y = log(1 + π2 ) + π 2π +4 asintoto orizzontale per x → ∞, y = log(1 + π2 ) − π+42πasintotoorizzontale perx → −∞

(1)

MATLT - MECMLT

IlNUMEROdellaFILAè ontenuto neltestodell'eser izio4edèla ostantesommataa

n ²

^nel^primo

denominatore della serie.

Fila 1

1.

domf = R

^,^non ⁱ^sono ^simmetrie.

lim ^x→±∞ f (x) = log(1 + ^π ₂ ) ± π ^2π +4

^,

y = log(1 + ^π ₂ ) + π ^2π +4

^asintoto orizzontale per

x → ∞

^,

y = log(1 + ^π ₂ ) − _π+4 ^2π

^asintotoorizzontale per

x → −∞

La funzione derivata prima è

f ^′ (x) = 1 1 + (x − 2) ²

4 π + 4 + 1

1 + arctan |x − 2|

|x − 2|

x − 2

domf ^′ = domf \ {2}

^,

x = 2

^punto ^angoloso.

f

^è ^res
ente ⁱⁿ

] − ∞, 1[∪]2, +∞[

^;

x = 1

^punto ^di ^massimo ^relativo;

x = 2

^punto ^di ^minimo

assoluto;

f

^è^limitata ^ma^non ^ammette^massimo^assoluto.

Dalla presenza dell'asintoto orizzontale per

x → −∞

^e ^del^punto ^di^massimo^relativo ⁱⁿ

x = 1

^è

evidente he ideve essere unpunto diesso in

] − ∞, 1[

^.

−20 −15 −10 −5 0 5 10 15 20

−0.5 0 0.5 1 1.5 2

PSfragrepla ements

x

f (x )

−0.5 0 0.5 1 1.5 2 2.5 3 3.5 4

−0.1

−0.05 0 0.05 0.1 0.15 0.2 0.25 0.3

PSfragrepla ements

x

f (x )

2. l'unione delle semirette

{y ≤ 0 , x = 0}

^,

{x ≤ − ³ ₄ , y = ³ ₄ }

^,

{x ≥ ³ ₄ , y = ³ ₄ }

^.

3.

ℓ = 1/7

4. La serie onverge per

0 ≤ α ≤ 7

5.

ℓ = 1/6

6.

lim

x →+∞ F (x) = π 4

7.

y(x) = ₃ ¹

₂

[(1 + x ² ) log(1 + x ² ) − x ² + 1] ²

(2)

1.

domf = R

lim ^x→±∞ f (x) = log(1 + ^π ₂ ) ± π ^2π +4

^,

y = log(1 + ^π ₂ ) + π ^2π +4

x → ∞

^,

y = log(1 + ^π ₂ ) − _π+4 ^2π

x → −∞

f ^′ (x) = 1 1 + (x − 3) ²

4 π + 4 + 1

1 + arctan |x − 3|

|x − 3|

x − 3

domf ^′ = domf \ {3}

^,

x = 3

^punto ^angoloso.

f

^è ^res
ente ⁱⁿ

] − ∞, 2[∪]3, +∞[

^;

x = 2

x = 3

^punto ^di ^minimo

assoluto;

f

x → −∞

x = 2

^è

] − ∞, 2[

^.

{y ≤ 0 , x = 0}

^,

{x ≤ − ⁵ ₆ , y = ⁵ ₆ }

^,

{x ≥ ⁵ ₆ , y = ⁵ ₆ }

^.

3.

ℓ = 1/6

0 ≤ α ≤ 6

5.

ℓ = 1/9

6.

lim

x →+∞ F (x) = π 6

7.

y(x) = ₅ ¹

₂

[(1 + x ² ) log(1 + x ² ) − x ² + 1] ²

Fila 3

1.

domf = R

lim x→±∞ f (x) = log(1 + ^π ₂ ) ± _π+4 ^2π

^,

y = log(1 + ^π ₂ ) + _π+4 ^2π

x → ∞

^,

y = log(1 + ^π ₂ ) − _π+4 ^2π

x → −∞

f ^′ (x) = 1 1 + (x − 4) ²

4 π + 4 + 1

1 + arctan |x − 4|

|x − 4|

x − 4

domf ^′ = domf \ {4}

^,

x = 4

^punto ^angoloso.

f

^è ^res
ente ⁱⁿ

] − ∞, 3[∪]4, +∞[

^;

x = 3

x = 4

^punto ^di ^minimo

assoluto;

f

x → −∞

x = 3

^è

] − ∞, 3[

^.

{y ≤ 0 , x = 0}

^,

{x ≤ − ⁷ ₈ , y = ⁷ ₈ }

^,

{x ≥ ⁷ ₈ , y = ⁷ ₈ }

^.

