Risposte Test Allenamento - dm.unipi.it · 92 Esercizi di Analisi Matematica II (Parte A) –...

Capitolo 4: Schede di Esercizi – Risposte 91

Risposte Test Allenamento

Inf-Sup-Max-Min 1

Max P.ti Max Min P.ti Min

1 1 !9 1

9 1 1 1

8 4 0 1

5 1 0 1

13 2 0 1

4 2 !4 2

4 1 !2 2

4 2 !4 2

5 2 !1 1

2 2 !1/4 2

3 2 0 1

6 2 !2 2

22 1 !3 1

4 2 !2 2

2 2 !1/4 2

4"

6/9 1 !4"

6/9 1

Inf-Sup-Max-Min 2


2 1 0 1

3 1 0 1

4 2 0 1

1 1 !1 1

4 3 0 1

1 2 1/2 1

1/4 1 0 +#"

2 1 !"

2 1

2 2 0 #

8 1 !8 1

"2 1 !

"2 1

"10 1 !2

"2 1

1 1 !1/4 1

0 3 !1 1

1/4 1 !3 1

1/4 2 !2 1


92 Esercizi di Analisi Matematica II (Parte A) – Versione 2009/2010


Inf-Sup-Max-Min 3


"2 2 !

"2 2

"2 2 !

"2 2

2 4 0 4

"2/ 4

"27 2 !

"2/ 4

"27 2

"5 4 0 1

"5 4 1 2

1 2 !2 2

3/8 2 !12 1

4"

6/9 2 !4"

6/9 2

3 1 !3 1

4 2 0 1

4 2 7/8 4

6 2 0 1

6 2 2 2

3 +"

3 2 0 1

175/27 2 !3 2

Inf-Sup-Max-Min 4


8 4 2!

3 2

16"

5/125 2 !16"

5/125 2

27 1 !27 1

1 2 0 #"

15/30 # !"

15/30 #

4/33 1 0 #

1/e 2 0 #

e2 4 0 #

3/8 1 0 1

4/27 1 !4 1

27/16 2 0 1

9 1 !9 1

9 1 9/5 1

!/4 4 0 #

!/4 4 0 4

2 cos(2"1/4) 4 1 + cos 1 4




Inf-Sup-Max-Min 5


2 2 0 #

3 1 0 #

4 1 0 1

5 1 1 1

5 1 1 1

3 1 0 #

7 1 3 1

1 2 0 #

6 1 2 1

2 1 0 #

2 1 0 1

3 1 0 #

e 1 0 #

17/4 2 0 #

2"

2 2 !2 1

2"

2 2 0 #

Inf-Sup-Max-Min 6

Sup P.ti Max Inf P.ti Min A.Staz.

+# 0 0 1 0

+# 0 !# 0 1

+# 0 !# 0 1

+# 0 0 1 0

+# 0 !# 0 1

+# 0 0 # 0

+# 0 !# 0 1

+# 0 !# 0 3

+# 0 0 1 0

+# 0 !# 0 3

+# 0 !94/8 2 1

1 0 0 # 0

1/2 1 !1/2 1 0

+# 0 0 1 0

+# 0 2 ! ! 2 1

+# 0 !!/2 0 1




Limiti 1

R2 D1 D2 D3 D4 D5

N.E. +# +# N.E. +# +#

N.E. N.E. N.E. N.E. +# N.E.

+# +# +# +# +# +#

N.E. N.E. +# N.E. N.E. +#

N.E. +# +# N.E. +# +#

+# +# +# +# +# +#

N.E. +# +# N.E. +# +#

N.E. N.E. +# N.E. N.E. +#

N.E. N.E. N.E. N.E. +# 0

N.E. N.E. +# N.E. N.E. +#

N.E. N.E. N.E. N.E. +# 1

0 0 0 0 0 0

N.E. N.E. 0 N.E. N.E. 0

N.E. N.E. 0 N.E. +# 0

0 0 0 0 0 0

N.E. N.E. +# N.E. N.E. +#

Limiti 2

R2 D1 D2 D3 D4 D5

N.E. N.E. +# N.E. +# +#

N.E. N.E. N.E. N.E. +# 0

+# +# +# +# +# +#

+# +# +# +# +# +#

N.E. +# N.E. N.E. N.E. +#

N.E. N.E. +# N.E. N.E. +#

+# +# +# +# +# +#

N.E. +# +# N.E. +# +#

N.E. N.E. +# N.E. N.E. +#

N.E. N.E. +# N.E. +# 0

N.E. +# N.E. N.E. N.E. +#

0 0 0 0 0 0

N.E. N.E. N.E. N.E. N.E. 0

N.E. N.E. N.E. N.E. N.E. 0

N.E. +# +# N.E. +# +#

N.E. +# +# N.E. N.E. +#




Inf-Sup-Max-Min 7

Sup P.ti Max Inf P.ti Min

+# 0 9/2 1

+# 0 !# 0

+# 0 !1/4 1

+# 0 !# 0

0 1 !# 0

+# 0 !# 0

+# 0 1 2

+# 0 !# 0

+# 0 8"

