Lösungen / Solutions

1

f[x_, t_] := 
      E^(x t)/x +  
        E^(Sin[Sqrt[x^2 + 1]])*(ArcTan[Cos[x - Pi]])^(2/
                5);     f[x, t]

D[f[x, t], t]

Integrate[Evaluate[D[f[x, t], t]], {x, 0, 1}]

2

Remove[f]

f[x_, y_] := Sin[x y] - Cos[x + y];  f[x, y]

a

?Dt

Dt[f[x, y]]

Collect[Dt[f[x, y]], {Dt[x], Dt[y]}]

b

v = {Cos[Pi/4], Sin[Pi/4]}

grad = {D[f[x, y], x], D[f[x, y], y]}

v . grad

v . grad /. {x -> Pi/2, y -> Pi}

N[%]

3

Remove[f]

f[x_, y_] := Sin[x - y];
g[x_, y_] := Sqrt[1 + 2 Cos[x - y]^2];  g[x, y]

Fe[x_, y_] := 
        Sqrt[1 + D[f[x, y], x]^2 + 
            D[f[x, y], y]^2];
Fe[x, y]

A = 
      Integrate[
        Integrate[g[x, y] Fe[x, y], {y, 0, -1/2  x + Pi/2}], {x, 0, 
          Pi}]

N[%]

4

Remove[f]

a

Plot[Pi/x, {x, 0, 5}];

b

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

Integrate[f[x, y], {y, 0, Pi/x}]

Integrate[
      Integrate[f[x, y], {y, 0, Pi/x}], {x, 0, Infinity}]

Integrate[Integrate[f[x, y], {y, 0, Pi/x}], {x, a, b}]

Integrate[Integrate[f[x, y], {y, 0, Pi/x}], {x, 1/b, b}]

Limit[Integrate[Integrate[f[x, y], {y, 0, Pi/x}], {x, 1/b, b}], 
      b -> Infinity]

5

a

Remove[x, y, f]

f[x_, y_] := 3 x^2 - 5  x y  + 4  y^2  - y + x - 1

exp1 = D[f[x, y], x]

{solvthis = Solve[exp1 == 0 /. y -> 1, x], 
      solvthis // N}

{solvthis = Solve[exp1 == 0 /. y -> -1, x], 
      solvthis // N}

exp2 = D[f[x, y], y]

{solvthis = Solve[exp2 == 0 /. x -> 1, y], 
      solvthis // N}

{solvthis = Solve[exp2 == 0 /. x -> -1, y], 
      solvthis // N}

{f[1, 1], f[-1, 1], f[1, -1], f[-1, -1]}

D[f[x, y], y]

s = 
      Solve[{1 + 6 x - 5 y == 0, -1 - 5 x + 8 y == 
              0}, {x, y}] // Flatten

num

f[x, y] /. s

num

Det[{{D[f[x, y], {x, 2}], D[f[x, y], x, y]}, {D[f[x, y], y, x], 
          D[f[x, y], {y, 2}]}}]

Plot3D[f[u, v], {u, -1, 1}, {v, -1, 1}];

b

num := % // N

gra[x_, y_] := Evaluate[{D[f[x, y], x], D[f[x, y], y]}]

vl[x_] := Sqrt[x . x]

lgra[x_, y_] := vl[gra[x, y]]

Solve[(D[lgra[x, y], y] /. x -> 1) == 0, {y}]

num

Solve[(D[lgra[x, y], y] /. x -> -1) == 
        0, {y}]

num

Solve[(D[lgra[x, y], x] /. y -> 1) == 0, {x}]

num

Solve[(D[lgra[x, y], x] /. y -> -1) == 
        0, {x}]

num

c

Show[
        Graphics[{Line[{{1, (83/89)}, {-1, -(57/89)}}], 
            Line[{{(59/61), 
                  1}, {-(81/61), -1}}], 
Line[{{-1, -1}, {-1, 1}}], 
            Line[{{1, -1}, {1, 
                  1}}], 
Line[{{-1.5, -1}, {1, -1}}], Line[{{-1, 1}, {1, 1}}], PointSize[0.03], 
            Point[{-(3/23), (1/23)}]}]];

Kontrolle des Schnittpunktes:

U1 = {1, (83/89)};  U2 = {-1, -(57/89)}; U3 = {(59/61), 1}; 
    U4 = {-(81/61), -1};

ger1[t_] := U1 + t (U2 - U1)

ger2[t_] := U3 + t (U4 - U3)

solv = Solve[{ger1[t] == ger2[tt]}, {t, tt}]

ger1[t] /. solv

ger2[tt] /. solv

O.k. !!!!!

d

Remove[gradi]

g[x_, y_] := y + x/2 - 1/4

g[x, y]

gradi[f_] :=  {D[f, x], D[f, y]}

?gradi

gradi[f[x, y]]

gradi[g[x, y]]

Remove[k]

g1 = ({1 + 6 x - 5 y, -1 - 5 x + 8 y} == 
          k {(1/2), 1})

g2 = (g[x, y] == 0)

Solve[{g1, g2}, {k, x, y}]

num