Zad 1 Korzystając z twierdzenia de l’Hospitala
obliczyć granicę:
a) lim
x→0
arcsin
x
2
sin 3x
b) lim
n→∞
ln(2x+1)
ln(x
2
+2
)
c) lim
x→1
x
3
−1
x
2
−4x+3
d) lim
x→∞
(x − ln x)
e) lim
x→∞
ln
2
x
√
x
f ) lim
x→0
+
ln x · ln (x + 1)
g) lim
x→1
x ln x
x
2
−4x+3
h) lim
x→1
x
10
−1
x
3
−1
i) lim
x→π
sin3x
sin5x
j) lim
x→0
+
ctgx
lnx
k) lim
x→∞
ln x−2x
ln x+x
l) lim
x→∞
ln(x
2
+x)
ln(4x+1)
Zad 2 Zbadać monotoniczność:
a)f (x) = (x
3
− 3x
2
) e
x
b)f (x) =
x
3
1−x
2
c)f (x) = 2x
3
− 9x
2
+ 12x
d)f (x) =
lnx
x
e)f (x) =
x
3
3
+
x
2
2
− 2x + 1 f )f (x) = x
3
e
x
g)f (x) = (x + 1)
2
(x − 2)
h)f (x) = x
4
· ln
3
x
i)f (x) = 3x
5
− 5x
3
j)f (x) = x
2
− lnx
k)f (x) = x
2
√
1 − x
2
l)f (x) = e
2x−x
2
m)f (x) = 2x − ln(2x + 3) n)f (x) = xln
2
x
o)f (x) = (x + 1)
3
· x
5
p)f (x) = x
3
+
3
x
q)f (x) = ln
5
x − ln
2
x
r)f (x) = x · ln
5
x
s)f (x) = x
6
− 6x
4
t)f (x) = x · ln
2
x
Zad 3 Znaleźć ekstrema lokalne funkcji:
a)f (x) = x
5
(x − 2)
5
b)f (x) = 2x +
3
e
x
−1
c)f (x) =
2x
2
+10x+13
xe
2
d)f (x) = x
3
· e
−x
e)f (x) = x
3
+ 3x
2
+ 4
f )f (x) =
lnx
x
g)f (x) = 2sinx + cos
2
x
h)f (x) =
lnx
(x−2)
4
3
i)f (x) = x
3
− 2x
2
+ x
j)f (x) =
2x
2
−1
x+1
k)f (x) =
x
3
3
− 2x
2
+ 3x
l)f (x) =
√
x
2
− 25
m)f (x) =
2x
2
−x+1
x+3
n)f (x) =
2x
x
2
+1
o)f (x) = 2 − (x − 1)
4
3
p)f (x) =
xlnx
1+2lnx
q)f (x) = (x
3
− 3x
2
)e
x
r)f (x) = e
2x−x
2
s)f (x) =
(x−2)
2
2x
t)f (x) =
√
x · e
x
u)f (x) =
sin
2
x
x
w)f (x) = x
3
− 2x + x
x)f (x) =
−π
2
x
y)f (x) =
x
2
2
e
x
z)f (x) =
1
(2x−1)
3
Zad 4 Największa i najmniejsza wartość
a)f (x) = x
4
− 2x
2
w [−2, 2]
bf (x) =)e
−x
− e
−2x
w [−1, 1]
c)f (x) = 4 ln x − x
w [1, e
2
]
d)f (x) = x
2
e
−5x
w [−1, 1]
e)f (x) = x − ln x
w [
1
e
, e]
f )f (x) = x +
√
x
w [0, 1]
Zad 5 Zbadać przedziały wklęsłości i wypukłości
funkcji oraz znaleźć jej punkty przegięcia:
a)f (x) = x
2
− 2x + 2x lnx
b)f (x) =
3
√
1 − x
2
c)f (x) = e
arctg
4
√
x
d)f (x) = arcsin(x
2
)
e)f (x) = ln(ln(x
2
+ 1))
f )f (x) =
sinx
x
2
g)f (x) = x
2
− ln x
h)f (x) = x
2
ln x
i)f (x) = 4x
2
+ x
−1
j)f (x) = ln(tg)
x
3
k)f (x) = x
2
− 2x + 2x ln x
l)f (x) =
3
√
sin2x
m)f (x) = x
4
− 6x
3
+ 12x
2
− 7x + 2
Zad 6 Zbadać przebieg zmienności i narysować
wykres funkcji:
a)f (x) =
x
lnx
b)f (x) =
lnx
√
x
c)f (x) =
1
1+x
2
d)f (x) = −x
3
+ 4x − 3
e)f (x) = sin x − sin
2
x
f )f (x) =
ln x
√
x
Przygotował: Andrzej Musielak