2 2

1/2 1 !# 0

0 2 !# 0

+# 0 !3 1

+# 0 !2 1

2 + !/2 0 !!/2 0

"5/2 1 !

"5/2 1

"5/2 1 0 #

Inf-Sup-Max-Min 8

Sup P.ti Max Inf P.ti Min

6 1 0 1

5 1 !9 1

6 1 0 #

1/2 2 !1/2 2

1 1 !1 1

3/2 2 !3/2 2

19/6 2 !"

2 2

3 +"

2 2 !"

2 1

49/12 2 !2 1

22/3 · 33/4 4 !22/3 · 33/4 4

1/432 8 0 #

+# 0 !# 0

2 1 !1/4 #

+# 0 3 1

2 2 !# 0

"2/2 1 !

"2/2 1




Punti stazionari 1

m.L. A.S.

A.S. M.L.

m.L. m.L.

A.S. A.S.

A.S. A.S.

m.L. N.S.

m.L. m.L.

A.S. N.S.

A.S. N.S.

A.S. A.S.

A.S. A.S.

A.S. M.L.

N.S. A.S.

M.L. m.L.

m.L. A.S.

A.S. A.S.

Punti stazionari 2

Min glob. Min loc. Max glob. Max loc. Altri staz.

1 1 1 1 0

2 2 0 0 1

0 0 0 0 1

0 0 0 1 2

0 0 0 0 2

1 1 0 0 0

0 1 0 0 4

0 0 0 0 1

2 2 2 2 1

2 2 2 2 1

# # 0 0 0

0 1 0 0 2

0 0 0 0 1

0 0 0 0 1

0 0 0 0 1

0 1 4 4 4




Integrali Doppi 1

1 9

1 0

9/2 19/2

3/2 35/6

1 9

2 log 2 ! 1 4

0 !3/2

0 !1

(e ! 1)(e4 ! 1)/2 (e ! 1)(e6 ! 1)/(2e)

2(e ! 1) + (e4 ! 1)/2 2(e2 ! 1)

log"

2 (3/2) log 2

1/4 !3/4

!/8 3!/2

!/12 !/24

8/3 + !/2 ! log 2 8/3 ! !/2 + log 2

0 0

Integrali Doppi 2

e ! 2 0

(e2 ! 1)/4 (e2 ! 1)/4

2 2/!

1 0

(e ! 1)2/2 0

0 9(1 ! 2 3"

2)/28

0 !(e3! + 1)/5

!/4 ! (1/2) log 2 !/2 ! log 2

a a

1/4 + log 6!

3/8 1/4 + log 6!

3/8

!/16 ! (1/8) log 2 0

0 4!

8!2 4!2

log(27/16) log(16/27)

1/2 + log(3"

3/4) 1/2 ! log(3"

3/4)

b b

a =3

2log2 3 ! 2 log2 2 + log

16

27, b = 14 log 2 !

9

2log 3 ! 3




Insiemi Normali 1

A = {(x, y) $ R2 : 0 % x % 1, 0 % y % x}

= {(x, y) $ R2 : 0 % y % 1, y % x % 1}

B = {(x, y) $ R2 : 0 % x % 1, x % y % 1}

= {(x, y) $ R2 : 0 % y % 1, 0 % x % y}

C = {(x, y) $ R2 : 0 % x % 1, 0 % y % 1 ! x}

= {(x, y) $ R2 : 0 % y % 1, 0 % x % 1 ! y}

D = {(x, y) $ R2 : 0 % x % 1, 1 ! x % y % 1}

= {(x, y) $ R2 : 0 % y % 1, 1 ! y % x % 1}

E = {(x, y) $ R2 : !1 % x % 1, |x| % y % 1}

= {(x, y) $ R2 : 0 % y % 1, !y % x % y}

F = {(x, y) $ R2 : 0 % x % 1, 0 % y % x} & [1, 2] ' [0, 1] &

& {(x, y) $ R2 : 2 % x % 3, 0 % y % 3 ! x}

= {(x, y) $ R2 : 0 % y % 1, y % x % 3 ! y}

G = {(x, y) $ R2 : 0 % x % 2, 0 % y % x/3} & {(x, y) $ R2 : 2 % x % 3, x ! 2 % y % x/3}

