Aufgabe 39

(a)

plot1 = 
      Plot[((5  x - 8)/(6  x + 7)), {x, -5, 5}, 
        PlotStyle -> RGBColor[1, 0, 0], AspectRatio -> Automatic, 
        PlotRange -> {-5, 5}]

sol = Solve[y == ((5  x - 8)/(6  x + 7)), x]

plot2 = 
      Plot[x /. sol, {y, -5, 5}, PlotStyle -> RGBColor[0, 1, 0], 
        AspectRatio -> Automatic, 
        PlotRange -> {-5, 5}]

Show[plot1, plot2]

(b)

plot1 = 
      Plot[(((x^2) - 3)/(2  x)), {x, -5, 5}, 
        PlotStyle -> RGBColor[1, 0, 0], AspectRatio -> Automatic, 
        PlotRange -> {-5, 5}]

sol = Solve[y == (((x^2) - 3)/(2  x)), x]

plot2 = 
      Plot[x /. sol[[1]], {y, -5, 5}, 
        PlotStyle -> RGBColor[0, 1, 0], AspectRatio -> Automatic, 
        PlotRange -> {-5, 5}]

plot3 = 
      Plot[x /. sol[[2]], {y, -5, 5}, 
        PlotStyle -> RGBColor[0, 0, 1], AspectRatio -> Automatic, 
        PlotRange -> {-5, 5}]

Show[plot1, plot2, plot3]

Aufgabe 40

Plot[ {1 - x, 
        (1/x), (Sqrt[1 - (x^2)]), ((1 - (x^3))^(1/3)), ((1 - (x^4))^(1/4)), 
        If[x < 1, (1/(x^2)), (1/(Sqrt[x]))]}, {x, 0, 5}, 
      PlotStyle -> RGBColor[0, 0, 1], AspectRatio -> Automatic, 
      PlotRange -> {0, 5}]

Aufgabe 41

p[x_] := 
      30  (x^5) + 73  (x^4) - 239  (x^3) + 211  (x^2) - 269  x + 138

Plot[p[x], {x, -5, 2}]

p[-5]

p[-4]

p[1]

p[2]

Dies liefert die Intervalle (-5,-4), (-4,1) und (1,2) für drei reelle Nullstellen. Alle Nullstellen findet man mit

Factor[p[x]]

da diese rational sind.

Aufgabe 42

Limit[Ln[1 + x]/x, x -> 0]

Limit[Log[1 + x]/x, x -> 0]

Aufgabe 43

Solve[{g == ((f[x] + f[-x])/2), u == ((f[x] - f[-x])/2)}, 
      f[x]]

Solve[{g == ((f[x] + f[-x])/2), 
        u == ((f[x] - f[-x])/2)}, {f[x], f[-x]}]

Solve[{z == x + I y, 
        zquer == x - I y}, {x, y}]

Solve[{x == ((z + zquer)/2), 
        y == ((z - zquer)/(2  I))}, {z, zquer}]

Aufgabe 44

Needs["Graphics`ImplicitPlot`"]

plot1 = 
      ImplicitPlot[x*y == 1, {x, -5, 5}, {y, -5, 5}, 
        PlotPoints -> 100, 
        PlotStyle -> RGBColor[1, 0, 0]]

sol = 
      Solve[{xtilde == x + y, ytilde == x - y}, {x, 
          y}]

gleichung = x*y - 1 == 0 /. sol[[1]]

plot2 = 
      ImplicitPlot[gleichung, {xtilde, -5, 5}, {ytilde, -5, 5}, 
        PlotPoints -> 100, 
        PlotStyle -> RGBColor[0, 1, 0]]

Show[plot1, plot2]

Map[Expand, gleichung]

Eliminate[ {xtilde == x + λ, 
        ytilde == x - λ}, x]

Eliminate[ {xtilde == μ + y, ytilde == μ - y}, 
      y]