[x],x]

Integrate[f[x],{x,a,b}]

N[%]

Integrate[f[x],{x,a,b}]/.Log->log

(3/10) (-5 log[5] + 
            5 log[19] + (Sqrt[5]) log[(1/19) (21 + 4 (Sqrt[5]))]) /. 
      a_ log[x_] + b_ log[y_] + c_ log[z_] -> 
        log[x^(a)  y^b z^c]

g

f[x_]:=x E^(-x^2); a = 0; b= 1;

Integrate[f[x],x]

Integrate[f[x],{x,a,b}]

N[%]

2

Remove["Global`*"]

f[x_]:=2 E^(-(x-1)^2); a = 0; b= 3/2;

Integrate[f[x],x]

Integrate[f[x],{x,a,b}]

N[%]

a

n=8

xt[k_]:= k(b-a)/n

sR = Sum[f[xt[k]] (b-a)/n,{k,1,n}]//N

b

sT = f[xt[a]]/2 (b-a)/n+Sum[f[xt[k]] (b-a)/n,{k,1,n-1}]+ f[xt[b]]/2 (b-a)/n // N

c

sT-sR

d

n=100;
Sum[f[xt[k]] (b-a)/n,{k,1,n}]//N

f[xt[a]]/2 (b-a)/n+Sum[f[xt[k]] (b-a)/n,{k,1,n-1}]+ f[xt[b]]/2 (b-a)/n // N

3

Remove["Global`*"]

a

h = 3.5;
b = 6.5;
l = 20000;
f[x_]:= a x^4+h;

solv = Solve[f[b/2]==0,{a}] // Flatten

f[x_]:= a x^4+h /.solv; f[x]

Plot[f[x],{x,-b/2,b/2},AspectRatio->Automatic,PlotRange->{0,h}];

b

A1 = Integrate[f[x],{x,-b/2,b/2}]

A1 = Integrate[f[x],{x,-b/2,b/2}]//N

r1 = Integrate[f[x],{x,-b/2,b/2}] l

c

a = a/.solv

f[x_]:= a x^4 + h1; f[x]

solv = Solve[f[b1/2]==0,{h1}] // Flatten

f[x_]:= a x^4+h1  /. solv; f[x]

Integrate[f[x],{x,-b1/2,b1/2}] == 2 A1

solv1 = Solve[Integrate[f[x],{x,-b1/2,b1/2}] == 2 A1,{b1}] // Flatten

solv1[[5]]

b1 = b1 /. solv1[[5]]

solv

4

Remove["Global`*"]

a

f[x_,c_]:= Integrate[t^3 /4+-3 t,t]+c /.t->x;  f[x,c]

f[0,c]==5

f[x_]:= f[x,5]; f[x]

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

b

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

N[%]

c

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

N[%]

d

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

e

Integrate[f[x],{x,-2,2}]

N[%]

5

Remove["Global`*"]

Integrate[3x/x/(x-1)/(x+1),{x,2,4}]

f[x_,y_]:=2 (x-y)^2+3 x y^2-5 (2 y+x)

f[x,y]//TeXForm

f[x_,y_,z_]:=f[x,y]

p1=Plot3D[f[x,y],{x,-4,4},{y,-4,4}];

p2=ParametricPlot3D[{x,2x-1,z},{x,-4,4},{z,-100,100}];

Show[p1,p2];

a

<<Calculus`VectorAnalysis`

gf=Grad[f[x,y,z], Cartesian[x, y, z]]

b

Solve[Evaluate[Grad[f[x,y,z], Cartesian[x, y, z]]]==0,{x,y,z}]

Solve[Evaluate[Grad[f[x,y,z], Cartesian[x, y, z]]]==0,{x,y,z}]//N

c

g[x_,y_]:=2x-1-y

g[x_,y_,z_]:=g[x,y]

gg=Grad[g[x,y,z], Cartesian[x, y, z]]

Solve[{gf[[1]]==λ gg[[1]],gf[[2]]==λ gg[[2]],g[x,y]==0},{λ,x,y}]

Solve[{gf[[1]]==λ gg[[1]],gf[[2]]==λ gg[[2]],g[x,y]==0},{λ,x,y}]//N