= {(x, y) $ R2 : 0 % y % 1, 3y % x % 2 + y}

H = {(x, y) $ R2 : 0 % x % 1, 1 ! x % y % 1} & {(x, y) $ R2 : 1 % x % 3, (x ! 1)/2 % y % 1}

= {(x, y) $ R2 : 0 % y % 1, 1 ! y % x % 2y + 1}




Insiemi Normali 2

A = {(x, y) $ R2 : !1 % x % 1, x2 % y % 1}

= {(x, y) $ R2 : 0 % y % 1, !"y % x % "

y}

B = {(x, y) $ R2 : !1 % x % 1, 0 % y %"

1 ! x2}

= {(x, y) $ R2 : 0 % y % 1, !!

1 ! y2 % x %!

1 ! y2}

C = {(x, y) $ R2 : !2 % x % 0, !"

4 ! x2 % y %"

4 ! x2}

= {(x, y) $ R2 : !2 % y % 2, !!

4 ! y2 % x % 0}

D = {(x, y) $ R2 : 0 % x % 1, x2 % y % x}

= {(x, y) $ R2 : 0 % y % 1, y % x % "y}

E = {(x, y) $ R2 : 0 % x % 1, x2 % y %"

x}

= {(x, y) $ R2 : 0 % y % 1, y2 % x % "y}

F = {(x, y) $ R2 : !2 % x % !"

3, 0 % y %"

4 ! x2} & [!"

3,"

3] ' [0, 1] &

& {(x, y) $ R2 :"

3 % x % 2, 0 % y %"

4 ! x2}

= {(x, y) $ R2 : 0 % y % 1, !!

4 ! y2 % x %!

4 ! y2}

G = [0, 1] ' [0, 1] & {(x, y) $ R2 : 1 % x %"

2, 0 % y % 2 ! x2}

= {(x, y) $ R2 : 0 % y % 1, 0 % x %"

2 ! y}

H = {(x, y) $ R2 : !1 % x % 0, !(1/2)"

1 ! x4 % y % 0}

= {(x, y) $ R2 : !1/2 % y % 0, ! 4!

1 ! 4y2 % x % 0}




Integrali Doppi 3

32/3 32/5

4 8/5

16/3 32/3

3/40 1/15

0 0

(2/3)!3 ! 4! 0

!8/15 0

1024/105 0

!1/6 0

5/28 1/5

1/2 1

1/3 1/2

(16"

2 ! 19)/15 7/12

125/12 125/6

(e ! 1)/6 1/15

!(1/4) sin(!2) 1

Integrali Doppi 4

3/2 3

2 5/2

1/3 4/3

4/3 4/3

11/30 11/5

11/30 23/15

3/2 3

2 5/2

5!/2 ! 4 !/8 ! 1 + 3"

3/4

!1 1 !"

2/4

10/9 5/12

!1 !2

1/3 7/3

1 11/60

3/2 188/15

4 ! ! 16 ! 4!




Integrali Doppi 5

Insieme x y |x| |y| x2 + y2 |x ! y| |x + y|

0 0 4/3 4/3 !/2 (4/3)"

2 (4/3)"

2

0 2/3 2/3 2/3 !/4 (2/3)"

2 (2/3)"

2

0 !2/3 2/3 2/3 !/4 (2/3)"

2 (2/3)"

2

2/3 0 2/3 2/3 !/4 (2/3)"

2 (2/3)"

2

!2/3 0 2/3 2/3 !/4 (2/3)"

2 (2/3)"

2

1/3 1/3 1/3 1/3 !/8 2("

2 ! 1)/3 2/3

!1/3 1/3 1/3 1/3 !/8 2/3 2("

2 ! 1)/3

!1/3 !1/3 1/3 1/3 !/8 2("

2 ! 1)/3 2/3

1/3 !1/3 1/3 1/3 !/8 2/3 2("

2 ! 1)/3

!"

2/3"

2/3 2/3 2/3 !/4 (2/3)"

2 (2/3)"

2

"2/3 !

"2/3 2/3 2/3 !/4 (2/3)

"2 (2/3)

"2

"2/3

"2/3 2/3 2/3 !/4 (2/3)

"2 (2/3)

"2

"2/6

1

3!

"2

6

"2/6

1

3!

"2

6!/16 !1/3 +

"2/3 1/3

"2/6

1

3+

"2

6

2

3!

