chap 4

IR IR IR
x y
IR2
A⊂IR IR2
F:A⊂IR −→ IR2
t7−→ F(t) = (x(t), y(t)).
A F
IR2
||(x, y)|| =qx2+y2,
(un= (xn, yn))n(α, β)||(xn−α, yn−β)||
n+∞(xn)n(yn)n
α β.
F IR2A
F M(t) = (x(t), y(t)) t
A F
IR2
t0∈A t0
A F t0F(t)F(t0)t t0

F t0
F(t)−F(t0)
t−t0
F0(t0)t t0
F t0F0(t0)
IR2
F t0t7−→ x(t)
t7−→ y(t)t0
F t0t7−→ x(t)t7−→ y(t)
t0
F0(t0) = (x0(t0), y0(t0)).
||F(t)−F(t0)|| =q(x(t)−x(t0))2+ (y(t)−y(t0))2.
F(t)F(t0) = (x(t0), y(t0)) x(t)x(t0)y(t)
y(t0)x(t)x(t0)y(t)y(t0)F(t)
F(t0)
F t0F0(t0) = (l, k)
x(t)−x(t0)
t−t0
y(t)−y(t0)
t−t0
l k
FC1A F
A
t7−→ F0(t)A
FCnA n F n
A n A
IR2

FCnI t0
A t ∈I
F(t) = F(t0)+(t−t0)F0(t0) + 1
2!(t−t0)2F(2)(t0) + ... +1 n!(t−t0)nF(n)(t0)+(t−t0)nε(t−t0),
ε I IR2
ε(t) (0,0) ∈IR2t t0
x y ε ε = (ε1, ε2)ε1
ε2x y
F(t0)F0(t0) = 0
F0(t0) =
(x0(t0), y0(t0)) = 0
p F (p)(t0)6= 0
p F (p)(t0) = (x(p)(t0), y(p)(t0))
F(t0).
F(t)−F(t0) = (t−t0)p(x(p)(t0) + ε1(t−t0), y(p)(t0) + ε2(t−t0)),
ε(t−t0) = (ε1(t−t0), ε2(t−t0)). ε1ε2
IR t t0
p= 1 F(t)−F(t0)
t−t0
=1
t−t0
[(t−t0)p(x(p)(t0)+ε1(t−t0), y(p)(t0)+ε2(t−t0))]
(x0(t0), y0(t0)) p > 1
(x(p)(t0), y(p)(t0))

x y
t(x(t), y(t))
t
x y t t
x y
x y
x(t) = 3t−t3
y(t) = t2+1
t2
.
IR∗x
y(x(t), y(t))
(x(−t), y(−t))
IR∗
+
x IR
y IR∗
x0(t) = 3(1 −t2)y0(t) = 2(t2−1)(t2+ 1)
t3.
t0 1 + ∞
x0(t) + 0 −
x(t) 0 %2& −∞
y0(t)−0 +

y(t) + ∞ & 2%+∞
t
x y
t+∞x
y
x y
x y
M(t) = M(1/t)t6= 0 ]0,1] [−1,0[
[1,+∞[ ] − ∞,−1]
x y
x y
x(t) = sin(t)+3cos(2t)
y(t) = sin2(t)−cos(4t),
T= 2π
(x(1), y(1))
t0t1t
x(t) = 3t−t3
y(t) = t2+1
t2
.
3t1−t3
1= 3t0−t3 0
t2
1+1
t2
1=t2
0+1
t2
0
.
(t2
1−t2 0)(t2
0t2 1−1) = 0
t2
1+t0t1+t2 0= 3 ,
t0=t1t0=−t1t0t1= 1 t0t1=−1