f(x) =

n

Beachte:
Bei kleiner werdendem Δx bleibt das größere Steigungsdreieck
mit den Katheten y und Subtangente (von B bis (x_0,0))
sichtbar.

Zoom in

Zoom out

Aus der Tabelle ist auch ersichtlich, dass die Folge der Sekantensteigungen 
 für  Δx=1/2^n →0  die Tangentensteigung (den Grenzwert) erzeugt.

n = 2 (slider from 1 to 26, step 1)
L = Sequence(1 / 2^(k), k, 1, 35)
f(x) = (1 / 2 * x^(2))
x_0 = Point(xAxis)
a = Line(x_0, yAxis)
P = Intersect(f, a, (2, 2))
h = Element(L, n)
Q: (x(x_0) + h, f(x(x_0) + h))
b = Line(P, xAxis)
c = Line(Q, yAxis)
A = Intersect(c, b)
Δx = Segment(P, A)
Δy = Segment(A, Q)
t = Line(P, Q)
m: Δy / Δx
Text2 = Text("G. Roolfs")
Eingabefeld1 = Textfield(f)
Text3 = LaTeX(P, true, true)
Text4 = LaTeX(Q, true, true)
A2: 1
A3: A2 + 1
A4: A3 + 1
A5: A4 + 1
A6: A5 + 1
A7: A6 + 1
A8: A7 + 1
A9: A8 + 1
A10: A9 + 1
A11: A10 + 1
A12: A11 + 1
A13: A12 + 1
A14: A13 + 1
A15: A14 + 1
A16: A15 + 1
B2: 1 / 2^(A2)
B3: 1 / 2^(A3)
B4: 1 / 2^(A4)
B5: 1 / 2^(A5)
B6: 1 / 2^(A6)
B7: 1 / 2^(A7)
B8: 1 / 2^(A8)
B9: 1 / 2^(A9)
B10: 1 / 2^(A10)
B11: 1 / 2^(A11)
B12: 1 / 2^(A12)
B13: 1 / 2^(A13)
B14: 1 / 2^(A14)
B15: 1 / 2^(A15)
B16: 1 / 2^(A16)
A17: A16 + 1
B17: 1 / 2^(A17)
A18: A17 + 1
B18: 1 / 2^(A18)
A19: A18 + 1
B19: 1 / 2^(A19)
A20: A19 + 1
B20: 1 / 2^(A20)
A21: A20 + 1
B21: 1 / 2^(A21)
A22: A21 + 1
B22: 1 / 2^(A22)
A23: A22 + 1
B23: 1 / 2^(A23)
A24: A23 + 1
B24: 1 / 2^(A24)
A25: A24 + 1
B25: 1 / 2^(A25)
A26: A25 + 1
B26: 1 / 2^(A26)
A27: A26 + 1
B27: 1 / 2^(A27)
C2: (f(x(P) + B2) - f(x(P))) / B2
C3: (f(x(P) + B3) - f(x(P))) / B3
C4: (f(x(P) + B4) - f(x(P))) / B4
C5: (f(x(P) + B5) - f(x(P))) / B5
C6: (f(x(P) + B6) - f(x(P))) / B6
C7: (f(x(P) + B7) - f(x(P))) / B7
C8: (f(x(P) + B8) - f(x(P))) / B8
C9: (f(x(P) + B9) - f(x(P))) / B9
C10: (f(x(P) + B10) - f(x(P))) / B10
C11: (f(x(P) + B11) - f(x(P))) / B11
C12: (f(x(P) + B12) - f(x(P))) / B12
C13: (f(x(P) + B13) - f(x(P))) / B13
C14: (f(x(P) + B14) - f(x(P))) / B14
C15: (f(x(P) + B15) - f(x(P))) / B15
C16: (f(x(P) + B16) - f(x(P))) / B16
C17: (f(x(P) + B17) - f(x(P))) / B17
C18: (f(x(P) + B18) - f(x(P))) / B18
C19: (f(x(P) + B19) - f(x(P))) / B19
C20: (f(x(P) + B20) - f(x(P))) / B20
C21: (f(x(P) + B21) - f(x(P))) / B21
C22: (f(x(P) + B22) - f(x(P))) / B22
C23: (f(x(P) + B23) - f(x(P))) / B23
C24: (f(x(P) + B24) - f(x(P))) / B24
C25: (f(x(P) + B25) - f(x(P))) / B25
C26: (f(x(P) + B26) - f(x(P))) / B26
C27: (f(x(P) + B27) - f(x(P))) / B27
B1 = Text("Δx = 1/2^n")
Sekantensteigung = Text("Δx = 1/n")
C1 = Text("m = Δy /Δx ")
A28: A27 + 1
B28: 1 / 2^(A28)
C28: (f(x(P) + B28) - f(x(P))) / B28
A29: A28 + 1
B29: 1 / 2^(A29)
C29: (f(x(P) + B29) - f(x(P))) / B29
A30: A29 + 1
B30: 1 / 2^(A30)
C30: (f(x(P) + B30) - f(x(P))) / B30
A31: A30 + 1
B31: 1 / 2^(A31)
C31: (f(x(P) + B31) - f(x(P))) / B31
A32: A31 + 1
B32: 1 / 2^(A32)
C32: (f(x(P) + B32) - f(x(P))) / B32
A33: A32 + 1
B33: 1 / 2^(A33)
C33: (f(x(P) + B33) - f(x(P))) / B33
A34: A33 + 1
B34: 1 / 2^(A34)
C34: (f(x(P) + B34) - f(x(P))) / B34
g = Segment(P, x_0)
Text1 = Text("Sekantensteigung  m = " + m)
B = Intersect(t, xAxis)
C = Midpoint(x_0, P)
Text7 = Text("  y", C, true, true)