3.

ℓ = 1/5

(3)

0 ≤ α ≤ 5

5.

ℓ = 1/12

6.

lim

x →+∞ F (x) = π 8

7.

y(x) = ₇ ¹

₂

[(1 + x ² ) log(1 + x ² ) − x ² + 1] ²

Fila 4

1.

domf = R

lim ^x→±∞ f (x) = log(1 + ^π ₂ ) ± _π+4 ^2π

^,

y = log(1 + ^π ₂ ) + _π+4 ^2π

x → ∞

^,

y = log(1 + ^π ₂ ) − _π+4 ^2π

x → −∞

f ^′ (x) = 1 1 + (x − 5) ²

4 π + 4 + 1

1 + arctan |x − 5|

|x − 5|

x − 5

domf ^′ = domf \ {5}

^,

x = 5

^punto ^angoloso.

f

^è ^res
ente ⁱⁿ

] − ∞, 4[∪]5, +∞[

^;

x = 4

x = 5

^punto ^di ^minimo

assoluto;

f

x → −∞

x = 4

^è

] − ∞, 4[

^.

{y ≤ 0 , x = 0}

^,

{x ≤ − ₁₀ ⁹ , y = ₁₀ ⁹ }

^,

{x ≥ ₁₀ ⁹ , y = ₁₀ ⁹ }

^.

3.

ℓ = 1/4

0 ≤ α ≤ 4

5.

ℓ = 1/15

6.

lim

x →+∞ F (x) = π 10

7.

y(x) = ₉ ¹

₂

[(1 + x ² ) log(1 + x ² ) − x ² + 1] ²

Fila 5

1.

domf = R

lim x→±∞ f (x) = log(1 + ^π ₂ ) ± _π+4 ^2π

^,

y = log(1 + ^π ₂ ) + _π+4 ^2π

x → ∞

^,

y = log(1 + ^π ₂ ) − _π+4 ^2π

x → −∞

f ^′ (x) = 1 1 + (x − 6) ²

4 π + 4 + 1

1 + arctan |x − 6|

|x − 6|

x − 6

domf ^′ = domf \ {6}

^,

x = 6

^punto ^angoloso.

f

^è ^res
ente ⁱⁿ

] − ∞, 5[∪]6, +∞[

^;

x = 5

x = 6

^punto ^di ^minimo

assoluto;

f

x → −∞

x = 5

^è

] − ∞, 5[

^.

(4)

{y ≤ 0 , x = 0}

^,

{x ≤ − ¹¹ ₁₂ , y = ¹¹ ₁₂ }

^,

{x ≥ ¹¹ ₁₂ , y = ¹¹ ₁₂ }

^.

3.

ℓ = 1/3

0 ≤ α ≤ 3

5.

ℓ = 1/18

6.

lim

x →+∞ F (x) = π 12

7.

y(x) = ₁₁ ¹

₂

[(1 + x ² ) log(1 + x ² ) − x ² + 1] ²

Fila 6

1.

domf = R

lim x→±∞ f (x) = log(1 + ^π ₂ ) ± _π+4 ^2π

^,

y = log(1 + ^π ₂ ) + _π+4 ^2π

x → ∞

^,

y = log(1 + ^π ₂ ) − _π+4 ^2π

x → −∞

f ^′ (x) = 1 1 + (x − 7) ²

4 π + 4 + 1

1 + arctan |x − 7|

|x − 7|

x − 7

domf ^′ = domf \ {7}

^,

x = 7

^punto ^angoloso.

f

^è ^res
ente ⁱⁿ

] − ∞, 6[∪]7, +∞[

^;

x = 6

x = 7

^punto ^di ^minimo

assoluto;

f

x → −∞

x = 6

^è

] − ∞, 6[

^.

{y ≤ 0 , x = 0}

^,

{x ≤ − ¹³ ₁₄ , y = ¹³ ₁₄ }

^,

{x ≥ ¹³ ₁₄ , y = ¹³ ₁₄ }

^.

3.

ℓ = 1/2

0 ≤ α ≤ 2

5.

