Тема 2. Символьные вычисления с использованием встроен-

Вариант

Задание

1

a² − b²

a + b

2

a² − b²

a – b

3

a³ + 3 ⋅ b² ⋅ a + 3 ⋅ b ⋅ a² + b³

a + b

4

a³ + 3 ⋅ b² ⋅ a − 3 ⋅ b ⋅ a² − b³

a – b

5

a⁴ − 4 ⋅ b³ ⋅ a + 6 ⋅ b² ⋅ a² − 4 ⋅ b ⋅ a³ + b⁴

a³ + 3 ⋅ b² ⋅ a − 3 ⋅ b ⋅ a² − b³

6

a⁴ + 4 ⋅ b³ ⋅ a + 6 ⋅ b² ⋅ a² + 4 ⋅ b ⋅ a³ + b⁴

a³ + 3 ⋅ b² ⋅ a + 3 ⋅ b ⋅ a² + b³

7

^cos ( ^x )2 ^{+ sin} ( ^x )2

8

_x² _{− 3 ⋅ x − 4}

_{+ 2 ⋅ x − 5}

^{x – 4}

9

_e^2⋅ln( ^a )

10

( ^{a + b}) ^⋅ ( ^{a + b}) ^⋅ ( ^{a + b} )

Вариант

Задание

1

( ^{x + 1}) ^⋅ ( ^{x −1}) ^⋅ ( ^{x + 2}) ^⋅ ( ^{x − 2})

2

( ^{x + 3}) ^⋅ ( ^{x − 3}) ^⋅ ( ^{x + 2}) ^⋅ ( ^{x − 2})

3

( ^{x + 1}) ^⋅ ( ^{x −1}) ^⋅ ( ^{x + 4}) ^⋅ ( ^{x − 4})

4

( ^{x + 1}) ^⋅ ( ^{x −1}) ^⋅ ( ^{x + 3}) ^⋅ ( ^{x − 3})

6

( ^{x + 1}) ^⋅ ( ^{x −1}) ^⋅ ( ^{x + 5}) ^⋅ ( ^{x − 5})

7

( ^{x + 5}) ^⋅ ( ^{x − 5}) ^⋅ ( ^{x + 2}) ^⋅ ( ^{x − 2})

8

( ^{x + 5}) ^⋅ ( ^{x − 5}) ^⋅ ( ^{x + 3}) ^⋅ ( ^{x − 3})

9

( ^{x + 5}) ^⋅ ( ^{x − 5}) ^⋅ ( ^{x + 4}) ^⋅ ( ^{x − 4})

10

( ^{x + 1}) ^⋅ ( ^{x − 2}) ^⋅ ( ^{x + 3}) ^⋅ ( ^{x − 4})

Вариант

Задание

1

x⁴ − 13 ⋅ x² + 36

2

x⁴ + 3 ⋅ x³ − 15 ⋅ x ² −19 ⋅ x + 30

3

x⁴ − x³ − 7 ⋅ x ² + x + 6

4

x⁴ − 10 ⋅ x ² +9

5

x⁴ − 17 ⋅ x ² +16

6

x⁴ − 26 ⋅ x ² +25

7

x⁴ − 29 ⋅ x ² +100

8

x⁴ − 40 ⋅ x ² +144

9

x⁴ − 45 ⋅ x ² +324

10

x⁴ − x³ − 11⋅ x ² +9 ⋅ x + 18

11Вариант

Задание

1

x² − a ⋅ y² ⋅ x² + 2 ⋅ y² ⋅ x − x + x ⋅ y

2

^{x2 − a ⋅ y2 −} (^{3 + x} ) ^{⋅ x2 + 2 ⋅ y2 ⋅ x − x + x ⋅ y}

3

^{x2 − a ⋅ y2 −} (^{3 + x} ) ^{⋅ x2 + 2 ⋅ y2 ⋅ x}

4

x² − a ⋅ y ² ⋅ x² + 2 ⋅ y ² ⋅ x − x + x ⋅ ₍ y − x + x² ₎

5

x² − ₍a + x − x² ₎ ⋅ y ² − ₍3 + x ₎ ⋅ x² + 2 ⋅ y ² ⋅ x

7

x² − ^{( a − y )} − a ⋅ y² ⋅ x² + 2 ⋅ y² ⋅ x − x + x ⋅ y x

8

^{x2 − a ⋅ y2 ⋅ x2 + 2 ⋅ y2 ⋅ x − x ⋅} ( ^{a + y} ) ^{+ x ⋅ y}

9

x² − a ⋅ y ² ⋅ x² + ₍2 ⋅ y ² + 1₎ ⋅ x − x + x ⋅ ₍ y + 1₎

10

x² − ₍a ⋅ y ² + 2₎ ⋅ x² + ₍2 ⋅ y ² − 3₎ ⋅ x − x + x ⋅ y

Вариант

Задание

1

( ^{x + a} ) ^⋅ ( ^{x + b} ) ^⋅ ( ^{x + c} ) ^⋅ ( ^{x + d} )

