Assymptote de la courbe de la composée
Author: Hédi Abderrahim
Date: 7 Novembre 2013

Courbe \;  et \;  asymptote \;  oblique  \; de \;  la \;  fonction  \; composée

\mathscr{D_1}

\left(  C_f \right) 

\left(  C_g \right) 

\mathscr{D_2}

M\left(x_{0},f\left( x_{0}  \right)    \right) 

M_4\left(x_{0} ,g\left( f\left( x_{0}  \right)   \right)   \right) 

\left( C_{gof} \right) 

\Delta_2

\Delta_1

M_{0}\left(  x_{0} ,0\right) 

M_{1 }\left( 0,f\left( x_{0}  \right)   \right) 

M_{2}\left(f\left(  x_{0} \right),0    \right) 

M_{3}\left( f\left( x_{0}  \right) ,g\left( f\left( x_{0}  \right)   \right)   \right) 

y=x

N

N_{0}\left( x_{0} ,0 \right) 

N_{1}\left(0 ,k\left(  x_{0} \right)   \right) 

y=x

N_{2}\left(k\left( x_{0} ,0 \right)    \right) 

N_{3}\left( k\left(  x_{0} \right),m\left(k\left(x_{0}   \right)    \right)    \right) 

N_{4}\left(x_{0} ,m\left( k\left( x_{0}  \right)   \right)      \right) 

\Delta'_2

\Delta'_1

\Delta

P

Déplacer \;  (ou \; animer  \; le \;  point \;  P)  \ pour \;  voir\; les  \; différentes \;  étapes \  des \;  constructions

N

N_{1}\left(0,k(x_{0)}   \right) 

N_0\left(x_{0},0   \right) 

N_2\left( k(x_{0}),0  \right) 

N_3\left(k\left( x_{0}  \right),m\left( k\left( x_{0}  \right)   \right)     \right) 

\Delta_1

\Delta_2

N_4\left(x_{0} ,m\left(k\left(x_{0}   \right)    \right)   \right) 

f(x) = -x + 2 + 3 / (x^(2) + 1)
g(x) = 0.5 * x + 1 + 3 * x / (x^(2) + 1)
M = Point(f)
B = Point(f)
E = Point(g)
a = OrthogonalLine(M, xAxis)
b = OrthogonalLine(M, yAxis)
M_0 = Intersect(a, xAxis)
M_1 = Intersect(b, yAxis)
c = Segment(M, M_1)
d = Segment(M, M_0)
M_2 = Mirror(M_1, e)
h = OrthogonalLine(M_2, xAxis)
M_3 = Intersect(g, h, (4.6, 3.922743682310469))
i = Segment(M_3, M_2)
j = Segment(M_1, M_2)
l = Line(M_3, xAxis)
M_4 = Intersect(l, a)
lieu1 = Locus(M_4, M)
A = Point(lieu1)
F = Point(l)
G = Point(a)
H = (10.0171113059178, 5.455477111341231)
k_1 = 8
I = Point(Circle[H, 7.5])
k_2 = Segment(H, I)
P = Point(k_2)
q = Distance(H, P)
J = Point(e)
m(x) = 0.5 * x + 1
s, t = AngularBisector(xAxis, yAxis)
K = Point(s)
D = Point(m)
k(x) = -x + 2
C = Point(k)
N = Point(k)
n = OrthogonalLine(N, yAxis)
p = OrthogonalLine(N, xAxis)
N_1 = Intersect(n, yAxis)
N_0 = Intersect(p, xAxis)
r = Segment(N, N_1)
a_1 = Segment(N, N_0)
L = Mirror(N_1, s)
b_1 = OrthogonalLine(L, xAxis)
N_3 = Intersect(m, b_1)
c_1 = Segment(N_3, L)
d_1 = Segment(N_1, L)
T = Point(m)
Δ_2 = Line(N_3, xAxis)
Δ_1 = Line(N_0, yAxis)
O = Point(Δ_1)
Q = Point(Δ_2)
N_4 = Intersect(Δ_2, Δ_1)
lieu2 = Locus(N_4, N)
R = Point(lieu2)
S = Point(lieu2)

Tool: Asymp Comp (command AsympComp)
Input: xAxis, yAxis, f, g, M, e
Output: lieu1

f(x) = -x + 2
g(x) = 0.5 * x + 1
a = OrthogonalLine(M, xAxis)
b = OrthogonalLine(M, yAxis)
M_1 = Intersect(b, yAxis)
M_2 = Mirror(M_1, e)
h = OrthogonalLine(M_2, xAxis)
M_3 = Intersect(g, h, (2.8099999999999996, 2.405))
k = Line(M_3, xAxis)
M_4 = Intersect(k, a)