Funktion und Ableitung (mit Fahrrad)
Author: Dörte Haftendorn

Bewege P. Schalte dann einzeln die Erklärugselemente hinzu.
Markieren die blaue Kurve und bewege sie mit dem Pfeiltasten. Achte auf die Ableitung.
Für f kann jede andere Funktion eingeben werden. (f doppelt anklicken zum Ändern.)
Den Funkrionsterm von f hann man mit der gepunkteten Geraden erklären.
Das schöne Fahrrad ist von Dieter Riebesehl.

Ortskurve

Steigung

Extrema von f

Wendepkte von f

Ha 2010

f(x) = (((1 / 10 * (x + 2)^(2)) * (x - 2)) * (x - 4)) + 2.0000000000000004
H, I, J = Extremum(f)
f_3(x) = f'(x)
H_x, I_x, J_x = Root(f_3)
Fahrradgroesse = 0.4 (slider from 0.4 to 1.2, step 0.1)
ny(x) = f(x) + (0.5 * Fahrradgroesse / sqrt(1 + (f'(x))^(2)))
nx(x) = x - ((0.5 * Fahrradgroesse) * f'(x) / sqrt(1 + (f'(x))^(2)))
P = Point(f)
e = Circle(P, (0.5 * Fahrradgroesse))
a = Tangent(P, f)
c = OrthogonalLine(P, a)
C, D = Intersect(e, c)
M = If(y(D) > y(P), D, C)
P_1 = Root(sqrt((nx(x) - x(M))^(2) + (ny(x) - y(M))^(2)) - (1.3 * Fahrradgroesse), x(P), x(P) + (2.5 * Fahrradgroesse))
B: (x(P_1), f(x(P_1)))
g = Circle(B, (0.5 * Fahrradgroesse))
b = Tangent(B, f)
d = OrthogonalLine(B, b)
E, F = Intersect(g, d)
N = If(y(F) > y(B), F, E)
G = Midpoint(M, N)
G' = Rotate(G, 59.99999999999999°, M)
l = Segment(G', M)
m_3 = Line(G, l)
i = Segment(M, G)
n = Line(G', i)
H_2 = Intersect(m_3, n)
a_1 = Ray(G, H_2)
h_4 = Circle(M, P)
k = Circle(N, B)
α = Angle(G, M, G')
p = Segment(G', H_2)
r = Segment(H_2, N)
b_1 = Ray(G, G')
f_1(x) = sqrt((nx(x) - x(M))^(2) + (ny(x) - y(M))^(2)) - (1.3 * Fahrradgroesse)
s = Circle(G, (0.8 * Fahrradgroesse))
I_2 = Intersect(s, b_1)
J_2 = Intersect(s, a_1)
j = Segment(J_2, G)
q = Line(J_2, p)
c_1 = Circle(J_2, (0.2 * Fahrradgroesse))
K, L = Intersect(c_1, q)
d_1 = Segment(K, L)
g_1 = Circle(G, (0.15 * Fahrradgroesse))
h_1 = Ray(G, M)
k_1 = Circle(G, (1.95 * Fahrradgroesse))
O = Intersect(k_1, h_1)
i_1, j_1 = Tangent(O, g_1)
e_1 = Segment(H_x, H)
Q = Intersect(e_1, j_1)
R = (0.5200090689193531, 1.0508706670333623)
S = Intersect(e_1, i_1)
T = (0.5828653525719789, 1.0999748193087284)
U = Intersect(g_1, i_1)
V = Intersect(g_1, j_1)
l_1 = Segment(Q, V)
m_1 = Segment(S, U)
H' = Rotate(H_2, 90°, N)
H'_1 = Rotate(H_2, 260°, N)
n_1 = Ray(N, H')
p_1 = Ray(N, H'_1)
q_1 = Circle(N, (0.53 * Fahrradgroesse))
W = Intersect(q_1, n_1)
Z = Intersect(q_1, p_1)
r_1 = CircleArc(N, Z, W)
G'' = Rotate(G', 130°, M)
β = Angle(G', M, G'')
G''_1 = Rotate(G', 280°, M)
s_1 = Ray(M, G'')
t_1 = Ray(M, G''_1)
c_2 = Circle(M, (0.53 * Fahrradgroesse))
A_1 = Intersect(c_2, s_1)
B_1 = Intersect(c_2, t_1)
d_2 = CircleArc(M, B_1, A_1)
G''_2 = Rotate(G', (-4 * x(P)), G)
γ = Angle(G', G, G''_2)
a_2 = Line(G''_2, G)
e_2 = Segment(I_x, I)
C_1 = Intersect(e_2, a_2)
D_1 = (0.6431098507226991, 0.9063193912966376)
b_2 = Segment(D_1, C_1)
f_2 = Line(C_1, i)
g_2 = Line(D_1, i)
h_2 = Circle(C_1, (0.05 * Fahrradgroesse))
E_1, F_3 = Intersect(h_2, f_2)
i_2 = Segment(E_1, F_3)
k_2 = Circle(D_1, (0.05 * Fahrradgroesse))
G_1, H_1 = Intersect(k_2, g_2)
j_2 = Segment(G_1, H_1)
p_2(x) = 3 - (x - 6)^(2) / 12 - (x - 6)^(3) / 54 + (2 * sin(x))
X_p: (x(P), 0)
n_2 = Segment(P, X_p)
h = Tangent(P, f)
o = Slope(h)
A: (x(P), Slope[h])
m = Segment((x(P) + 1, y(P)), (x(P) + 1, y(P) + Slope[h]))
c_3 = Segment(A, X_p)
u = false
W_1, W_2 = TurningPoint(f)
F_1, F_2 = Extremum(f_3)
w_1 = Segment(F_1, W_1)
w_2 = Segment(F_2, W_2)
e_3 = Segment(J_x, J)
t = true
v = false
w = false
q_2: y = y(H)
m_2: y = y(H)
I_1 = (-4.599623671496721, 0.20184924539065907)
J_1 = (-3.2664113560667607, 0.20184924539065907)
Text1: "f(x)=" + (LaTeX[f]) + ""