[ITA] Esercizi di Analisi MAtematica 1 Ingegneria.pdf

8/10/2019 [ITA] Esercizi di Analisi MAtematica 1 Ingegneria.pdf

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4/50

7!

4!

3! 4!5!

n!

(n+ 1)!

(n!)2

n n! .

210

6

5

1

n + 1

(n 1)!

11

8

73

2018

10

7


5/50

A= (3, 1] (0, 2] B = (, 1] [1, +)

A= [0, 1) [2, 3) B= (, 3) [2, 4) A= (, 3] (1, +) B= [1, 3] (2, 4)

A= (, 1) (1, +)

B= [0, 4] (0, 3)

max A= 2 min A max B= min B= 1

max A min A= 0 max B min B= 2

max A= 3 min A max B min B= 1

max A min A max B min B

A =

(n,

n); n= 1, 2, 3, . . .

B =

n, 12n

; n= 0, 1, 2, . . .

C =

n,

1

n 1

; n= 1, 2, 3, . . .

D =

n,

n

n+ 1

; n= 1, 2, 3, . . .

.

sup A = + infA = min A = 1 max A A sup B = 0 infB = min B =1 max B B sup C = max C = 0 infC =1 min C sup D = 1 infD =min D= 1/2 max D

0 5 10 15 200

1

2

3

4

5

A

x

y

0 5 10 15 201

0.5

0

0.5

1

B

x

y

0 5 10 15 201

0.8

0.6

0.4

0.2

0

C

x

y

0 5 10 15 200

0.2

0.4

0.6

0.8

1

D

x

y


6/50

{an} n1

an= (

2)n

10

an= (1)nn0 an= ln

n 1n

0 an= 10

1/n >1

n {an} n

n {an} n

nN an0 n 1n

1 1n

>0

{bn} n1

n

bn= 3n 100

9

bn= 3 ln n < 0

bn= 2n 1000

bn= e

n 100< 0

3n 1009

nlog3100

9 n3

ln n > 3 n > e3 n21 nlog2100 n7

en


7/50

limn

21/n = 1

limn ln

n+ 1

= +

> 0

ln

1 +1

n

< n > N

ln1 + 1n< 1 + 1

n

< e n > (e

1)1 =N

>0

11 n2

< 1n2 1 < n > N

n >

1 + 1 =N

> 0 |21/n 1|< n > N 21/n 10 n 21/n < + 1 n > (log2( + 1))

1 =N

M >0 ln

n + 1

> M

n+ 1 > eM

n >(eM 1)2 =N

an= n2 +n +

M > 0 an > M

n2 + n M >0 n >1 +

1 + 4M

2 =N

{an} {an} {an}

limn

an= l, an> 1 nl >1 anan+1 n {an}

an+1an n lim

nan=

an = (1)n 1 1 an = 1 +

1

n an > 1 n l= 1

an =

1

n

{an} {bn} limn anbn= 0

an= n sin(

2 +n)

an= ln n cos(n)

an= (1)n(n n2)

an= n+ (1)nn

sin(2 +n) = (1)n an = (1)n n + (M, +) M >0 n bn = 1/n2

cos(n) = (1)n an = (1)n ln n bn = 1/ ln

2 n

(1)n+1n2 bn = 1/n

3

+

bn = 1/n2


8/50

an= n+

(1)nn

sup inf max min

sup an = + infan= min an = 0 max an limn an = +

an limn an = 0

limn 2an = 1

limn(an+1 an) = 0 limn 1an = +

supnN an< +

an = 1

2

n

kN nsin

k n

nN ksin

k n

k= 1 k >1 n+ n = mk m N n= 2mk+ 1 sin

k= 0

nN limk sin

k n

= sin 0 = 0

3

2n4 + 3n3 + 1

n+ ln n

n nn+en

ln n nn ln n

n1/2 +n1/3 + 1

n1/4 +n1/5

3

2n4 + 3n3 + 1

n + ln n 2

1/3n4/3

n =

3

2n

n nn + en

nn

= 1

ln n nn ln n

nn

=n


9/50

n1/2 + n1/3 + 1

n1/4 + n1/5 n

1/2

n1/4 = 4

n

limn

1 +n

n n2 3nn

limn

1

1 + (

1)nn

limn

n ln nn

limn

n n

limn

n

1 (2/3)n lim

n

2nen en3n lim

nen 2n

3n

limn

3n sin(n/2)

2n

1 + n

n n2 3nn

n7/3

n =n4/3 lim

n(n4/3) =

limn

|1 + (1)nn|= + limn

1

1 + (1)nn = 0

n ln nn

n limn

n= +

n n n

1 (2/3)n 1 limn

n

1 (2/3)n = +

2< e


10/50

q >1 limn1

qn

nk=0

qk

nk=0

qk = 1 qn+1

1 q ;

limn1

qn

nk=0

q

k

= limn1

qn+1

qn(1 q) = limnqn(qn

q)

qn(1 q) = limn q

1 q = q

q 1

q >1

1

+ 1

an= 1 +(1)n

n

an= (1)nn an= n + (1)n

an=1 1n

{an} lim

nan3n

= 0, limn

ann3

= +

limn

ann

= +, limn

ann

n= 0

limn a

n2n

= +, limn a

n3n

= 0

limn

anln n

= 0, limn

a2nln n

= +

an= n4

an= n 3

n

an= en

an= (ln n)2/3

a, bR+

a, b

an=2a+b 1a

n

ab (a, b)

2a + b 1a

1 0a + b 10.

