$\circ$

$\vartriangleright$

x

I_a

l

v

h

V(t)

z(t)

t

t

B3

B1

B6

f1

f3

f2

f4

f5

B2

B5

f6

B8

f7

B4

f8

B7

F9

F10

Nächste Bestandsentwicklung

a

L

G

a =

$\color{#7D7DFF}\blacklozenge$

f

I_a

f

t

I_a(x+h)-I_a(x)

h

I_a(x+h)-I_a(x)

\frac{I_a(x+h)-I_a(x)}{h} \
\
\qquad\quad\downarrow h \rightarrow 0\
\
\qquad I_{a}'(x) 

\frac{I_a(x+h)-I_a(x)}{h} \
\
\qquad\quad\downarrow h \rightarrow 0\
\
\qquad f(x)

s

t

Sekante / Tangente

I_a(x+h)-I_a(x)

(I_a(x+h)-I_a(x))/h

m_s = \frac{I_a(x+h)-I_a(x)}{h}

I_a(x+h)-I_a(x)

z

f(z) = (I_a(x+h)-I_a(x))/h

(I_a(x+h)-I_a(x))/h

x+h

m_s = (I_a(x+h)-I_a(x))/h

f(x)

m_t = I_a'(x)

Rechteck

I_a(x+h)-I_a(x)

h

m_t = I_a'(x)

t = -2 (slider from -2 to b, step dt)
a = 1 (slider from 0 to 5, step 1)
b: 5
g1: x = 1
g2: x = 2
g3: x = 3
g4: x = 4
P1 = Point(g1)
P2 = Point(g2)
P3 = Point(g3)
P4 = Point(g4)
g0: x = 0
g5: x = 5
P0 = Point(g0)
P5 = Point(g5)
m5: (y(P5) - y(P4)) / (x(P5) - x(P4))
m2: (y(P2) - y(P1)) / (x(P2) - x(P1))
m1: (y(P1) - y(P0)) / (x(P1) - x(P0))
m4: (y(P4) - y(P3)) / (x(P4) - x(P3))
P6: P2
m3: (y(P3) - y(P6)) / (x(P3) - x(P6))
f = Polynomial({P0, P1, P2, P3, P4, P5})
tListe = Sequence(i, i, a, b, 1)
TPunkte = Zip((i, 0), i, tListe)
T = Point(TPunkte)
ta: x(T)
V0 = 0
F = Integral(f)
I(t) = F(t) - F(a)
F0(x) = If[a ≤ x ≤ b, F(x)]
V: F(ta) - F(0)
S1, k, l, m, n = Polygon((-1.5, -500), (-1, -500), (-1, F0(ta)), (-1.5, F0(ta)))
S2, k1, k2, k3, k4 = Polygon((-1.5, -500), (-1, -500), (-01, 500), (-1.5, 500))
gL: -2.5
gR: 6
gO: 600
gU: -100
gO2: 250
faktor = 0 (slider from -5 to 5, step 0.1)
VListe: {}
dt = 0.1 (slider from -5 to 5, step 0.1)
zaListe: {}
gU2: -50
Vmax: 1000
Vmin: 0
zMin: -100
zMax: 100
Z0 = Point(Segment[(0, zMin), (0, zMax)])
Text5: "" + (y(Z0)) + ""
Va: (0, V)
za: f(t)
ft = Segment((t, 0), (t, f(t)))
zaListeDiskret = Sequence((i, f(i)), i, a, ta, 1)
c_{1} = 0
index = 2
Text6: "B" + index + ""
VListeDiskret = Sequence((i, F0(i)), i, a, ta, 1)
A: (a, 0)
c = true
sIt = Segment((ta, 0), (ta, F0(ta)))
d = true
It = Integral(f, a, ta)
e: (t * 4)
Eingabefeld1 = Textfield(a)
I2 = Integral(f, Max[a, 4], Min[7, ta])
I3 = Integral(f, Max[a, 7], ta)
I1 = Integral(f, a, Min[4, ta])
nullmarke = Segment((-2, 0), (-0.5, 0))
S3, h1, h2, h3, h4 = Polygon((-1.5, 0), (-1, 0), (-1, F0(ta)), (-1.5, F0(ta)))
p = false
f0(x) = If[a ≤ x ≤ b, f(x)]
F0a(x) = If[a ≤ x ≤ b, F0(x) - F0(a)]
f0a(x) = If[a ≤ x ≤ b, (f0(x) * x)]
Q: (ta, F0a(ta))
tQ = Tangent(Q, F0a)
mQ = Slope(tQ)
t0(x) = If[2 ≤ x ≤ 6, tQ(x)]
h = 1 (slider from -1 to 1, step 0.1)
Idt = Integral(f0, ta, ta + h)
dh = Vector((ta, F0a(ta)), (ta + h, F0a(ta)))
dI = Vector((ta + h, F0a(ta)), (ta + h, F0a(ta + h)))
R: Q + dh + dI
sekante = Line(Q, R)
tangente = Tangent(Q, F0a)
bSekante = false
bDifferenz = false
bDifferenzenquotient = false
S: Q + dh
bDifferenz2 = false
hRechteck: Idt / h
gRechteck(x) = hRechteck
R1: (ta, 0)
R2: (ta + h, 0)
R3: (ta + h, hRechteck)
R4: (ta, hRechteck)
Rechteck = Polygon({R1, R2, R3, R4})
Schnittstellen = NSolve(f0(x) = gRechteck(x))
Schnittpunkt1 = Intersect(Element[Schnittstellen, 1], xAxis)
z1: x(Schnittpunkt1)
Schnittpunkt2 = Intersect(Element[Schnittstellen, 2], xAxis)
z2: x(Schnittpunkt2)
x1Wert: x1(0)
x2Wert: x2(0)
zStelle = If(h > 0, If[ta ≤ x1Wert ≤ ta + h, x1Wert, x2Wert], If[ta + h ≤ x1Wert ≤ ta, x1Wert, x2Wert])
B: (zStelle, f0(zStelle))
B0: (zStelle, 0)
sDifferenzenquotient = Segment(B0, B)
bDifferenzenquotient2 = false
Xh: (ta + h, 0)
ms = Slope(sekante)
sx = Segment((ta, 0), (ta, f0(ta)))
mt = Slope(tangente)
bRechteck = false
Z: (zStelle, 0)
C = Intersect(f0, gRechteck, ta - h, ta)
D = Intersect(gRechteck, f0, ta, ta + h)
PosH = If(h > 0, (Q + S) / 2 + (-0.05, -15), (Q + S) / 2 + (-0.05, 50))
PosIa = If(h > 0, (S + R) / 2 + (0.05, 20), (S + R) / 2 + (-1.25, 20))
PosMs = If(h > 0, Q + (-1.6, 80), Q + (0.1, 20))