Lösungen

1

a

f[z_] := z^2;
g[x_] := Log[x]; h[x_] := f[g[x]];
f[g[x]]

b

u[s_] := Sin[s];
v[t_] := 1 - Cos[t]; w[t_] := u[v[t]];
u[v[t]]

Plot[w[t], {t, -Pi, 4  Pi}];

Solve[u[v[t]] == 0, {t}]

Table[w[k 2 Pi], {k, -5, 5}]

2

a

f1[x_] := a x^2 + b x + c;  D[f1[x], x]

b

f2[x_] := 4 x^6 - 9 x^5 + x^3 - 2  x^2 + 8  x - 7;  
    D[f2[x], x]

c

f3[x_] := (x^6 - x^5) E^x;  D[f3[x], x]

D[f3[x], x] // Simplify

d

f4[x_] := (4 x^6 - 9  x^5)/(3 E^x);  D[f4[x], x]

D[f4[x], x] // Simplify

e

f5[x_] := Cos[Log[x]];  D[f5[x], x]

3

a

Remove[g]

g[x_] := (x - 1) (x - 2) - Sin[x^2] + 4  x^(-1);  
    g[x] // Expand

D[g[x], x]

D[g[x], x] /. x -> Pi

D[g[x], x] /. x -> Pi  // N

b

tanV = D[g[x], x] /. x -> Pi  // N

ArcTan[tanV]

ArcTan[tanV]/Degree // N

2  Pi/Degree // N

4

a

Remove[f]

f[x_] := (x - 4) (x - 2) (x + 2); f[x] // Expand

Plot[f[x], {x, -4, 6}, 
        PlotStyle -> {{Thickness[0.01], GrayLevel[0.1]}}];

D[f[x], x]

D[f[x], x] // Expand

Plot[Evaluate[D[f[x], x]], {x, -3, 6}, 
        PlotStyle -> {{Thickness[0.01], GrayLevel[0.1]}}];

Solve[Evaluate[D[f[x], x]] == 0, x]

Solve[Evaluate[D[f[x], x]] == 0, x] // N

b

Plot[Evaluate[D[f[x], {x, 2}]], {x, -3, 6}, 
        PlotStyle -> {{Thickness[0.01], GrayLevel[0.1]}}];

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

% // N

c

Tan[45 Degree]

Tan[30 Degree]

Solve[Evaluate[D[f[x], x]] == Tan[30 Degree], x]

% // N

Plot[f[x], {x, -1, 4}, AspectRatio -> Automatic, 
        Epilog -> {Line[{{-0.484, 0}, {-0.484, f[-0.484]}}], 
            Line[{{3.1509, 0}, {3.1509, f[3.1509]}}]}];