Lösungen/ Solutions

Probl/Ex 1      W

a

f00[x_] := x^4 +  x^6;  D[f00[x], x]

b

f01[x_] := x^4 Log[x];  D[f01[x], x]

c

f02[x_] := x^4 / Log[x];  D[f02[x], x]

d

f03[x_] := Log[x^4];  D[f03[x], x]

e

f04[x_] := x E^(1/x);  D[f04[x], x]

Probl/Ex 2      W

f2[x_] :=  
        7  x^4  +  12 x^3  +  -x^2  + 8  x  + 9;

f21[x_] = Evaluate[D[f2[x], {x, 1}]]; f21[x]

f22[x_] = Evaluate[D[f2[x], {x, 2}]]; f22[x]

f23[x_] = Evaluate[D[f2[x], {x, 3}]]; f23[x]

Plot[{f2[x], f21[x], f22[x], f23[x]}, {x, -10, 10}];

Plot[{f2[x], f21[x], f22[x], f23[x]}, {x, -2.1, 
          1.8}];

Probl/Ex 3      W

f3[x_] := Sin[x];

Plot[f3[x], {x, -6, 10}];

a

D[f3[x], x] == 0.9

x0 = ArcCos[0.9]

b

(D[f3[x], x]  /.  x -> x0)  ==  
      Tan[α0]

α0 = ArcTan[0.9]

c

x1 = 0.9

(D[f3[x], x]  /.  x -> x1)

Cos[0.9]

t3[x_] := Cos[x1] (x - x1)  + Sin[x1];  t3[x]

Plot[{Sin[x], t3[x]}, {x, 0, 2}];

y1 = t3[0]

x2 = Solve[t3[x] == 0, {x}] //  Flatten

Probl/Ex 4      W

Remove[f4, a, b, c, d, x, solv]

f4[x_] := a x^3  +  b x^2  +  c x  +  d

g1 =  (f4[2] == 0)

g2 =  (D[f4[x], x] == 1 /.  x -> 1)

g3 =  (D[f4[x], {x, 2}] == 0 /.  
          x -> -3)

g4 =  (f4[3] == -1)

solv = Solve[{g1, g2, g3, g4}, {a, b, c, d}] // Flatten

f4[x] /. solv

Plot[{f4[x] /. solv},  {x, -20, 8}];