Lösung Übung 1.1

Mathe 1 Rommelfanger, Blatt 5, Nr. 16, SS 1998 a. f'(x)=8x³ - 9x² + 12, f''(x) = 24x² - 18x;

b. g'(x) = 7

5 3 ²

( − x)

, g''(x) = 42

5 3 ³

( − x)

; c. h'(x) = -3e^-3x, h''(x) = 9e^-3x; d. p'(x) = 2

x , p''(x) = ₋ ² x2

; e. q p

p ' ( )

( )

= −

+ 10

10 3 ²

, q''(p) = ²⁰⁰

10 3 ³

( p+ )

f. u'(x) = 10 (2x - 3)⁴, u''(x) = 80 (2x - 3)³; g. v'(x) = ln4 ⋅ 4^x, v''(x) = (ln4)²⋅ 4^x; h. w'(x) = 3

5

x ln , w''(x) = ⁻³

2 5 x ln

.

Mathe 1 Rommelfanger, Blatt 5, Nr. 17, SS 1998 a. f'(x)= -18x² + 6x + 2; f"(x)= -36x + 6; stets ist D = R, b. g'(x) = ^{-2 x} - 3 x - 8 x

( x

4 2

3 −2)²

; g"(x)= 4 x + 12 x + 56 x + 12 x + 16 ( x

6 4 3

3 −2)³

; jeweils ist D = R\{³ 2 }, c. h'(x) = ^x

x +

+ 2

2 ( 1)³

, h"(x) = ^{− −} + x x

4

4 ( 1)⁵

; jeweils D = ]-1, +∞[,

d. p'(x) = −⁷₃x⁶3−x , p"(x) = ^{2 7 4 9}

9 3

2 6 2 5

( )

( )

x x

−

⋅ − , jeweils D = [- 3, 3], e. q'(x) = 2x + 1

3

x+ , q"(x) = 2 - 1 3 ²

(x+ ) , jeweils D = ]-3, +∞[,

f. v'(x) = 2^x⋅ln2 - e^−x2 (1 - 2x²), v"(x) = 2^x (ln2)² - e^−x2 (4x³ - 6x), jeweils D = R.

Mathe 1 Rommelfanger, Blatt 6, Nr. 2, SS 1998 Mathe 1 Rommelfanger, Blatt 6, Nr. 14, SS 1998 Mathe 1 Rommelfanger, Blatt 6, Nr. 15, SS 1998 a. h'(2) = 404; b. H'(3) = ³

2 3 + 28.

Lösung Übung 1.1

( )

1-x 2 (1-x) 2

(1-x)

2

1-x 2 1-x 2

a) f(x) = e 1 x u = e v = 1+x u' = -e v'=2x 1

2 1+x

f'(x) = -e 1 x e 1 1 x

2

−

⋅ +

⋅ + + ⋅ +

1

1-x 2

2

2

2x = e 1 x

1 x

 x 

 

⋅ − + + 

 + 

 

( )

( ) ( )

( )

x 2

x 2 x x x x x

4 3 3

e 3

b) f(x) = x 3

e x 3 (e 3) 2(x 3) e x 3 (e 3) 2 e (x 3) 2(e 3)

f'(x) = =

(x 3) (x 3) x + 3

+ +

+ − + ⋅ + + − + ⋅ = + − +

+ +

! Lösung Übung 1.2

( ) ( )

3 2

2

2

(1) f(x) = ax bx cx + d Zeichnung

f'(x) = 3ax 2bx + c f''(x) = 6ax + 2b f'''(x) = 6a

(2) f(0) = 0 d = 0

(3) f' 3 = 0 0 = 3a 3 2b 3 c

+ + →

+

→

→ + +

0 = 9a + 2b 3 + c

(4) f'' (0) = 0 2b = 0 b = 0 (5) f''' (x) = 12 6a = 12 a = 2 (5) in (3) : 18 + c = 0

→ →

→ →

c= -18→

−

Lösung Übung 1.3

2

2 2

2

2

3

a) f(x) = x sin (1 + x )

f'(x) = 1 sin (1 + x ) x 2x cos(1 + x )

b) f(x) = x a mit a

x f'(x) = 2x - a

x f''(x) = 2 +2a

x

es muß gelten: f'(3) = 2 3 -a 0 9

R

⋅ + ⋅ ⋅

+ ∈

⋅ = a = 54

da f'' (3) = 2 + 108 6 0 Min 27

→

= ≥ →

! Lösung Übung 1.4

3 2

a) f(x) = x 9x = x (x 9) = x (x + 3) (x-3) 3. Binomische Formel

− − Grund: Schnelles Ablesen der Nullstellen

( ) ( )

2 2 1

2

3

b) f'(x) = 3x 9 0 x 3 = 0 x 3 f''(x) = 6x

(1) f'' 3 6 3 0 Min (rel.) (2) f'' 3 6 ( 3) 0 Max (rel.) (3) 6x = 0 x 0

− = → − = ±

= ⋅ ≥ →

− = ⋅ − < →

→ =

f'''(x) = 6≠0 WP bei x→ 3=0 c) Zeichnung

d) Randextrema? f(-1) = 8

f(4) = 28 absolutes Max bei x = 4 f( 3) 6 3 abs

→

= − → olutes Min bei x = 3

Lösung Übung 1.5

( ) ( ) ( )