A(1/3, 0) B(1, 0) C(0, 1) AB AC BC


11/50

n=0

n

n!

n=1

(1 en)

n=1

21/n

nen

n=1

(1)n nn

n=0

1 n+n21 +n2 +n4

n=1

(1)n 1n2 +n

n=2

(1)nn ln n

n=1

3 +en

n!

n=0

en +

1

2

n

n=0

an limn

an= 0

n + 1

(n + 1)! n!

n=

1n(n + 1)

n0

limn(1en) = 1 +

n2

2n

ne

n 1e


12/50

n=1

1

n2

1n2 + n

1

n ln n

3

n=0

1

n!

n=0

1

2

n

n=21

ln n

n=1

1

n2 + sin n

n=1

(1)n 1n1/3

n=1

ln n nn+ 1

n=21

ln n >

n=21

n

n=1

1

n2

1

n1/3

ln n nn + 1

1n

n=1

1

n1/2

n=0

n2 + 1

n!

n=1

1

n+ ln n

n=0

(1)n(1 +en)

n=0

n+ sin n

1 +n2


13/50

(n + 1)2 + 1

(n + 1)! n!

n2 + 1 =

n2 + 2n + 2

n3 + n2 + n + 1 1

nn0

n=11

n

limn(1 +

en) = 1 n=0

(1)n

n=1

1

n

n=1

1

nlnn

n=2

1(ln n)n

n=1

1

nlnn 1 qn

1 + qn qn

qn = 1

n=0

1 = + 0< q


14/50

n=1

1

n 1

n + 3

=

1 14

+

1

2 1

5

+

1

3 1

6

+

1

4 1

7

+

1

5 1

8

+ ...=

= 1 +1

2+

1

3+

1

4 1

4

+

1

5 1

5

+ ...= 1 +

1

2+

1

3=

11

6 .

a

b

a

b

n=1

na

1 +nb

n=1

1

nba

ba > 1 b > a+ 1 + b < a+ 1 a, b b > a + 1

an= sin(n/2) + | cos(n/2)|

limn an

n=1an

1;1;1; 1;1; ... limn an

n=1

1

n1+ n1 0< 1


15/50

1

1

ln x

1x x3

11 x2

x2 4x+ 3

ln

1

1 |x|

11 ln |x|

(0, e) (e, +) (, 1) (1, 0) (0, 1) (1, +) (1, 1)

(, 1) [3, +) (1, 1)

(e, 0) (0, e)

arcsin

1

x2

arccos(2

x2)

(, 1] [1, +) [3, 1] [1, 3]

1 xex

ex ex

sin x

1 +x2


16/50

x x2

x2 x3

ex +ex

x|x| sin x cos x

1 x x2 (x 2)3

1 +x+ cos x

|x3| ln x ex

1 x2

ex x sin(x /4)

ln x sin x 1 e|x|

1 (x+ 1)2 ln x x

1/

x 1/x 1/x2 (0, 1]

ex/2

ex

e2x

sin(x/2) sin x sin(2x) [0, 4] ln(2x) ln x ln(x/2) (0, +)

f1(x) = x2 1

f2(x) = (x+ 1)2 1

f3(x) =|x2 1|

f4(x) = 1 (x 1)2


17/50

4 2 0 2 420

15

10

5

0

51xx

2

x

y

2 0 2 4 6100

50

0

50

100(x2)

3

x

y

10 5 0 5 1010

5

0

5

10

151+x+cos(x)

x

y

4 2 0 2 40

10

20

30

40

50

60

70|x

3|

x

y

0 0.5 1 1.5 27

6

5

4

3

2log(x)e

x

x

y

4 2 0 2 415

10

5

0

51x

2

x

y

4 2 0 2 40

10

20

30

40

50

60e

xx

x

y

5 0 51

0.5

0

0.5

1

sin(x/4)

x

y


18/50

0 2 4 6 82

1

0

1

2

3log(x)sin(x)

x

y

4 2 0 2 40

0.2

0.4

0.6

0.8

11e

|x|

x

y

10 5 0 5 10120

100

80

60

40

20

0

201(x+1)

2

x

y

0 2 4 6 87

6

5

4

3

2

1log(x)x

x

y

0 0.2 0.4 0.6 0.8 10

2

4

6

8

10

1/x1/2

, 1/x, 1/x2

x

y

1/x1/2

1/x

1/x2

1 0.5 0 0.5 10

2

4

6

8

x

y

ex/2

, ex, e

2x

ex/2

ex

e2x

5 0 52

1

0

1

2

3

4sin(x/2), sin(x), sin(2x)

x

y

sin(x/2)sinxsin(2x)

0 0.2 0.4 0.6 0.8 18

6

4

2

0

2

x

y

log(x/2), log(x), log(2x)

log(x/2)log(x)log(2x)


19/50

2 1 0 1 21

0

1

2

3x

21

x

y

2 1 0 1 22

0

2

4

6

8(x+1)

21

x

y

2 1 0 1 20

0.5

1

1.5

2

2.5

3|x

21|

x

y

2 1 0 1 28

6

4

2

0

21(x1)

2

x

y

f f

f(x) =

x sex[0, 1]2x 1 sex(1, 2]

f(x) =

x se x[1, 0]2x se x(0, 1]

f(x) = x se x[1, 0]x/2 se x(0, 1]

f(x) =

x+ 1 se x[1, 0]1 +x/2 se x(0, 1] .

f1(y) =

y se y[0, 1](y+ 1)/2 se y(1, 3]

f1(y) =

y se y[0, 1]y/2 se y[2, 0)

f1(y) =

y se y[1, 0]2y se y(0, 1/2]

f1(y) =

y 1 se y[0, 1]2y 2 se y(1, 3/2]

f1(x) =

x/2 x02x x < 0;

f2(x) =

x/2 + 1 x02x 1 x < 0.


