ESERCIZI 1. Calcolare le seguenti derivate

ESERCIZI
C ONTENTS
1.
2.
3.
Calcolare le seguenti derivate (semplificando il piú possibile le espressioni)
Calcolare i seguenti limiti con i teoremi di de l’Hôpital
Formule di Taylor e Mac Laurin
1
2
3
1. C ALCOLARE LE SEGUENTI DERIVATE ( SEMPLIFICANDO IL PIÚ POSSIBILE LE ESPRESSIONI )
p
(1) D[h (x + a)(x + b)(x + c)] i
a
(2) D sin(x2 − 5x + 1) + tan
.
h
√ xi
x
(3) D sin 3x + cos + tan x .
5
(4) D h[sin(2x) cos(3x)].
i
x
(5) D sin sin(2x) .
2
h
i
x
x
(6) D 2 tan3 − 6 tan + 3x .
2
2
h
x
xi
(7) D sin2 cot
.
3
2
1
1
6
8
sin 3x −
sin 3x .
(8) D
24
18
1
1
2
2
(9) D − cos(5x ) − cos(x ) .
20x
4
tan 2 + cot x2
(10) D
.
x
h
i
p
3
(11) D sin(7x2 ) + 1 + x2 .
(12) D x log3 x − 3xlog2 x + 6x log x − 6x .
q
(13) D
1 + log2 x .
hp
i
√
(14) D
log x + 1 + log( x + 1) .
"
#
r
√
√
1+x− 1−x
1−x
√
(15) D log √
+ 2 arctan
.
1+x
1+x+ 1−x
h
i
p
(16) D 2 log(2x − 3 1 − 4x2 ) − 6 arcsin(2x) .
x
xe arctan x
(17) D
.
log5 x
h
i
(18) D (tan 2x)cot(x/2) .
x−2 p
2
(19) D 2 arcsin √ − 2 + 4x − x .
6
(20) D |x2 + 1| .
(21) D [| sin x| + | cos x|] .
(22) D [(arctan
x)x ] .
h
i
(23) D 2arcsin(3x) + (1 − arccos(3x))2 .
"
r !#
1
b
(24) D √ arcsin x
.
a
b
1
2
ESERCIZI
p
3x2 − 1
+
log
1 + x2 + arctan x .
3
3x
"
√ #
√
x 2
1 + x 2 + x2
√
+ 2 arctan
(26) D log
.
1 − x2
1 − x 2 + x2
h
i
p
(27) D exp(arctan 1 + log(2x + 3)) .
"
√ #
x 3
1
.
(28) D √ arctan
1 − x2
3
(25) D
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
(11)
(12)
(13)
(14)
(15)
(16)
(17)
(18)
(19)
(20)
(21)
(22)
(23)
2. C ALCOLARE I SEGUENTI LIMITI CON I TEOREMI DI DE L’H ÔPITAL
log cos x
lim
[0].
x→0 sin 2x
x − arctan x
lim
[1/3].
x→0
x3 m
m
x −a
lim
[m/n] se m 6= n, a 6= 0.
x→a xn − an
2x
e −1
lim
[2/3].
x→0 arcsin(3x)
√
√
3
3
√
x− 5
6
√
[2/(3 5)] .
lim √
x→5
x− 5
log cos(ax)
lim
[a2 /b2 ].
x→0 log cos(bx)
e3x − 3x − 1
lim
[9/50].
x→0
sin2 (5x)
1
x sin(2x)
lim 1 −
.
x→0
2
2
log(x − 3)
lim 2
[4/7].
x→2 x + 3x − 10
x
−x
2 − (e + e ) cos x
[1/3].
lim
x→0
x4
log x
a
−x
lim
[log a − 1].
x→1
log x
2 π
√
1 − 4 sin 6 x
lim
[π 3/6].
2
x→1
1−x
sin 3x − 3xex + 3x2
lim
[18].
x→0 arctan x − sin x − x3 /6
√
√
ea x − 1
lim p
[a b/b] .
x→0
sin(bx)
x
e2 − 1 + x2
lim
[2e2 ] .
x→+∞
1/x
etan x − ex
lim
[1] .
x→0 tan x − x
sin(−2x)
lim
.
x→0
3x
log x
lim
[0].
x→0+ cot x
cot 5x
lim
[1/5].
x→0 cot x
log sin x
lim
[0].
x→0+ cot x
log(x − a)
lim+
[1].
x→a log(ex − ea )
2
log(1 + ex )
lim
[0] .
x→+∞
cot πx
tan(πx/2)
lim
[∞] .
x→1 log(1 − x)
ESERCIZI
(24)
(25)
(26)
(27)
(28)
(29)
(30)
3
log sin 2x
[1] .
log sin x
log x
lim
[1].
x→0+ log sin x
√
log 3 x3 + x2
lim
[1].
x→+∞
log x
log sin ax
lim
[1] .
x→0 log sin bx
log x
[1/2].
lim
x→0 1 + 2 log sin x
cos x log(x − 3)
lim
[cos 3].
x→3 log(ex − e3 )
π/x
lim
[π 2 /2].
x→0 cot(πx/2)
lim
x→0+
3. F ORMULE DI T AYLOR E M AC L AURIN
Scrivere i polinomi approssimanti dell’ordine n indicato e centrati in x0 delle seguenti funzioni.
(1) f (x) = x3 − 2x2 + 3x + 5,
n = 3,
x0 = 2.
(2) f (x) = x5 − 2x4 + x3 − x2 + 2x − 1,
n = 5,
x0 = 1.
(3) f (x) = x4 − 5x3 + x2 − 3x + 4,
n = 4,
x0 = 4.
(4) f (x) = x3 + 3x2 − 2x + 4,
n = 3,
x0 = −1 .
(5) f (x) = x10 − 3x5 + 1,
n = 10,
x0 = 1.
(6) f (x) = ex ,
n = 3,
x0 = −1 .
(7) f (x) = log x,
n = 4,
x0 = 1.
1
n = 3,
x0 = a .
(8) f (x) = ,
xx
(9) f (x) = xe ,
n = 4,
x0 = 0.
x
,
n = 4,
x0 = 2.
(10) f (x) =
x−1

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