Lösungen

1

Remove["Global`*"]

a

i

f[x_] := 
        1/5 x^5 + 1/4 x^4 + 1/3 x^3 + 1/2  x^2 + x - 
          1;
f[x]

f'[x]

Evaluate[f'[x]] (x - 1) // Expand

ii

(Evaluate[f'[x]] // Expand) /. x -> 1

iii

ArcTan[(Evaluate[f'[x]] // Expand) /. x -> 1]

N[%]

b

i

f1[x_] := Log[3 x^2 + 1];  f1[x]

f1'[x] // Simplify

r1 = (f1'[x] // Simplify) /. x -> 0

f2[x_] := E^(a x);  f2[x]

f2'[x] // Simplify

r2 = (f2'[x] // Simplify) /. x -> 0

f3[a_] := E^(a x);  f3[a]

f3'[a] // Simplify

r3 = (f3'[a] // Simplify) /. x -> 0

f4[x_] := E^(2  a);  f4[x]

f4'[x] // Simplify

r4 = (f4'[x] // Simplify) /. x -> 0

f5[x_] := Tan[x + 1] + Sin[x^2];  f5[x]

f5'[x] // Simplify

r5 = (f5'[x] // Simplify) /. x -> 0

N[%]

r1 + r2 + r3 + r4 + r5

r1 + r2 + r3 + r4 + r5 /. a -> 1

N[%]

ii

(f1'[x] // N) /. {x -> Pi, a -> 2}

s[1] = 
      ArcTan[(f1'[x] // N)] /. {x -> Pi, a -> 2}

(f2'[x] // N) /. {x -> Pi, a -> 2}

s[2] = 
      ArcTan[(f2'[x] // N)] /. {x -> Pi, a -> 2}

(f3'[x] // N) /. {x -> Pi, a -> 2}

s[3] = 
      ArcTan[(f3'[a] // N)] /. {x -> Pi, a -> 2}

(f4'[x] // N) /. {x -> Pi, a -> 2}

s[4] = 
      ArcTan[(f4'[x] // N)] /. {x -> Pi, a -> 2}

(f5'[x] // N) /. {x -> Pi, a -> 2}

s[5] = 
      ArcTan[(f5'[x] // N)] /. {x -> Pi, a -> 2}

{Max[Table[s[k], {k, 1, 5}]], s[3]}

a = 2

(f1'[x] // N) /. x -> Pi

s[3] = ArcTan[(f1'[x] // N)] /. x -> Pi

(f2'[x] // N) /. x -> Pi

s[2] = ArcTan[(f2'[x] // N)] /. x -> Pi

(f3'[x] // N) /. x -> Pi

s[3] = ArcTan[(f3'[a] // N)] /. x -> Pi

(f4'[x] // N) /. x -> Pi

s[4] = ArcTan[(f4'[x] // N)] /. x -> Pi

(f5'[x] // N) /. x -> Pi

s[5] = ArcTan[(f5'[x] // N)] /. x -> Pi

{Max[Table[s[k], {k, 1, 5}]], s[3]}

c

g1[x_] := 
        E^(2  x)/(x^2 - 1) + 
          x Cos[x^2];
g1'[x] // Simplify

(g1'[x] // Simplify) /. x -> 0

d

g2[
          x_] := -1/((x - 1) (x + 
                  2));
g2'[x] // Simplify

(g2'[x] // Simplify) /. x -> 0

2

Remove["Global`*"]

a

f[x_] := 
      x (x - 1) (x - 3) (x - 6) + 40 // ExpandAll

f[x]

b

Plot[f[x], {x, -1, 7}];

c

Evaluate[f'[x]] /. x -> 0

ArcTan[(Evaluate[f'[x]] /. x -> 0)]/Degree // N

ArcTan[(Evaluate[f'[x]] /. x -> 6)]/Degree // N

d

solv = Solve[Evaluate[f'[x]] == 0, {x}] // N // 
        Chop

Table[f[x] /. solv[[k]], {k, 1, 3}]

e

solv1 = Solve[Evaluate[f''[x]] == 0, {x}] // N // 
        Chop

Table[f[x] /. solv1[[k]], {k, 1, 2}]

f

Plot[f[x], {x, 4.8, 5.2}];

FindRoot[f[x] == 0, {x, 4.9}]

FindRoot[f[x] == 0, {x, 5.0}]

f[5]

3

Remove["Global`*"]

a

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

Series[f[x], {x, 0, 1}]