20/50

1 0 1 2 3 41

0

1

2

3

4(a)

x

y

3 2 1 0 1 2 33

2

1

0

1

2

3(b)

x

y

2 1 0 1 22

1

0

1

2(c)

x

y

2 1 0 1 22

1

0

1

2(d)

x

y

f11 (y) =

2y se y0y/2 se y


21/50

f(x) = e2x3

f(x) =

1 2x

f(x) = arctg(2x 1) f(x) = ln(2 + 3x)

f : R

(0, +

) f1 : (0, +

)

R f1(y) = ln y+ 3

2

f : (, 1/2][0, +) f1 : [0, +)(, 1] f1(y) = (1 y2)/2 f : R (/2, /2) f1 : (/2, /2)R f1(y) =tan y+ 1

2

f : (2/3, +)R f1 : R (2/3, +) f1(y) = ey 2

3

1 0 1 2 3 41

0

1

2

3

4(a)

x

y

2 1 0 1 2 32

1

0

1

2

3(b)

x

y

2 1 0 1 2

2

1

0

1

2(c)

x

y

2 1 0 1 2 3

2

1

0

1

2

3(d)

x

y

limx+

x sin x

1 +x2

limx+

x 3x1 +x

limx0

x2

1 ex

limx1

ln x

ex e

x+

x sin x

1 + x2 sin x

x

limx+

sin x

x = 0

x+

x 3x1 + x

1x

limx+

1x

= 0

x0 1 ex x x2

1 ex x


22/50

x 1 =y

limx1

ln x

e(ex1 1)= limy0ln(y+ 1)

e(ey 1) .

y0 ln(y + 1)y ey 1y ln(y+ 1)e(ey 1)

1

e

1

e

limx0+x ln(x)

limx1

(x 1)2sin(1 x)

limx0

tg x

x2

limx

x4ex

limx1

1 x2 ln x

limx+

arccos

1

x

arctg x

limx0

e3x 1ln(1 2x)

limx

2

x 2cos x

x= y lim

x0+

x ln(

x) = lim

y0+y ln y = 0

y= 1 x limx1

(x 1)2sin(1 x) = limy0

y2

sin y= lim

y0y = 0

x

0 tan x

x lim

x0

tg x

x2

= limx0

1

x

=

x= y lim

xx4ex = lim

y+y4

ey = 0

x 1 =y

limx1

1 x2 ln x

= limy0

y2ln(1 + y)

= limy0

y2y

=12

;

limx+

arccos

1

x

arctg x

=arccos 0

/2 =

/2

/2= 1

x0 e3x 13x ln(1 2x) 2x 32

x

/2 =y

limx

2

x 2

cos x = lim

y0y

cos(y+ /2)= lim

y0y

sin y =1.

x+

1

2 + sin x

ln x sin x

x2 sin x

cos2 x


23/50

xn = (2n + 1)/2

xn = n/2

xn = (2n + 1)/2

xn = n/2

limx0sin x

ln(1 +x2)

limx

2

x 2cos2 x

limx0

xe1/x

limx0

arctg(1/x)

x2

x0 sin xx ln(1 + x2)x2 limx0

sin x

ln(1 + x2) = lim

x01

x =

x

2

=y

limx

2

x 2

cos2 x = lim

y0y

cos2(y+ /2)= lim

y0y

sin2 y = lim

y01

y =;

1

x=y lim

yey

y lim

y+ey

y = + lim

yey

y = 0

x0 arctg(1/x) /2 1x2

+ limx0

arctg(1/x) 1x2

=

limx0

sin(2x)

1 e3x

limx0sin(2x)

|1 e3x|

limx0

ex e2xx

limx0

x

ln(1 +x2)

limx0

sin x2 sin xx2 x

limx0

| sin x|x

limx0+

xe1/x

sin(2x)

1 e3x 2

3

2/3

|1 e3x|=

1 e3x x0e3x 1 x < 0

2/3 2/3

x0 ex(1 ex)

x ex 1

x

ln(1 + x2) 1

x


24/50

sin x2 sin xx2 x = =

sin x2

x2 x

x 1sin x

x 1

x 1 1 1

y= 1/x

limx0+

xe1/x = limy+

ey

y = +.

x0+

x

x+ tg 3

x

x0 x + tg 3xx + 3x 3x x/ 3x= 6x

limx+

1 +

1

2x2

3x2

limx+

1 +

1

2x2

3x2= lim

x+

1 +

1

2x2

2x23/2 2x2 = y

limy+

1 +

1

y

y3/2=e3/2 =e

e.