ℓ = 1/21

6.

lim

x →+∞ F (x) = π 14

7.

y(x) = ₁₃ ¹

₂

[(1 + x ² ) log(1 + x ² ) − x ² + 1] ²

y = log(1 + π2 ) + π 2π +4 asintoto orizzontale per x → ∞, y = log(1 + π2 ) − π+42πasintotoorizzontale perx → −∞

n 2

domf = R

lim x→±∞ f (x) = log(1 + π 2 ) ± π 2π +4

y = log(1 + π 2 ) + π 2π +4

x → ∞

y = log(1 + π 2 ) − π+4 2π

x → −∞

f ′ (x) = 1 1 + (x − 2) 2

 4

π + 4 + 1

1 + arctan |x − 2|

|x − 2|

x − 2



domf ′ = domf \ {2}

x = 2

f

] − ∞, 1[∪]2, +∞[

x = 1

x = 2

f

x → −∞

x = 1

] − ∞, 1[

x

f (x )

x

f (x )

{y ≤ 0 , x = 0}

{x ≤ − 3 4 , y = 3 4 }

{x ≥ 3 4 , y = 3 4 }

ℓ = 1/7

0 ≤ α ≤ 7

ℓ = 1/6

lim

x →+∞ F (x) = π 4

y(x) = 3 1

[(1 + x 2 ) log(1 + x 2 ) − x 2 + 1] 2

domf = R

lim x→±∞ f (x) = log(1 + π 2 ) ± π 2π +4

y = log(1 + π 2 ) + π 2π +4

x → ∞

y = log(1 + π 2 ) − π+4 2π

x → −∞

f ′ (x) = 1 1 + (x − 3) 2

 4

π + 4 + 1

1 + arctan |x − 3|

|x − 3|

x − 3



domf ′ = domf \ {3}

x = 3

f

] − ∞, 2[∪]3, +∞[

x = 2

x = 3

f

x → −∞

x = 2

] − ∞, 2[

{y ≤ 0 , x = 0}

{x ≤ − 5 6 , y = 5 6 }

{x ≥ 5 6 , y = 5 6 }

ℓ = 1/6

0 ≤ α ≤ 6

ℓ = 1/9

lim

x →+∞ F (x) = π 6

y(x) = 5 1

[(1 + x 2 ) log(1 + x 2 ) − x 2 + 1] 2

domf = R

lim x→±∞ f (x) = log(1 + π 2 ) ± π+4 2π

y = log(1 + π 2 ) + π+4 2π

x → ∞

y = log(1 + π 2 ) − π+4 2π

x → −∞

f ′ (x) = 1 1 + (x − 4) 2

 4

n ²

lim ^x→±∞ f (x) = log(1 + ^π ₂ ) ± π ^2π +4

y = log(1 + ^π ₂ ) + π ^2π +4

y = log(1 + ^π ₂ ) − _π+4 ^2π

f ^′ (x) = 1 1 + (x − 2) ²

4

domf ^′ = domf \ {2}

{x ≤ − ³ ₄ , y = ³ ₄ }

{x ≥ ³ ₄ , y = ³ ₄ }

y(x) = ₃ ¹

[(1 + x ² ) log(1 + x ² ) − x ² + 1] ²

lim ^x→±∞ f (x) = log(1 + ^π ₂ ) ± π ^2π +4

y = log(1 + ^π ₂ ) + π ^2π +4

y = log(1 + ^π ₂ ) − _π+4 ^2π

f ^′ (x) = 1 1 + (x − 3) ²

4

domf ^′ = domf \ {3}

{x ≤ − ⁵ ₆ , y = ⁵ ₆ }

{x ≥ ⁵ ₆ , y = ⁵ ₆ }

y(x) = ₅ ¹

[(1 + x ² ) log(1 + x ² ) − x ² + 1] ²

lim x→±∞ f (x) = log(1 + ^π ₂ ) ± _π+4 ^2π

y = log(1 + ^π ₂ ) + _π+4 ^2π

y = log(1 + ^π ₂ ) − _π+4 ^2π

f ^′ (x) = 1 1 + (x − 4) ²

4

domf ^′ = domf \ {4}

{x ≤ − ⁷ ₈ , y = ⁷ ₈ }

{x ≥ ⁷ ₈ , y = ⁷ ₈ }

y(x) = ₇ ¹

[(1 + x ² ) log(1 + x ² ) − x ² + 1] ²

lim ^x→±∞ f (x) = log(1 + ^π ₂ ) ± _π+4 ^2π

y = log(1 + ^π ₂ ) + _π+4 ^2π

y = log(1 + ^π ₂ ) − _π+4 ^2π

f ^′ (x) = 1 1 + (x − 5) ²

4

domf ^′ = domf \ {5}

{x ≤ − ₁₀ ⁹ , y = ₁₀ ⁹ }

{x ≥ ₁₀ ⁹ , y = ₁₀ ⁹ }

y(x) = ₉ ¹

[(1 + x ² ) log(1 + x ² ) − x ² + 1] ²

lim x→±∞ f (x) = log(1 + ^π ₂ ) ± _π+4 ^2π

y = log(1 + ^π ₂ ) + _π+4 ^2π

y = log(1 + ^π ₂ ) − _π+4 ^2π

f ^′ (x) = 1 1 + (x − 6) ²

4