Verhalten im Unendlichen
0<x<14,44723
0<y<6,9
f
= x
D
f
W
= y
a > 0
a < 0
x -> -∞
von II nach I von III nach I
von III nach IV von II nach IV
lim f(x) = -∞
x -> ∞
III
lim f(x) = ∞
II
n gerade n ungerade
IV
I
0,0006612672x
a
Definitions- und Wertebereich
5
n
6 Verhalten im Unendlichen
1,71973<x<4,31656
10,2512<x<15,6394
x<1,71973
4,31656<x<10,2512
15,6304<x
Monotonie
wenn:
f´(x) < 0
monoton fallend:
f´(x) > 0
f´´´(x)= 0.039676x -0.633189x + 2.14173
monoton steigend:
2
4 Monotonie
2
x = 0
x = 14.44723
1
x I y
0 0
1,53 2,67
4,2 1,81
8,57 5,87
10,58 6,9
14,45 0
Mathematica:
3
f´´(x)= 0.0132253x -0.316594x +2.14173x -3.83313
0 = 0.0006612672x⁵ - 0.0263828592x⁴ + 0.3569558021x³ - 1.9165654583x² + 3.9327139744x
f(x)=0
2
3
f´´(x)= 4*0.0033064x -0.105531x +1.07087x -3.83313x+3,93271
2
1 Nullstellen
4
3
f´(x)= 0.0033064x -0.105531x +1.07087x -3.83313x+3,93271
2
4
f´(x)= 5*0,0006612672x -4*0,0263828592x +3* 0,3569558021x -2*1,9165654583x+3,9327139744
3
2
Vorbereitung
0
= 52.7864 m²
f(x)dx
6
=
5
0
0.0006612672
4
6
0.0263828592
3
x - x + x - x + x
5
[ ]
0.3569558021
2
4
1.9165654583
3
3.9327139744
2
14.44723
bestimmtes Integral
6
5
0.0006612672
4
6
0.0263828592
3
F(x)= x - x + x - x + x + c
5
0.3569558021
2
4
1.9165654583
3
3.9327139744
2
Stammfunktion:
s= 21,77m
W3
0
f(2.84178) = 2.29214
f(7.50238) = 4.49768
f(13.5943) = 2.01512
W3
W2
W3
W2
x = 2.84178
x = 7.50238
x = 13.5943
W2
f´´´(x )= 0.662764
f´´´(x )=-0.375493
f´´´(x )= 0.866308
W1
s=
= 0
= 0
= 0
x = 2.84178
x = 7.50238
x = 13.5943
W1
4
3
1+(0.0033064x -0.105531x +1.07087x -3.83313x+3,93271)² dx
2
W1
Mathematica:
14,44723
W3
A= ?
a
W2
f´´´(x )= 0.662764
f´´´(x )=-0.375493
f´´´(x )= 0.866308
0 = 0.0132253x -0.316594x +2.14173x -3.83313
3
= 0
= 0
= 0
2
s=
W1
1+(f´(x))² dx
Mathematica:
f´´(x) = 0
b
notwendige Bedinung
f´´´(x) = 0
hinreichende Bedingung
s = ?
3 Wendepunkte
f(1.71973) = 2.6897
f(4.31656) = 1.80612
f(10.2512) = 6.95243
(f(15.6304) =-1.47243
E4
E3
x = 1.71973
x = 4.31656
x = 10.2512
(x = 15.6304)
E3
E2
0.576495 > 0 TIP
-0.900506 < 0 HOP
2.79908 < 0 TIP)
E4
f´´(x ) =
f´´(x ) =
f´´(x ) =
(f´´(x ) =
E2
E1
-1.01898 < 0
HOP
E4
E3
x = 1.71973
x = 4.31656
x = 10.2512
(x = 15.6304)
E2
0.576495 > 0 TIP
-0.900506 < 0 HOP
2.79908 < 0 TIP)
f´´(x ) =
f´´(x ) =
f´´(x ) =
(f´´(x ) =
E2
Mathematica: solve, x
E1
-1.01898 < 0
HOP
Mathematica:
4
0 = 0.003306336x -0.105531437x +1.070867406x -3.833130917x+3,9327139744
3
2
f´(x)=0
f´´(x)=0
notwendige Bedingung
hinreichende Bdingung
keine Symmetrie
f(x) = 0.0006612672x⁵ - 0.0263828592x⁴ + 0.3569558021x³ - 1.9165654583x² + 3.9327139744x
2 Extrema
f(x)=-f(-x)
f(x)=f(-x)
5 Symmetrie
Nullstellen
-0.026208=
5
1,81= 0*4,2 - 0,03*4,2 + 0,36*4,2 -1,92*4,2 + 3,93*4,2
4
3
2
Vorbereitung
Wendepunkte
Extrema
Symmetrie
0
x I y
0 0
1,53 2,67
4,2 1,81
8,57 5,87
10,58 6,9
14,45 0