3x2 + 1

1 x

x2 +ex

2x+ 3

x2 1ln |x| 2x

x x1/31 +ex

y =3x 3

+ y = 1/2x 3/4

y = 0 x+ x

f(x) = x(1 +ex) + ln x x+

limx+

f(x)

x = 1 lim

x+(f(x) x) = lim

x+xex + lim

x+ln x = +

n=1

1 cos

1

n

1 cos xx2

2 x0

1 cos1n 1

2

1

n

2=

1

2n2 pern ;

1

2

n=1

1

n2


25/50

ln3(1 +x2)

x

tg x

sin2(3 2x)

ex

ln x

cos3(1 x2)

ln x

1 x2 exx

2

x

1 + sin x

ln3(1 + x2)

=

6x ln2(1 + x2)

1 + x2

x

tg x

=

tg x x(1 + tg2 x)tg2 x

sin2(3 2x)

=4 sin(3 2x)cos(3 2x)

ex

ln x

=ex x ln x 1

x ln2 x

cos3(1 x2)

= 6x cos2(1 x2) sin(1 x2)

ln x

1 x2

=1 x2 + 2x2 ln x

x(1 x2)2

ex

x2

=ex

x2

(1 2x)

x

1 + sin x

=

1 + sin x x cos x(1 + sin x)2

D

x tg(1 +x2)

D

arcsin(1 + 3x3)

D

1 + sin x

1 cos2 x

D

ln

ln(x+ 1)


26/50

D

x tg(1 + x2)

= tg(1 + x2) + 2x2(1 + tg2(1 + x2))

D

arcsin(1 + 3x3)

= 9x21 (1 + 3x3)2

D

1 + sin x

1 cos2 x

=cos x(2 + sin x)sin3 x

D

ln

ln(x + 1)

= 1

(x + 1)ln(x + 1)

D

2x2

D x

ln2 x

D

x2 sin x

D sin(ex)

D2x2= 2x2 ln 2 2x D

x

ln2 x=

ln x 2ln3 x

D

x2 sinx

= x2 sinx

2cos x ln x + 2sin x

x

D sin

ex

=ex cos ex

f

f(x) = ex

2

x=

1

f(x) =

1

1 x x= 0

f(x) =

1

tg x x= /4

f(x) = 1

ln x x= e

f(x) x0 y f(x0) = f(x0)(x x0)

y= 2x/e + 3/e

y= x + 1

y=2x + /2 + 1

y= 2 x/e

(0, +) f(x) = ln x g(x) = arctg x

x0 f g

x0, f(x0)

x0, g(x0)

f (x0, f(x0)) f(x0) = 1/x0

g (x0, f(x0)) g(x0) = 1/(1 + x20)

1

x0=

1

1 + x20 x20x0+ 1 = 0

x0


27/50

f(x) = x3 x0 [0, 1]

f

(x0, f(x0)) 1

(x, f(x)) f(x) = 3x2 f(x) = 1

x = 13

x0 = 1

3

1

3,

1

3

3

y =

x 23

3

D 1

f

g(x)

D

f(x)g(x)

D

1

f(x)g2(x)

D

f

1

g(x)

D 1

f

g(x) =f(g(x)) g(x)

f2(g(x))

D

f(x)g(x)

= f(x)g(x)

g(x) ln f(x) + g(x)f(x)f(x)

D 1

f(x)g2(x)=f

(x)g(x) + 2f(x)g(x)f2(x)g3(x)

D

f

1

g(x)

=f

1

g(x)

g

(x)g2(x)

D

f(1 +x+g(x))

D

f(1 +x) g(1 x)

D

f

g(x) g(x+ 1) D

f(xg(x))

D

f(1 + x + g(x))

= f(1 + x + g(x)) (1 + g(x))

Df(1 + x) g(1 x)= f(1 + x) g(1 x) f(1 + x)g(1 x)

D

f

g(x) g(x + 1)

= f (g(x) g(x + 1)) (g(x) g(x + 1))

D

f(xg(x))

= f(xg(x)) (g(x) + xg(x))

D

f(x)g(x)h(x)

D

f(x)g(x)h(x)

= f(x)g(x)h(x)

g(x)h(x) + g(x)h(x)

ln f(x) + g(x)h(x)f(x)f(x)


28/50

f(x) = xex

f

f

0 f1(y) y= f(0)

(, 0] [0, 1] [1, +) f1(0) =

0

f1

(0) = 1

f(0) = 1

2 1 0 1 2 3 410

9

8

7

6

5

4

3

2

1

0

1

xex

x

y

f(x) = x3 +x f1(y)

f1(2)

f1

(2)

f(x) = 3x2 + 1 > 0x R f(x) R f1(y) R

f1

(y) = 1

f(x) x =

f1

(y)

f(x)

f1

(2)

x

3

+ x 2 = 0

x= 1 f1(2) = 1

f1

(2) = 1

f(1) =

1

4

sin |x|

3

x 1 x|x 1|

| sin x|

x= 0 D (sin |x|) = cos |x| D(|x|) |x| x= 1 3

y y = 0

x= 1 |y| x= k kZ y= sin x

f(x) =

1 x


29/50

f R x=1 x=1 g x= 1 x=1 x=1

3 2 1 0 1 2 3

4

2

0

2

4

6

f

x

y

3 2 1 0 1 2 3

4

2

0

2

4

6

g

x

y

x= 1

x < 0 x > 0

(0, 1)