2

2 2 2

2 2 2 2

2

a) f(x) = ln x

x 3x

(x 3x) (x 3x) - x (2x - 3) x 3x x(2x-3) x 3 - 2x+ 3 x 1

f'(x) = = -

x x 3x x x 3x x 3x x 3x x - 3

−

− ⋅ − = ⋅ − − = − = −

− − −

−

Schnellere Lösung durch andere Schreibweise:

2 2

2

f(x) = ln x ln (x) ln(x 3x)

x 3x

1 2x - 3

f'(x) =

x x 3x

= − −

−

− −

( ) ( )

( )

( ) ^{( )} ( )

43

13

2 1

3 3

2 4 2

2

2 2 2

b) f(x) = 3 x x x x

f'(x) = (x4 x) 2x + 1 3

4 1 4

f''(x) = x x 2x + 1 x x 2

3 3 3

−

+ = +

+ ⋅

⋅ + ⋅ + + ⋅

Lösung Übung 1.6

( ) ( )

( ) ( )

2

a a

a

ln (3x + 2)

a) y = log 3x +2 log 3x +2

lna

y' = 2 log 3x + 2 1 3

3x+2 lna

 

  = 

   

⋅ ⋅

 

 

( ) ^{( )} ( )

1-x

1-x 1-x

b) f(x) =1-x e

- e 1 x (e ) 1 1 x -x

f'(x) = stimmt überein.

+ − = − + − =

! Lösung Übung 1.7

3 2

2

23

a) f(x) = ax bx cx + d

f'(x) = 3ax 2bx + c f''(x) = 6ax + 2b f'''(x) = 6a

(1) f(0) = 0 d = 0 (2) f' (0) = -15 c= -15 (3) f''(- ) 0

+ +

+

→

→

=

3 2

4a + 26 = 0 b = 2a

(4) f'''(x) = 6 6a = 6 a = 1 in (3) b = 2

f(x) = x 2x 15x

→ − →

→ → →

+ −

23

3

2 2

1

b) f(x) = x 2x - 15x

(1) x(x 2x - 15) = 0 x 0 x 2x - 15 = 0 x = -1 1 15

+

+ → = +

± +

2 3

3

2

23

x 3 x 5

(2) x 2 15 ( 3)( 5)

(3) f'(x) = 3x 4x - 15

f''(x) = 6x + 4 = 0 x = - (siehe Teil a!)

x x x x x

= = −

+ − = − +

+

→ f'''(x) = 6≠0 WP→

Lösung Übung 1.8

( ) ( ) ( )

x

x x x x x x

2 2 2

x x x

f(x) = x+3

e 3

(e 3) (x + 3)e e 3 xe 3e 3 e (x 2)

f'(x) =

e 3 e 3 e 3

+

+ − = + − − = − +

+ + +

( ) ( )

( ) ( )

( ) ( ) ( )

( )

12

1 1

2 2

12

1 1 1

2 2 2

12

2 2 2 2

2 2 1 2

2

2 3 2 2 2 3

2 2

2 3 3

2 2

g(x) = x x 1 x x 1

g'(x) = 2x x 1 x x 1 2x

x 1 x

= 2x x 1 x x 1 2x x 1

x 1 x 1

2x (x 1) x 3x 2x

=

x 1 x 1

−

−

+ = +

+ + ⋅ + ⋅

 + 

+ + + = + ⋅ +  + +

+ + = +

+ +

! Lösung Übung 1.9

2 2

2 2 2 2 2

4 3 3 3

3 2

6 4 4 4 4

3x - x 2 (i) f(x) =

x

(3 - 2x)x (3x - x 2) 2 (3 - 2x)x (3x - x 2) 2 3x -2x 6x +2x 4 4 3x

f'(x) =

x x x x

-3x (4 3x) 3x -3x (4 3x) 3 -3x - 12 + 9x 6x - 12 (x - 2)

f''(x) = 6

x x x x x

x

−

− − = − − = − + = −

− − ⋅ = − − ⋅ = = =

4 3

8 5 5 5

x (x 2) 4x x (x 2) 4 x - 4x + 8 8 3x

f'''(x) = 6 6 6 6

x x x x

− − = − − = = −

f

2 2

1 2

43

( ) a) D \{0}

b) NS: Zähler = 0 0 = 3x - x 2 0 = x -3x + 2 x = 2 x = 1 c) Extrema: f' (x) = 0 4 - 3x = 0 x =

ii =

−

→

¡

( )

23

43 4

4 3

5

f''( ) = 6 0 Max-

d) WP: f''(x) = 0 0 = 6x - 12 x = 2 8-6 12

f'''(2) = 6 0 2 32

≤ →

→ →

⋅ = ≠

]

P e) (- ; 0) f'(x) 0 f(x) fällt monoton (0; 4/3 f'(x) 0 f(x) steigt monoton (4/3; ) f'(x) 0 f(x) fällt monot

→W

∞ < →

≥ →

∞ < → on

f) (- ; 0) f''(x) 0 f(x) ist konkav gekrümmt (0; 2) f''(x) 0 f(x) ist konkav gekrümmt 2; ) f''(x) 0 f(x) ist konvex

∞ < →

< →

∞ > →

[