Vorkurs Mathematik

86 6. Lösungen
11. (a) lim f(x) = 0 (b) lim f(x) = 3 (c) lim f(x) = ∞ , lim f(x) =
x→±∞ x→±∞ x→−∞ x→∞
−∞ (d) lim
x→−∞
(f) lim
x→∞
f(x) = ∞
f(x) = −∞ , lim
x→∞
f(x) = ∞ (e) lim
x→±∞
f(x) = 0
12. (a) lim f(x) = 1 , lim f(x) = −1 (b) lim f(x) = ∞ , lim f(x) = −∞
x→−∞ x→∞ x→−∞ x→∞
(c) lim
x→±∞
f(x) = 0 (d) lim
x→±∞
13. (a) −6 (b) 1 (c) 6 (d) −1
4
f(x) = 0
(e) 4 (f) 4 (g) −9 (h)
6.5.2 Lösungen zum Abschnitt ” Differentialrechnung“
Aufgaben:
1. (a) f ′ (x) = −2 x 3 + x 2 − 4 x − x −2
(c) f ′ (x) = 10 x −6 − 9 x −4 + 24 x −3
1 (f) f ′ (x) = − x −2 + 5 x −6 + 2 x + 2 x −3
(h) f ′ (x) = 4 a 2 x 3 − 4 a x b (i) f ′ (x) = −1
3 x−2 − 2
(k) f ′ (x) = 1 −5
x 6 (l) f
6 ′ (x) = −1
3
2
3
(b) f ′ (x) = 3 a 2 x 2 − 2 √ b x + 1
2 c
(d) f ′ (x) = −x −3
2 − 2 x −1
3 − 4 x −5
3 (e) f ′ (x) =
(g) f ′ (x) = 3 x 2 + 5 2
x 3 − 2 x −
2 3 1
x 2
2
3 x−3
−4
x 3 − 1 −13
x 12 (m) f
12 ′ (x) =
(n) f ′ (x) = 1 −1
x 2 sin(x) + x
2 1
2 cos(x) (o) f ′ (x) = ln(5) 5 x + ln(2) 2 x
(p) f ′ (x) = cos(x) ln(x) + sin(x)
x
(s) f ′ (x) = x (x 2 − 1) −1
2 (t) f ′ (x) =
(j) f ′ (x) = 2 + 6 x −3
1 − x
ex (q) f ′ (x) = 2 x cos(x 2 ) (r) f ′ (x) = 2 x sin(x 2 )
1
2 (1 − x 2 ) 1
2
2 sin(x) cos(x) (v) f ′ (x) = e sin(x) cos(x) (w) f ′ (x) = 1
1
ln(7) x
+ (1 + x) 1
2
2 (1 − x) 3
2
(u) f ′ (x) =
2 cos(x
2 ) (x) f ′ (x) =
2. (a) f ′ (x) = n x n − 1 , f ′′ (x) = n (n − 1) x n − 2 , f ′′′ n − 3
(x) = n (n − 1) (n − 2) x
(b) f ′ (x) = 1
2 √ x , f ′′ (x) = −1
3
(c) f ′ (x) =
3 x3
2
(1 − x2 ) 5
(d) f ′ (x) =
−x
√ 1 − x 2 , f ′′ (x) =
1
1 − x +
4 2√ x 3 , f ′′′ (x) =
−1
√ 1 − x 2 −
x
(1 − x) 2 , f ′′ (x) =
8 2√ x 5
x 2
2
(1 − x2 ) 3
, f ′′′ (x) =
2 2 x
2 +
(1 − x) (1 − x) 3 , f ′′′ (x) =
−3 x
2
(1 − x2 ) 3
−
6
3 +
(1 − x)
6 x
(1 − x) 4
(e) f ′ (x) = −2 cos (1 − 2 x) , f ′′ (x) = −4 sin (1 − 2 x) , f ′′′ (x) = 8 cos (1 − 2 x)
Vorkurs Mathematik FB A/I H Harz