R

f(x) =|x 1| f(x) = arctg x

f(x) = 1/x

f(x) = 11 + x2

(, +)\{0} 0 1 f(x) =|x 1| |x + 1| + sgn x

2 1.5 1 0.5 0 0.5 1 1.5 21

0.5

0

0.5

1

1.5

2

2.5

3

3.5

4|x1||x+1|+sign(x)

x

y

f

f >0 f(0)< 0 f(1)> 0

f(x) = x2 x + 1


30/50

f

f(x) =ex

f

g

f(x) =

aebx sex01 +x2 sex


31/50

x= 0 x= 0 x+ y = 0 x x =2 e2/4

R

x+ y = 0 x x= 3/2

1

2e3 x= 2

R x y = 0 x+ x=2 4/e2 x= 0

x=2 2 R x y = 0 x+

x= 1/2 1/2e x= 0

4 2 0 2 4

0

2

4

6

8

101/(x

2e

x)

x

y

2 1 0 1 2

10

8

6

4

2

0

(x1)/e2x

x

y

1 0 1 20

2

4

6

8

10x

2e

x

x

y

1 0 1 22

0

2

4

6

8

10(x1)e

2x

x

y

e2x ex

x2 ln x

ln(x+ 1) x arctg(1 x)

R y = 0 x x = ln 2 x= ln 4 x > ln 4 x 1 x= 1

1

x 1

x 1

x

1 + |x|


32/50

1 0 1 210

0

10

20

30

40

50e

2xe

x

x

y

0 0.5 1 1.5 20

1

2

3

4

5x

2log(x)

x

y

1 0 1 2 3 44

3

2

1

0

1log(x+1)x

x

y

4 2 0 2 41.5

1

0.5

0

0.5

1

1.5arctg(1x)

x

y

ln(1 +x2)

1

x+ ln x

1

x 1

x 1= 1

x(1 x) xR \ {0, 1} x= 0x= 1 y = 0 x= 1/2 (0, 1)

R

y = 1 + y =1

x >0

x < 0

R x = 0 x=1 (1, 1)

(0, +) x= 0 x= 1 x= 2 x 2

1 x 2x2

exx

x

ln x

x(2 x)ex

[1, 1/2] (1/2, 1/2) x=1/4 x=1, 1/2

x 0 x > 0 x= 1/4 x= 0

(0, +) \ {1} x = 1 limx0+f(x) = 0 x= e (1, e

2) x= e2


33/50

2 1 0 1 210

5

0

5

101/x1/(x1)

x

y

2 1 0 1 21

0.5

0

0.5

1x/(1+|x|)

x

y

10 5 0 5 100

1

2

3

4

5log(1+x

2)

x

y

0 2 4 6 8 100

2

4

6

8

101/x+log(x)

x

y

R y = 0 + x= 2 2 x= 2 +

2 (3 3, 3 + 3)

x= 3 3

1.5 1 0.5 0 0.5 10.5

0

0.5

1

1.5(1x2x

2)

1/2

x

y

0 1 2 30

0.5

1

1.5

2

2.5

3e

xx1/2

x

y

0 2 4 6 85

0

5x/log(x)

x

y

2 0 2 4 6 83

2

1

0

1

2x(2x)e

x

x

y

1

x 1 x2

1

x2 2x+ 2


34/50

R

y= 0 x= 1/2 (0, 1) x= 0, 1

R

y = 0

x= 1

(0, 2) x= 0, 2

2 1 0 1 22

1.5

1

0.5

0

0.5

11/(x1x

2)

x

y

2 0 2 40

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

21/(x

22x+2)

x

y

f f

f(x) = |x|x 1

f(x) =

|x| 1x

R

\{1}

x= 1 y = 1 +

y=1 x= 0 (0, 1) f x= 0 x= 0

x= 0 x= 0 y = 1 + y =1 x < 0 f

x3 2x2 +x+ 1 1 x+x2 x3

x(4 x+x2

/2) 1 3x+x2 x3

x 2/3 limx

f(x) =

x > 1/3 x 2/3 limx

f(x) =

x > 1/3 x


35/50

4 2 0 2 43

2

1

0

1

2

3|x|/(x1)

x

y

2 1 0 1 22

1.5

1

0.5

0

0.5

1

1.5

2(|x|1)/x

x

y

1 0 1 23

2

1

0

1

2

3x

32x

2+x+1

x

y

2 1 0 1 25

0

5

10

151x+x

2x

3

x

y

1 0 1 2 310

5

0

5

10

15

20x(4x+x

2/2)

x

y

4 2 0 2 4100

50

0

50

10013x+x

2x

3

x

y


36/50

f(x) = ln x ln(x 1) f

(1, +) x = 1 y = 0

f(x) = 1

1 +ex

R y = 0 x

+

y = 1

x x= 0

4 3 2 1 0 1 2 3 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

11/(1+e

x)

x

y

f(x) = x2

a2 x2 a f

f

a

f

a

[a, a] [0, +) x= 0 x=a x=

2/3a a

f(b) f(a)b a = f

(c) c (a, b) c

f(x) = 1 x3 a= 0 b= 1 f(x) = x2 x a= 1 b= 2

f(x) = 1/x3 a= 1 b= 2

f(x) = 1

1 +x

a= 0

b= 2

f(c) =1 3x2 =1 x=1/3 c= 1/3(0, 1) f(c) = 2 2x 1 = 2 c= 3/2 f(c) =7/8 3/x4 =7/8 x= 4

