Lösungen VD B 2003

1

a

D[2  x^4 + 5 x^0 - 4  x^2 + x - 1/x, x]

D[2  x^4 + 5 x^0 - 4  x^2 + x - 1/x, x] /. x -> 1

D[2  x^4 + 5 x^0 - 4  x^2 + x - 1/x, {x, 2}]

b

Integrate[2  x^4 + 5 x^0 - 4  x^2 + x - 1/x, x]

Integrate[2  x^4 + 5 x^0 - 4  x^2 + x - 1/x, {x, 1, 2}]

N[%]

c

D[1/Cos[x/2 - 1]^2, x]

Integrate[1/Cos[x/2 - 1]^2, x]

2 Tan[(x/(2 - 1))]

d

Integrate[1/((x) (x - 1) (x + 1)), x]

Integrate[
      1/((x) (x - 1) (x + 1)), {x, 3, 5}]

% // N

Log[27/25]/2

% // N

Integrate[
      1/((x) (x - 1) (x + 1)), {x, 3, Infinity}]

% // N

Log[9/8]/2

% // N

e

4 Integrate[(E^x - E^(-x))/E^(2  x), x] // 
      Simplify

% // Together

4 Integrate[(E^x - E^(-x))/E^(2  x), {x, Pi, E}] // 
      Simplify

% // Together

% // N

2

p1 = {0, 0, 0}; p2 = {1, 0, 0}; p4 = {0, 1, 0}; 
    p3 = {1, 1, 0};
p5 = {0, 0, 1}; p6 = {1, 0, 1}; p8 = {0, 1, 1}; 
    p7 = {1, 1, 1};
p1234 = (p1 + p2 + p3 + p4)/4; p5678 = (p5 + p6 + p7 + p8)/4; 
    p1256 = (p1 + p2 + p5 + p6)/4; p3478 = (p3 + p4 + p7 + p8)/4; 
    p1458 = (p1 + p4 + p5 + p8)/4; p2367 = (p2 + p3 + p6 + p7)/4; 
    q1 = p1234; q2 = p5678; q3 = p1256; q4 = p3478; q5 = p1458; q6 = p2367; 
    p = Graphics3D[{PointSize[0.02], Point[p1], Point[p1], Point[p2], 
          Point[p3], Point[p4], Point[p5], Point[p6], Point[p7], Point[p8], 
          Point[q1], Point[q2], Point[q3], Point[q4], Point[q5], Point[q6], 
          Point[{0.5, 0.5, 0.5}]}, 
        Boxed -> False];
lin = 
      Graphics3D[{Line[{p4, p1, p2, p3, p4, p8, p7, p3, p7, p6, p2, p6, p5, 
              p1, p5, p8, p4}], 
          Line[{q1, q2, q3, q4, q1, q3, q4, q2, q5, q6, q1, q5, q6, q2, q3, 
              q5, q6, q3, q4, q5, q6, q4}], 
Line[{p1, 
              p7}], Line[{p4, p6}], Line[{p3, p5}], Line[{p2, p8}]}, 
        Boxed -> False]
Show[p, lin, 
        ViewPoint -> {2.686,  1.686,  0.877}];

Options[Graphics3D]

a

Winkel zwischen Diagonalen des Würfels, 1. Variante

p1 = {1, 1, 1}; p2 = {-1, 1, 1}

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

winkel = p1 . p2/(vecLen[p1] vecLen[p2])

ArcCos[%] // N

w1 = %/Degree

Winkel zwischen Diagonalen des Würfels, 1. Variante

p1 = {1, 1, 1}; p2 = {-1, -1, 1}

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

winkel = p1 . p2/(vecLen[p1] vecLen[p2])

ArcCos[%] // N

w2 = %/Degree

w1 + w2

b

Flächenwinkel Oktaeder (Verwendung Vektorprodukt und Normalenvektoren)

p1 = {1, 0, 0}; p2 = {0, 1, 0}; p3 = {0, 0, 1}; 
    p4 = {0, -1, 0};

n1 = Cross[(p1 - p2), (p3 - p2)]

n2 = Cross[(p1 - p4), (p3 - p4)]

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

winkel = n1 . n2/(vecLen[n1] vecLen[n2])

ArcCos[winkel]

ArcCos[-(1/3)] // N

ArcCos[winkel]/Degree // N

c

Interpretation? (Dualität)

d

vwu = a^3

vok = 4 *(a/2)^2 /2  * (a/2)/3 *2

vwu/vok

swu = a^2

sokC = 
      Cross[{a/2, 0, 0} - {0, a/2, 0}, {a/2, 0, 0} - {0, 0, a/2}]

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

sok = vecLen[sokC]/2

% // N

swu/sok  // Simplify

8 /(Sqrt[3]) // N

1/%

3

fa[x_] := (x - a) x^2;  D[fa[x], x] // Expand

a

f[x_] := (x - 1) x^2;  Plot[f[x], {x, -1, 2}];

b

fa[a]

