Lösungen/Solutions/E1S1AnUe/Ex12.1

Ueb/Ex 1      W

(* Old Form: Remove["Global`@*"] *)

Remove["Global`*"]

f[x_] := (x^2 - 4)/(x - 1)^2

Plot[f[x], {x, -1, 8}, PlotRange -> {5, -20}];

D[f[x], x]

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

{4, f[4]}

{4, f[4]} // N

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

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

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

Ueb/Ex 2      W

(* Old Form: Remove["Global`@*"] *)

Remove["Global`*"]

f[x_] := x^2

f'[x]

t[x_, x1_] := 2  x1 x  +  u;  
    solv1 = Solve[f[x1] == t[x1, x1], u] // Flatten

t[x_, x1_] := 2  x1 x  +  u /. solv1;   t[x, x1]

t[x_, x1_] := 2 x x1 - (x1^2)

t[1, x1] == -8

Solve[t[1, x1] == -8, {x1}]

{f[-2], t[-2, -2]}

{f[4], t[4, 4]}

Ueb/Ex 3      W

(* Old Form: Remove["Global`@*"] *)

Remove["Global`*"]

f[x_] := x^4 - 12  x^3 + 46  x^2 - 60  x + 25

D[f[x], x]

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

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

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

% // N

Ueb/Ex 4      W

(* Old Form: Remove["Global`@*"] *)

Remove["Global`*"]

mantellinie[x_] := -12/4 x  + b;  
    mantellinie[4] == 0

solv2 = Solve[mantellinie[4] == 0, {b}] //  Flatten

mantellinie[x_] := -12/4 x  + b /. solv2;  
    mantellinie[x]

volumenZylinder[x_] := x^2 Pi *mantellinie[x]; 
    volumenZylinder[x]

volumenZylinder'[x]  //  Simplify  //  Expand

D[volumenZylinder[x], x] //  Simplify  //  Expand

solv4 = 
      Solve[24 π x - 9 π (x^2) == 0, {x}] // 
        Flatten

N[%]

volumenZylinder''[x]  //  Simplify  //  Expand

24 π - 18 π x  /.  solv4[[1]]

24 π - 18 π x  /.  solv4[[2]]

===> Maximum bei    x -> 
      (8/3)

Ueb/Ex 5      W

(* Old Form: Remove["Global`@*"] *)

Remove["Global`*"]

f[x_] := Sin[x]

rechteck[x_] :=  ((Pi - x) - x)*f[x];   rechteck[
      x]

Solve[rechteck'[x] == 0, {x}]

===> Funktioniert nicht!!!!

gleich =  (a == b)

(gleich)[[1]]

(gleich)[[2]]

gleich1 =  (rechteck'[x] == 0)

gleichAdd[x_, 
        y_] := (x[[1]] + y == x[[2]] + y)

gleichMult[x_, 
        y_] := (x[[1]]*y == x[[2]]*y)

gleich2 = gleichAdd[(gleich1), 2 Sin[x]]

gleich3 = gleichMult[gleich2, (1/Cos[x])]

Plot[{π - 2 x, 2 Tan[x]}, {x, 0, Pi},  
        PlotRange -> {-4, 4}];

FindRoot[π - 2 x == 2 Tan[x], {x, 0.5}]

Plot[{π - 2 x, 2 Tan[x]}, {x, 0.70, 0.72}];