24/7 c= 4

24/7(1, 2)

f(c) =1/3 (1 + x)2 =1/3 c= 3 1

R


37/50

4 2 0 2 40

5

10

15

20

25

a=1

x

y

4 2 0 2 40

5

10

15

20

25

a=2

x

y

4 2 0 2 40

5

10

15

20

25

a=3

x

y

4 2 0 2 40

5

10

15

20

25

a=4

x

y

ex

ex

ex

ex

1 0 1 2 33

2.5

2

1.5

1

0.5

0

ex

x

y

1 0 1 2 30

0.5

1

1.5

2

2.5

3

ex

x

y

3 2 1 0 13

2.5

2

1.5

1

0.5

0e

x

x

y

3 2 1 0 10

0.5

1

1.5

2

2.5

3e

x

x

y


38/50

f

1 1 f(x) =|1 x2|

f

(1, f(1))

f

1 1

f(x) = ex

23x+2

f(x) = ln(2 +x2)

f(x) = sin(x )

f(x) = e3x2

y =x+ 2 f(1) = 3 f 1

y = 2

3(x1) + ln3 f(1) = 2/9 f 1

y = (x1) f(1) = f 1

y=2e2x + 3e2 f(1) = 2e2 f 1

0 0.5 1 1.5 20

0.5

1

1.5

2(a)

x

y

0 0.5 1 1.5 20.4

0.6

0.8

1

1.2

1.4

1.6

(b)

x

y

0.5 1 1.51

0.5

0

0.5

1(c)

x

y

0.5 1 1.50

5

10

15

20(d)

x

y


39/50

sin2(x 1) cos(x 1) dx

ex

(ex 6)4 dx

ln(1 +x)

2(1 +x) dx

arctg2 x

1 +x2 dx

(sin(x 1)) = cos(x 1) sin2(x 1) cos(x 1) dx= sin

3(x 1)3

+ c, cR;

(ex 6) = ex (ex 6)4ex dx= 1

3(ex 6)3 + c, cR;

(ln(1 + x))= 1

1 + x

ln(1 + x)

2(1 + x) dx=

(ln(1 + x))2

4 + c, cR;

(arctg x) = 1

1 + x2

arctg2 x1 + x2 dx=

arctg3 x

3 + c, cR.

1

x ln xdx

2x

1 +x4dx

cos(1/x)

x2 dx

11 3x2 dx


40/50

(lnln x)= 1

ln x 1

x lnln x + c cR

(arctg x2) = 2x

1 + x4 arctg x2 + c cR

sin

1

x

= cos(1/x) 1

x2

sin1x

+ c cR

(arcsin

3x) = 1

1 3x2

3 1

3arcsin

3x + c c

R

x

x2 + 4x+ 3

ln x

x

x sin x

(arcsin x)2

x3

1 +x2dx

1

cos x

x

x2 + 4x + 3=

A

x + 1+

B

x + 3 A=1/2 B= 3/2

12

ln |x + 1| +32

ln |x + 3| + c= ln

(x + 3)3

x + 1 + c, cR;

ln x

x dx= ln2 x

ln x

x dx,

ln x

x dx=

ln2 x

2 + c, cR;

x sin xdx=x cos x +

cos xdx=x cos x + sin x + c, cR;

x= sin y (arcsin x)2dx=

y2 cos ydy,

y2 cos ydy = y2 sin y+ 2y cos y 2sin y+ c, cR

(arcsin x)2dx= x(arcsin x)2 + 2 arcsin x

1 x2 2x + c;

x3

1 + x2

= x(x2 + 1) x

1 + x2

x3

1 + x2dx=

x dx 1

2

2x

1 + x2dx=

x2

2 ln

1 + x2 + c, cR;

t= tgx

2

2

1 t2 dt 2

1 t2 = 1

1 t + 1

1 + t

2

1 t2 dt= ln1 + t

1 t+ c

1

cos xdx= ln

1 + tgx

2

1 tgx2

+ c, cR.


41/50

3x2 ln xdx

1

x+ 1dx

1

1 +exdx

ex sin xdx

3x2 ln xdx= x3 ln x

x2dx= x3 ln x x

3

3 + c, cR;

x= t2 dx= 2t dt

2

t

t + 1dt= 2

dt 2

1

t + 1dt= 2t ln2(t + 1) + c, cR,

1

x + 1dx= 2

x ln2(x + 1) + c;

ex =t dx= 1/t dt 1

t(1 + t)dt=

1

t 1

t + 1

dt= ln

tt + 1

+ c, cR,

1

1 + exdx= ln

ex

ex + 1+ c;

2

e

x

sin xdx= e

x

(sin x cos x)

ex sin xdx=ex(sin x cos x)

2 + c, cR.

sin2 x cos2 x dx

x2

1 +xdx

sin x cos x= (sin 2x)/2 2x= y sin2 x cos2 x dx=

1

4

sin2 2x dx=

1

8

sin2 y dy,

sin2 x cos2 x dx=

1

8

y2

4 sin y cos y

2

+ c=

1

8

x2 1

2sin 2x cos2x

+ c, cR;

x2 = (x + 1)(x 1) + 1

x2

1 + xdx =

(x 1) dx +

1

1 + xdx =

x2

2 x + log |x + 1| + c, cR.