2

( ^{x + a} ) ^⋅ ( ^{x − b} ) ^⋅ ( ^{x + c} ) ^⋅ ( ^{x + d} )

3

( ^{x + a} ) ^⋅ ( ^{x + b} ) ^⋅ ( ^{x − c} ) ^⋅ ( ^{x + d} )

4

( ^{x + a} ) ^⋅ ( ^{x + b} ) ^⋅ ( ^{x + c} ) ^⋅ ( ^{x − d} )

5

( ^{x − a} ) ^⋅ ( ^{x + b} ) ^⋅ ( ^{x + c} ) ^⋅ ( ^{x + d} )

6

( ^{x − a} ) ^⋅ ( ^{x − b} ) ^⋅ ( ^{x + c} ) ^⋅ ( ^{x + d} )

7

( ^{x + a} ) ^⋅ ( ^{x − b} ) ^⋅ ( ^{x − c} ) ^⋅ ( ^{x + d} )

8

( ^{x + a} ) ^⋅ ( ^{x + b} ) ^⋅ ( ^{x − c} ) ^⋅ ( ^{x − d} )

9

( ^{x − a} ) ^⋅ ( ^{x − b} ) ^⋅ ( ^{x − c} ) ^⋅ ( ^{x + d} )

10

( ^{x − a} ) ^⋅ ( ^{x − b} ) ^⋅ ( ^{x − c} ) ^⋅ ( ^{x − d} )

