Lösungen

1

f[x_] := x (x - 2)^2;

a

Plot[f[x], {x, -2, 3}];

b

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

c

f'[x]

f'[x] //  Expand

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

d

Integrate[f[x], x]

Integrate[f[x], {x, 0, 2}]

e

Integrate[f[x], x]

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

2

?A

Remove[f, A];
f[x_] := 2 x^2;
A[t_] := 
        Integrate[f[x], {x, -1, t}];
A[t]

a

Solve[A[t] == 2/3, {t}]

b

Solve[A[t] == 10, {t}]

Solve[A[t] == 10, {t}] //  N

3

?V

Remove[f, V, A];
f[x_] := Sqrt[x];
V[x_] := 
        Integrate[f[t]^2 Pi, {t, 0, x}];
V[x]

V[10]

V[10] // N

4

a Skizze

b

r = 6400; φ = 21.5 Degree;  θ = 
      44.2 Degree;  φ

x =  r Sin[θ] Cos[φ]

y =  r Sin[θ] Sin[φ]

z =  r Cos[θ]

c

φ = (21.5 +  3/24 360) Degree

x =  r Sin[θ] Cos[φ]

y =  r Sin[θ] Sin[φ]

z =  r Cos[θ]

5

vVec = {2, 1};  wVec = {5, 6};

a Skizze

b

vecLen[v_] := Sqrt[v . v]

α = ArcCos[vVec . wVec/vecLen[vVec]/vecLen[wVec]]

% // N

%/Degree

α1 = 
      ArcCos[vVec . {1, 0}/vecLen[vVec]/vecLen[{1, 0}]]

d

uVec1 = {1, 
          Tan[2 α + α1]};
uVec = uVec1/vecLen[uVec1]  vecLen[vVec]

uVec // N

c

PVec = uVec + vVec

PVec // N

Kontrolle

Solve[t wVec[[1]] == PVec[[1]], {t}] // N

Solve[t wVec[[2]] == PVec[[2]], {t}] // N

Show[
Graphics[{PointSize[0.05], Point[{0, 0}], 
            Point[vVec], Point[wVec], Point[uVec], Point[PVec], 
            Line[{{0, 0}, wVec}], Line[{{0, 0}, vVec}], Line[{{0, 0}, uVec}], 
            Line[{PVec, vVec}], Line[{PVec, uVec}]}], 
        AspectRatio -> Automatic];

6

OA = {1, 1}; OB = {7, 2.5}; OC = {2, 6};

Solve[(OA +  t (OB - OA))[[1]] == 
        4, {t}]

OC1 = OA +  (1/2) (OB - OA)

Solve[(OB +  t (OC - OB))[[1]] == 
        5, {t}]

OA1 = OB + (2/5) (OC - OB)

b

solv = 
      Solve[OA + λ (OA1 - OA) == 
          OC + μ (OC1 - OC), {λ, μ}]

solv = 
      Solve[OA + λ (OA1 - OA) == 
            OC + μ (OC1 - OC), {λ, μ}] // 
        Flatten

OS = OA + λ (OA1 - OA) /. solv[[1]]

OS = OC + μ (OC1 - OC) /. solv[[2]]

c

solv1 = 
      Solve[OB + λ (OS - OB) == 
            OA + μ (OC - OA), {λ, μ}] // 
        Flatten

OB1 = OB + λ (OS - OB) /. solv1[[1]]

OB1 = OA + μ (OC - OA) /. solv1[[2]]

Kontrolle (Ceva)

(4 - 1)/(4 - 7) (5 - 2)/(5 - 7) (1.4 - 
            1)/(1.4 - 2)