42/50

(x 2)2 ln x dx

x2 sin x dx

x arctg x dx

ln(1 +x2) dx

(x 2)2 ln x dx =(x 2)

3

3 ln x

(x 2)3

3 1

xdx

=(x 2)3

3 ln x 1

3

x2 6x2 + 12x 8

x

dx

=(x 2)3

3

ln x

1

3x

3

3 2x3 + 6x2

8 ln x+ c

=x3 6x2 + 12x

3 ln x +

5

9x3 2x2 + c, cR;

x2 sin x dx =x2 cos x +

2x cos x dx

=x2 cos x + 2x sin x 2

sin x dx

=x2 cos x + 2x sin x + 2 cos x + c, cR;

x arctg x dx =

x2

2 arctg x

x2

2(1 + x2)dx

=x2

2 arctg x 1

2

1 1

1 + x2

dx

=x2

2 arctg x 1

2(x arctg x) + c, cR;

ln(1 + x2) dx =x ln(1 + x2) 2 x2

1 + x2dx

=x ln(1 + x2) 2

1 11 + x2

dx

=x ln(1 + x2) 2 (x arctg x) + c, cR.

sin

1x

x2

dx

1

3

x(1 + 3

x2)dx


43/50

1

2 x2 dx

x22xdx

x1 =t

sin

1x

x2 dx=

sin t dt= cos t + c= cos1

x+ c, cR;

x= t3 1

3

x(1 + 3

x2)dx=

3t

1 + t2dt=

3

2

2t

1 + t2dt=

3

2ln(1+t2)+c=

3

2ln(1+

3

x2)+c, cR;

1

2 x2 = 1

(

2 x)(2 + x) = 1

2

2

12 x +

12 + x

,

12 x

2dx=

1

22ln

2 + x

2 x + c, c

R;

x22xdx =

2x

ln 2x2

2x

ln 22x dx

= 2x

ln 2x2 2

ln 2

2x

ln 2x

2x

ln 2dx

= 2x

ln 2x2 2

x+1

ln2 2x +

2x+1

ln3 2+ c, cR.

2 x2dx

1x(1 +x)

dx

sin

x

dx

1

2 cos x dx

x=

2sin t dx=

2cos t dt

2 x2 = 2cos t

2 x

2

dx= 2

cos2

t dt= t + sin t cos t + c= arcsin

x

2 +x

2

2 x2

+ c, cR;

x= t2 1x(1 + x)

dx= 2

1

1 + t2dt= 2 arctg t + c= 2 arctg

x + c, cR;

x= t2 sin

x dx = 2

t sin t dt= 2

t cos t +

cos t dt

= 2t cos t + 2 sin t + c=2x cos x + 2 sin x + c, cR;


44/50

t= tanx

2 x= 2 arctant dx=

2

1 + t2dt cos x=

1 t21 + t2

1

2 cos x dx= 2

1

3t2 + 1dt= 2 arctan

3t + c= 2 arctan

3tan

x

2

+ c, cR.

ex ex

ex +exdx

ex ex

ex + exdx=

sinh x

cosh xdx = ln(cosh x) + c, cR.

21

1x(1 +

x)

dx

10

x1 +x2

dx

1

0

6x+ 3x2 + 1 +x

dx

10

x3

1 +x4dx

x= t2 dx= 2t dt

2

21

1

1 + tdt=

ln(1 + t)2

21

= ln3 + 2

2

4 ;

1

2 10

2x

1 + x2 dx= 1 + x21

0 =

2 1;

3

10

2x + 1

x2 + 1 + xdx=

ln(x2 + x + 1)3

10

= ln27;

1

4

10

4x3

1 + x4dx=

ln

4

1 + x410

= ln 4

2.

10

1

1 +

4xdx

10

x arctg xdx

/20

sin2 x x sin x2

dx

1/20

x1 x2 dx

4x= t2 10

1

1 +

4xdx=

1

2

20

t

1 + tdt=

1

2

t ln |t + 1|

20

= 1 ln

3;


45/50

x2

2 arctg x

10 1

2

10

x2

1 + x2dx=

8 1

2

10

1 1

1 + x2

dx=

8 1

2

x arctg x

10

=

4 1

2;

/2

0

sin2 xdx 12

/2

0

2x sin x2dx= x sin x cos x

2 /2

0+

1

2cos x2/2

0=

4 1

2;

12

1/20

2x1 x2 dx=

1

2

2

1 x21/20

= 1

3

2 .