Ansteigen der Tangente bei a

steig =  D[fa[x], x] /. x -> a

ArcTan[steig]

c

d

Minimum in Abhängigkeit von a

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

fa[((2 a)/3)]

e

Fläche in Abhängigkeit von a

flA = Integrate[fa[x], {x, 0, a}]

f

flA/fa[((2 a)/3)]

4

Remove[f, x, a, g, h, res1, res2]

Problem mit Rotation, Volumen

a

g[x_, a_] := a - x^2;  g[x, a]

Solve[g[x, a] == 0, {x}]

h[x_, a_] := a - x^4;  h[x, a]

Solve[h[x, a] == 0, {x}]

Plot[{g[x, 16], h[x, 16]},  {x, -4, 4}];

b

resg[a_] := Integrate[Pi g[x, a]^2, {x, -(Sqrt[a]), (Sqrt[a])}]; 
    resg[a]

resh[a_] := Integrate[Pi h[x, a]^2, {x, -(a^(1/4)), (a^(1/4))}]; 
    resh[a]

c

resg[a] == resh[a]

Solve[resg[a] == resh[a], {a}]

Solve[resg[a] == resh[a], {a}] // N

d

i1 = Integrate[g[x, a], {x, -(Sqrt[a]), (Sqrt[a])}]

i2 = Integrate[h[x, a], {x, -(a^(1/4)), (a^(1/4))}]

i2/i1

% // N

5

a

Plot[x^2 + x + 1, {x, -2, 2}];

f[x_, a_, b_] := x^2 + a x + b;
v1[x_, a_, b_] := {x, f[x, a, b]}; 
    v2[x_, a_, b_] := {x + 1, f[x + 1, a, b]}

Plot[f[x, 1, 1], {x, -2, 2}, 
        Epilog -> {PointSize[0.03], Point[p1 = v1[-0.5, 1, 1]], 
            Point[p2 = v2[-0.5, 1, 1]], Line[{{0, 0}, p1}], 
            Line[{{0, 0}, p2}]}];

b

s[x_, a_, b_] := v1[x, a, b] . v2[x, a, b] // Expand;  
    s[x, a, b]

c

s[x, 1, 1]

Plot[s[x, 1, 1], {x, -2, 2}];

d

D[s[x, 1, 1], x]

Solve[D[s[x, 1, 1], x] == 0, {x}]

Solve[D[s[x, 1, 1], x] == 0, {x}] // N

Minimum!

D[s[x, 1, 1], {x, 2}]

Solve[D[s[x, 1, 1], {x, 2}] == 0, {x}] // N

Keine Extremwerte!

e

Solve[v1[x, 1, 1] . v2[x, 1, 1] == 0, {x}] // N

Keine reellen NS!

Solve[v1[x, a, 0] . v2[x, a, 0] == 0, {x}] // N

x = -1,   x = 0

6

Graphik

Remove[a, b, f, g, h, p, m, res, u, x0]

f[x_] := x^2

pG = Plot[f[x], {x, -2, 2}];

Show[pG, 
        Graphics[{PointSize[0.02], 
            Point[{-1, f[-1]}]}]];

a = -0.8; b = 1.8; mit = (a + b)/2

g[x_] := (f[b] - f[a])/(b - a) (x - a)  + 
          f[a] ;
h[x_] := (f[b] - f[a])/(b - a) (x - mit)  + 
        f[mit]

Show[pG, 
        Graphics[{PointSize[0.02], 
            Point /@ {{a, f[a]}, {b, f[b]}, {mit, f[mit]}}, 
            Line[{{a - 0.2, g[a - 0.2]}, {b + 0.2, 
                  g[b + 0.2]}}], 
Line[{{a - 0.2, 
                  h[a - 0.2]}, {b + 0.2, h[b + 0.2]}}]}], 
        AspectRatio -> 1];

Remove[a, b, f, g, h, p, m, res, u, x0]

s(x)

p1 = {x0, p[x0]}; p2 = {x0 + k, p[x0 + k]};

{p1, p2}

m[x0_] := (p[x0 + k] - p[x0])/((x0 + k - x0));  
    m[x0]

p[x_] :=  a x^2;  
    s[x_] := m[x0] (x - x0) + p[x0]; {p[x],  s[x]} // 
      Simplify

a (k (x - x0) + (2 x - x0) x0)  //  Expand

x_m

res[xm_] := Evaluate[D[p[xm], xm]];  res[xm]

res[xm] == m[x0]

res[xm] == ExpandAll[m[x0]]

solv1 = Solve[res[xm] == m[x0], {xm}] // Simplify // 
        Flatten

xm = xm /. solv1[[1]]

p[xm]

===>Mitte zwischen x0 und x0+k

t(x)

t[x_] := m[x0] (x - xm) + p[xm]; t[x] // Simplify

A

Integrate[s[x] - p[x], {x, x0, x0 + k}]

Trapez

Integrate[s[x] - t[x], {x, x0, x0 + k}]

Verhältnis

Integrate[s[x] - t[x], {x, x0, x0 + k}]/
      Integrate[s[x] - p[x], {x, x0, x0 + k}]