"2

6

1

3+

"2

63!/16 !1/3 + 2

"2/3 1/3 +

"2/3

!"

2/61

3!

"2

6

"2/6

1

3!

"2

6!/16 1/3 !1/3 +

"2/3

"2/6 !

1

3!

"2

6

2

3+

"2

61 !

"2

65!/16 !1/3 +

"2 !1/3 +

"2




Integrali Doppi 6

4"

2/3 !4"

2/3

! 2!"

2/3

0 0

28"

2/3 28"

2/3

38!/3 76/3

e4(e5 ! 1)!/8 19"

2/6

2! 2!

0 2"

3

128/15 128/15

32/3 32/3

"3 + !/6 1

8"

3/3 + 4!/3 8"

3/3 + 4!/3

5!/4 0

2"

3 ! 2!/3 11/12

32/9 3!/2

(1/6) log(3 + 2"

2) +"

2/3 2/3

Integrali Doppi 7

4/15 4

0 1024/9

2! 3!

48! 40!

2! !

!13!/8 !65!/16

2! !/2

16/3 8/3

!

3/2 9/16

!6"

2 4"

2

9 · 54 · 2"13! 125!/256

!36 124

45/16 295/96

0 8

"3 ! 1/3 6

3/2 a

a =16

3

"2 !

35

24

"5 !

17

24

"17




Attenzione: le risposte di “Integrali Tripli 3” sono nella pagina successiva.

Integrali Tripli 1

3 6

9 8

5/2 15/8

5/2 25/2

1 (e + 1)(e ! 1)3

2!2 0

!3/2 !8

2 0

1/6 2/9

12! 36!

0 6!

0 64/3

0 64/3

16!/3 4!2

!/12 !/12

!/16 0

Integrali Tripli 2

8!/3 4!

16!/3 8/3

2! 32!"

2/21

32!"

2/5 0

0 203!/60

!32!/3 71!/6

!/2 !

0 243!/5

!2 0

!2/2 !3/32

!/3 8/9

!2/4 8/9

!32!/3 256/15

8! !8!

1/6 1/120

1/2 7/12

Integrali Tripli 4

!32! 32!/3

!32!/3 1144!/15

280!"

10/3 40!"

10/3

85!"

34/96 17!"

34/24

2!"

30/45 !"

6/60

4!"

3/81 0

2! !2

2!/3 0

16 · 35!/5 144!

!4! 4!"

2

10(! ! 1)/3 8(! ! 1)

22!"

3/5 0

512/3 ! 2! 4(16 ! !)

8!/3 + 40/3 ! 12"

3 0

16/3 64/45

16 ! 8"

2 0




Attenzione: le risposte di “Integrali Tripli 4” sono nella pagina precedente.

Integrali Tripli 3

Funzione S1 S2 S3 S4 S5 S6

1 4!/3 2!/3 !/3 !/6 28!/3 2!/3

x 0 0 !/8 !!/16 0 !"

2/8

y 0 0 !!/8 !/16 0 !"

2/8

z 0 !/4 0 !!/16 0 0

|x| !/2 !/4 !/8 !/16 15!/2 !/4

|y| !/2 !/4 !/8 !/16 15!/2 !/4

|z| !/2 !/4 !/8 !/16 15!/2 !/4

x2 + y2 + z2 4!/5 2!/5 !/5 !/10 124!/5 2!/5

x2 4!/15 2!/15 !/15 !/30 124!/15 2!/15

y2 4!/15 2!/15 !/15 !/30 124!/15 2!/15

z2 4!/15 2!/15 !/15 !/30 124!/15 2!/15

x2 + y2 8!/15 4!/15 2!/15 !/15 248!/15 4!/15

|y|3 !/3 !/6 !/12 !/24 21! !/6

x4 4!/35 2!/35 !/35 !/70 508!/35 2!/35

|x + y| !"

2/2 !"

2/4 ("

2 ! 1)!/4 ("

2 ! 1)!/8 15!"

2/2 !"

2/4

|z ! y| !"

2/2 !"

2/4 !"

2/8 !/8 15!"