10

ex

1 +exdx

/20

sin x

1 + sin xdx

2

1

1

x2

+x

dx

10

x

ex2dx

ex =t e1

1

1 + tdt=

ln |t + 1|

e1

= lne + 1

2 ;

t= tgx

2

1

0

2t

1 + t2 1

1 + 2t

1 + t2

21 + t2

dt=

1

0

4t

(1 + t2)(t + 1)2dt;

4t

(1 + t2)(t + 1)2 =

A

t2 + 1+

B

(t + 1)2

A= 2 B=2

2

10

1

t2 + 1dt 2

10

1

(t + 1)2dt= 2

arctg t

10

+ 2

1

t + 1

10

=

2 1;

1

x2 + x =

1

x 1

x + 1

21

1

x2 + x dx=

ln x21 ln(x + 1)

2

1 = ln

4

3 ;

x

ex2 =xex

2

=12

ex

2

12

10

ex

2

dx= 1 e1

2 ;

x2 =t

21

x3 +x

dx


46/50

10

3x

4 x2 dx

21

x

x+ 1dx

1/30

x

1 4x2 dx

3 + x= t 21

x3 + x

dx=

54

t 3t

dt=

54

(t1/2 3t1/2) dt=

2

3t

t 6

t54

=4

3(5 2

5);

x= 2sin t dx= 2cos t dt

4 x2 = 2 cos t 10

3x

4 x2 dx= 24 /60

cos2 t sin t dt=24

cos3 t

3

/60

= 8 3

3;

x= t2

2

1

x

x + 1dx =

2

1

2t2

t2 + 1dt = 2

2

1 1 1

t2 + 1 dt= 2 [t arctg t]21 = 2(2 arctg 2 1 + /4);

(1 4x2) =8x 1/30

x

1 4x2 dx = 18

1/30

8x

1 4x2 dx

= 18

2

3(1 4x2)3/2

1/30

= 1

12

1 5

5

27

.

10

2x2 31 +x2

dx

/40

sin x+ cos xcos3 x

dx

94

x

1 x dx

2x2 31 + x2

= 2 51 + x2

10

2x2 31 + x2

dx=

10

2 5

1 + x2

dx= [2x 5 arctanx]10 = 2

5

4;

/40

sin x + cos xcos3 x dx = /40

cos3 x sin x dx + /40

cos2 x dx

=

1

2cos2 x + tan x

/40

=3

2;

x= t2 94

x

1 xdx = 2 32

t2

t2 1dt =2 32

1 +

1

t2 1

dt

= 2 32

1 +

1

2

1

t 11

2

1

t + 1

dt =

32

2 1

t 1+ 1

t + 1

dt

=

2t ln |t 1| + ln |t + 1|

3

2=2 + ln2

3.


47/50

f g

f(x) = sin x g(x) = 2 sin x x[0, /2] f(x) =

1

1 +x2 g(x) = 2 +x x[0, 1]

f(x) = ln x g(x) = x+ 1 x[1, e] f(x) = 2x g(x) = 3x x[0, 1]

A=

/20

2(1 sin x)dx= 2

x + cos x/20

= 2

A=

10

2 + x 1

1 + x2

dx=

2x +

x2

2 arctg x

10

=5

2

4

A =

e1

(1 + x ln x)dx =

x+x2

2

e1

e1

ln xdx

ln xdx= x ln x x A= 2x +x2

2 x ln x

e

1

=e2

2

+ e

5

2

A=

10

2x 3x

dx=

2x

ln 2 3

x

ln 1/3

10

= 1

ln 2 2

ln27

0 0.5 1 1.5

0

0.5

1

1.5

2a

x

y

fg

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4b

x

y

fg

1 1.5 2 2.50

1

2

3

4c

x

y

fg

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2d

x

y

fg

D={(x, y); x0, x2 y2x2, y2}

D

x

D={(x, y); 0y2,

y/2xy}

A(D) =

20

yy/2

dxdy =

20

y

y/2

dy=

1

2

2

20

ydy

=

1

2

2

2

3

y

y20

=4

3(

2 1).


48/50

+1

1

(x+ 1)2dx

+

1

1

x(x+ 1)dx

+1/5

e5xdx

0

e1xdx

+1

1

(x + 1)2dx = lim

r+

r1

1

(x + 1)2dx = lim

r

1

x + 1

r1

= 1

2

1

x + 1

+1

limr

1

x + 1

r1

+

1

1

xdx

+

1

1

x + 1dx= ln x

x + 1+

1

= ln2

t=5x 15

1

etdt=1

5

et1

= 1

5e

1 x= t +1

etdt= +

21

1x 1 dx

30

1

x 3 dx

2

0

1(x 2)2 dx

10

e1/x

x2 dx

21

13

x 1 dx

21

(x 1)1/2dx= 2

x 121

= 2

30

1

x 3 dx= ln |x 3|3

0 =

20

1

(x 2)2 dx= 1

x 220

= +

1

x =t

+1

etdt=

et+1

= +

21

13

x 1 dx= 3

2

(x 1)2/3

21

= 3

2


49/50

d

dx

x0

et2

dt

F(x) =

x0

et2

dt

f(x) = ex2

F(x) = f(x) = ex2

f 11f(x) dx= 0

f

10

f(x) dx=

21

f(x) dx=

32

f(x) dx= . . .= 0 ;

f 11f(x) dx= 0

f(x) =

1 |x| 1/21 1/2


50/50

f(x) = x

1 +x2 [0, 1]

f [a, b]

M(f) = 1

b a

ba

f(x)dx,

10

x

1 + x2dx=

1

2

2x

1 + x2dx=

1

2ln(1 + x2)1

0= ln

2;

M(f) = ln

2

[ITA] Esercizi di Analisi MAtematica 1 Ingegneria.pdf

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Transcript of [ITA] Esercizi di Analisi MAtematica 1 Ingegneria.pdf