Вариант

Задание

1

^d _{( x}² _{+ y} ² ₎

^dx

2

^{d 2} _{( )}

_{a ⋅ x}² _{+ b ⋅ x + c}

^dx2

4

_d ² _{⎛ x}² _⎞

dx2 ⎜ _{ln ( x ) ⋅} sin ( x ) _{+ 2 ⋅ x + e} ⎟

^x

⎝ ⎠

5

_d ³ _{⎛ x}³ _⎞

_{⎜ x}² _{⋅ +2 ⋅ + x ⎟}

_dx^{3 ⎜} _{tg ( x )} ^⎟

^{⎝ ⎠}

6

^{d 2} _{( )}

_{a ⋅ x}² _{+ b ⋅ x + c}

^dx2

7

_d ⁴ _{⎛ 1 ⎞}

_{⎜ ⋅ x}² _{+ b ⋅ x + c ⎟}

_dx^{4 ⎜} _{ln ( x )} ^⎟

^{⎝ ⎠}

8

^d _(e^2⋅x _{+ e}^−2⋅x ₎

^dx

9

^{d 2} _{( )}

_e^2⋅x _{+ e}^−2⋅x

^dx2

10

d ³ _{( ( ) )}

_e^{sin x} _{+ e}^−2⋅x

^dx3

Вариант

Задание

1

_{∫ ∫∫} ^{( x}2 ^{+ y} 2 ^{+ z} 2 ^{) ⋅ dx ⋅ dy ⋅ dz}

2

_{∫ (} ² + 1₎ ⋅ dx

b

x

a

3

4

b

_∫ ^{π ⋅ x ⋅ dx}

A

4

_{sin ( x )}³

^∫ _{cos ( x )}5 ^{⋅ dx}

5

_{sin ( x )}

^∫ _{cos ( x )}4 ^{⋅ dx}

6

_y ²

_{∫ 4} ^{⋅ dy}

^{1 – y}

7

_e^x _{− e}^{− x}

^∫ _e^x _{+ e}^{− x ⋅ dx}

8

∫ ^e x ^{⋅ x ⋅ dx}

9

∫ ^{ln ( x ) ⋅ x ⋅ dx}

10

^h _r ² _{⋅ x}

^{∫ π ⋅} _h ^{⋅ dx}

⁰

Вариант

Задание

1

X

_∫ ^{cos (t ) ⋅ dt = a}

0

2

a ⋅ x² + b ⋅ x = −c

3

_∑ ² 385

x

i =

i =1

4

a ⋅ x³ + b ⋅ x² + c ⋅ x = 0

5

^sin ( ^x ) ^{+ cos} ( ^x ) ^{= 0}

6

k ⋅ x + b − y = 0

7

k ⋅ x² + b − y = 0

8

_e ^x ₊ ¹ _{+ x}⁴ _{= 0}

^x3

9

_{tg ( x ) +} ¹ _{= 0}

^x

10

¹ ₊ ¹ ₌ ^π

^{ln ( x ) x 2}

Вариант

Задание

1

_{(cos ( x )}² _{+ sin ( x )}² ₎2

_{2 2}

_{y ( x ) = , f ( x ) ⇒ cos ( x ) + sin ( x ) = 1}

^{cos ( x ) ⋅ 2!+ sin ( x ) ⋅ 3!} 1

2

_{(1 + tg ( x )}² _{)⋅ cos ( x )}²

_{y ( x ) = , f ( x ) ⇒ 1 + tg ( x )}² ₌ ¹

^{sin ( x )} 1 ^cos2 ^{( x )}

3

_{(1 + ctg ( x )}² _{)⋅ sin ( x )}²

_{y ( x ) = , f ( x ) ⇒ 1 + ctg ( x )}² ₌ ¹

^{cos ( x )} 1 ^sin 2 ^{( x )}

4

_{tg ( x )}² _{sin ( x )}

_{y ( x ) = ⋅ cos}³ _{( x ) , f ( x ) ⇒ tg ( x ) =}

^{sin ( x )}2 1 ^{cos ( x )}

5

_{ctg ( x )}² _{cos ( x )}

_{y ( x ) = ⋅ sin} ² _{( x ) , f ( x ) ⇒ ctg ( x ) =}

^{cos ( x )}3 1 ^{sin ( x )}

6

_{y ( x ) =} ^{sin ( x + y )} _,

^{sin ( x ) ⋅ cos ( y ) + sin ( y ) ⋅ cos ( x )}

^f₁ ( ^x ) ^{⇒ sin} ( ^x ) ^{⋅ cos} ( ^y ) ^{+ sin} ( ^y ) ^{⋅ cos} ( ^x ) ^{= sin} ( ^{x + y} )

7

_{4 ⋅ cos ( x )}² _{+ 4 ⋅ sin ( x )}² _{− cos ( x )}² _{− sin ( x )}²

_{y ( x ) = ,}

^{cos ( x )}2 ^{+ sin ( x )}2

_{f ( x ) ⇒ cos ( x )}² _{+ sin ( x )}² _{= 1}

¹

8

_{cos ( x )}² _{+ 2 ⋅ sin ( x )}² _{+ sin ( x )}² _{⋅ tg ( x )}²

_{y ( x ) = ,}

^{cos ( x )}

_{f ( x ) ⇒ cos ( x )}² _{+ 2 ⋅ sin ( x )}² _{+ sin ( x )}² _{⋅ tg ( x )}² ₌ ¹

_{1 2}

^{cos ( x )}

9

_{sin (3 ⋅ x )}² _{cos (3 ⋅ x )}²

₂ − ₂

_{y ( x ) =} ^{sin ( x ) cos ( x )} _,

^{8 ⋅ cos ( 2 ⋅ x )}

_{2 2}

_{f ( x ) ⇒} ^{sin (3 ⋅ x )} ₋ ^{cos (3 ⋅ x )} _{= 8 ⋅ cos ( 2 ⋅ x )}

_{1 2 2}

^{sin ( x ) cos ( x )}

Вариант

Задание

1

f(x)=e^x, x₀=5

2

f(x)=e^2x, x₀=2

3

f(x)=cos(x), x₀=π/4

4

f(x)=sin(x), x₀=π/2

5

f(x)=sin²(x), x₀=π/2

6

f(x)=cos² (x), x₀=π/4

7

f(x)=tan² (x), x₀=π/4

8

f(x)=sin³(x), x₀=π/4

9

f(x)=arcsin(x), x₀=π/4

10

f(x)=arcsin² (x), x₀=π/4

Вариант

Задание

1

1 − x

_x⁴ ₋₁

2

2 ⋅ x² − 3 ⋅ x + 1

x³ + 2 ⋅ x² − 9 ⋅ x −18

3

1

_x⁴ ₋₁

4

x + 1

_x³ _{+ 2 ⋅ x}² _{− 9 ⋅ x – 18}

5

1 − x²

x³ + 2 ⋅ x² − 9 ⋅ x −18

6

1 − x⁴

x³ + 2 ⋅ x² − 9 ⋅ x −18

7

1 − x²

x⁴ −1

8

1 − x³

x⁴ −1

9

1 − x² − x³

x⁴ −1

10

1 − x² − x³

x² −1

Вариант

Задание

1

4 ⋅ x² −1 lim

_x→ ¹ 2 ⋅ x −1

²

2

4 ⋅ x² −1 lim

^{x→6 2 ⋅ x −1}

3

lim ₍1 − cos ₍ x ₎₎

^{x →0}

4

_lim ¹

^{n→∞ n!}

5

⎛ ¹ ⎞n lim _⎜1 + _⎟ n→∞ _{⎝ n ⎠}

6

_lim ¹

^{x →∞ x}

7

_{sin ( 2 ⋅ x )}

_lim

^{x→0 x}

8

_{sin ( x )}

_lim

^{x→0 x}

9

_{sin ( 2 ⋅ x )}

_lim

^{x→0 sin (5 ⋅ x )}