2/2 " "




Integrali Impropri 1

No No

No No

No S̀ı

No S̀ı

No No

S̀ı S̀ı

No No

S̀ı S̀ı

No No

S̀ı No

S̀ı S̀ı

No No

No S̀ı

No No

S̀ı S̀ı

S̀ı No


No S̀ı

S̀ı No

S̀ı No

No S̀ı

No No

S̀ı S̀ı

No S̀ı

S̀ı S̀ı

S̀ı No

S̀ı No

No No

S̀ı No

No No

No S̀ı

No No

No No





No No

No No

No S̀ı

No No

No No

S̀ı No

No S̀ı

S̀ı No

No No

No No

No S̀ı

No No

No S̀ı

No No

No No

No No


No No

S̀ı No

No No

No S̀ı

No No

S̀ı No

S̀ı No

No S̀ı

No S̀ı

S̀ı S̀ı

S̀ı No

S̀ı S̀ı

No No

No No

No S̀ı

No No




Domini due dimensionali

Area Perimetro xG yG

2! 2! + 8 0 0

2! 2! + 4 !4"

2/(3!) !4"

2/(3!)

3!/2 + 9"

3/4 ! + 6 + 3"

3 (24 ! 9"

3)/(2! + 3"

3) 0

50 20"

2 0 0

81 18("

2 + 1) 0 6

1/2 2"

2 3 !1/2

32/3 C 2 8/5

9/2 C 3/2 12/5

"2 + 1 "" [!(4 +

"2) + 4]/[4(

"2 + 1)] 1/[4(

"2 + 1)]

!/3 +"

3/2 2! (8! + 3"

3)/(4! + 6"

3) 0

7/6 C 15/14 62/35

1/2 + log 2 "" 5/[6(1 + log 4)] 8/[3(1 + log 4)]

(e ! 2)/2 a [2(3 ! e)]/[3(e ! 2)] (e2 ! 3)/[6(e ! 2)]

5 log 2 "" 19/(15 log 2) 38/(15 log 2)

2! 2! + 4 8/(3"

e2 + !2) 8e/(3!"

e2 + !2)

! "" 32/(15!) 4/(5!)

a = 2"

e2 + 1 !"

2 + log"#"

2 + 1$#"

e2 + 1 ! 1$%




Solidi di Rotazione 1

Volume Baricentro Superficie

2! (0, 0, 1) 6!

2!/3 (0, 0, 3/8) (3 +"

2)!

16!/3 (3/4, 0, 0) 4(2 +"

2)!

!/3 (3/4, 0, 0) (1 +"

2)!

!/2 (0, 0, 1/3) (17 + 5"

5)!/6

32!/5 (0, 5/3, 0) C

2! (0, 0, (log 2)/2)

&

6 +"

17 !"

2 + log4(1 +

"2)

1 +"

17

'

!

4!/3 (0, 0, 0) 4!

16!/3 (0, 0, 3/4) 12!

14!/3 (0, 45/56, 0) 13!

! (0, 0,!1/6) 2(1 +"

2)!

3!/2 (2 log 2, 0, 0) C

23!/3 (95/92, 0, 0) (11 +"

2)!

88!"

3/5 (0, 0, 0) C

9!/2 (0, 0,!7/9) 27!"

2/2

(2!4 ! 3!2)/12 3!(!2 ! 3)/(4!2 ! 6) ""




Solidi di Rotazione 2

Solido Volume zG Iz Superficie

S1 32!/3 0 256!/15 16!

S2 4!"

2/3 3"

2/8 16!"

2/15 6!

C1 2! 1 ! 6!

C2 8!/3 3/2 16!/5 4!(1 +"

2)

P 2! 4/3 4!/3 19!/3

S1 ( C1 2!(8/3 !"

3) 21/[8(8 ! 3"

3)] (128/15 ! 22"

3/5)! (9 ! 2"

3)!

S2 ( C2 4!("

2 ! 1)/3 3/[8("

2 ! 1)] (16"

2/15 ! 4/13)! (4 !"

2)!

S2 ( P (8"

2 ! 7)!/6 7/[2(8"

2 ! 7)] (16"

2 ! 19)!/15 (5"

5 ! 12"

2 + 23)!/6

S1 ( C2 8!(2 !"

2)/3 3/[4(2 !"

2)] 16(8 ! 5"

2)!/15 2!(4 !"

2)

C1 ( C2 4!/3 21/16 3!/5 (3 +"

2)!

C1 ( P 3!/2 11/9 2!/3 (17 + 5"

5)!/6

C2 ( P 11!/6 31!/22 19!/15 (13/2 +"

2 ! 5"

5/6)!

S2 & C2 4! 9/8 68!/15 (10 + 3"

2)!

C1 & C2 10!/3 51/40 18!/5 (7 + 3"

2)!

S2 \ C2 4!/3 3/8 4!/3 (3"

2 + 2)!

C2 \ C1 4!/3 27/16 13!/5 (5 + 3"

2)!


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