Series[f[x], {x, 0, 1}] // Normal

b

n3b = Normal[Series[f[x], {x, 0, 1}]] /. x -> 0.1

f3b = f[0.1]

(n3b - f3b)/n3b*100 // Abs

4

Remove["Global`*"]

f[x_] := 
        x (x - 1) (x - 3) (x - 6);
h[x_] := -12 x (x - 3);
Plot[{f[x], h[x]}, {x, 0, 3}];

Integrate[h[x] - f[x], {x, 0, 3}]

Integrate[h[x] - f[x], {x, 0, 3}] // N

5

Remove["Global`*"]

a

f[x_] := x Sin[x];

Integrate[f[x]^2/2, {x, 0, Pi}]/Integrate[f[x], {x, 0, Pi}]

ky = N[%]

Plot[{ky, f[x]}, {x, 0, Pi}, 
        Epilog -> {PointSize[0.03], 
            Point[{Pi/2, f[Pi/2]}]}];

b

g[x_] := 
      Evaluate[f'[h] /. h -> Pi/2] (x - Pi/2) + 
        f[Pi/2]

Plot[{ky, f[x], g[x]}, {x, 0, Pi}, 
        Epilog -> {PointSize[0.03], 
            Point[{Pi/2, f[Pi/2]}]}];

g[0]

6

Remove["Global`*"]

h[x_] := -12 x (x - 3);

Plot[{h[x]}, {x, 0, 3}];

a

Laenge  =  
      Integrate[Evaluate[Sqrt[1 + (h'[x])^2]], {x, 0, 3}]

Laenge  =  
      Integrate[Evaluate[Sqrt[1 + (h'[x])^2]], {x, 0, 3}] // 
        N

b

Inhalt  =  Pi Integrate[h[x]^2, {x, 0, 3}]

Inhalt  =  Pi Integrate[h[x]^2, {x, 0, 3}] // N

c

Oberflaeche  =  
      2 Pi Integrate[
          Evaluate[h[x]  Sqrt[1 + (h'[x])^2]], {x, 0, 
            3}]

Oberflaeche  =  
      2 Pi Integrate[
            Evaluate[h[x]  Sqrt[1 + (h'[x])^2]], {x, 0, 3}] // 
        N

7

Remove["Global`*"]

DSolve[y'[x] == Cos[x]/y[x], y, x]

DSolve[{y'[x] == Cos[x]/y[x], y[0] == 0}, y, 
      x]

Simplify[D[((Sqrt[2]) (Sqrt[Sin])[x]), x]] == 
      Cos[x]/((Sqrt[2]) (Sqrt[Sin])[x])

8

Remove["Global`*"]

1+1/3+(1/3)^2+(1/3)^3+...

Sum[(1/3)^k, {k, 0, Infinity}]

1/(1 - 1/3)

N[%]

(1)^(1/3)+(1/3)^(1/3)+(1/3)^(2/3)+(1/3)^(3/3)+... = ((1/3)^(1/3))^0+((1/3)^(1/3))^1+((1/3)^(1/3))^2+((1/3)^(1/3))^3+...

Sum[((1/3)^(1/3))^k, {k, 0, Infinity}]

N[%]

1/(1 - (1/3)^(1/3))

N[%]

9

Remove["Global`*"]

Solve[4000 == r^2 Pi h  +  4/3/2 r^3 Pi, {h}]

Oberfl[r_] := 
      4/2 r^2 Pi  +  r^2 Pi  +  2 r Pi h /. 
        h -> -((2 (-6000 + π (r^3)))/(3 π (r^2)))

Oberfl[r]

Oberfl'[r]

Solve[Evaluate[Oberfl'[r] == 0], {r}]

N[%]

Plot[Oberfl[r], {r, 1, 15}];

10

Remove["Global`*"]

EmissA = 2 EmissB;  rA = 5 - rB;
EmissA/rA^2 + EmissB/rB^2

f[rB_] := ((2 EmissB)/(((5 - rB))^2)) + 
          (EmissB/(rB^2));
f'[rB] == 0

Solve[Evaluate[f'[rB] == 0], {rB}]

Solve[Evaluate[f'[rB] == 0], {rB}] // N

Plot[(f[rB] /. EmissB -> 1), {rB, 0, 5}];

Plot[(f[rB] /. EmissB -> 1), {rB, 1, 4}];

Von B aus:   2.2124666701222107`

Von A aus:

5 - 2.2124666701222107`