Aufgabe 3;

x = {0, 0.25, 0.5, 0.75, 1};

y = N[Exp[x]]

R = 
      Sum[{(a*x[[i]]^2 + b*x[[i]] + c - y[[i]])^2}, {i, 1, 
          5}]

Simplify[R]

p = 
      Fit[{{x[[1]], y[[1]]}, {x[[2]], y[[2]]}, {x[[3]], 
            y[[3]]}, {x[[4]], y[[4]]}, {x[[5]], 
            y[[5]]}}, {1, u, u^2}, u]

a = {0, 0.25, 0.5, 0.75, 1}

d = 
      Abs[Exp[a] - (1.00514 + 0.864277 a + 
              0.843538 (a^2))]

p1 = 
        Plot[Exp[x], {x, 0, 1},  
          PlotStyle -> Dashing[{0.01}]];

p2 = 
        Plot[p, {u, 0, 1},  PlotStyle -> Dashing[{0.03}]];

Show[p1, p2]

p3 = 
        Plot[Exp[x], {x, 0, 5},  
          PlotStyle -> Dashing[{0.01}]];

p4 = 
        Plot[p, {u, 0, 5},  PlotStyle -> Dashing[{0.03}]];

Show[p3, p4]

p5 = Plot[Exp[u] - p, {u, 0, 1}]

FindRoot[(Exp[v] - 0.864277  - 2*0.843538 v), {v, 
        0.2}]

FindRoot[(Exp[v] - 0.864277  - 2*0.843538 v), {v, 
        0.8}]