математика 1

𝑧 = 𝑥 + √𝑥𝑦 funksiyaning aniqlanish sohasini toping. Найдите область

2

определения функции 𝑧 = 𝑥 + √𝑥𝑦. Find the domain of function 𝑧 = 𝑥 + √𝑥𝑦.

• {𝑥 ≥ 0

  𝑦 ≥ 0

  𝑣𝑎 {𝑥 ≤ 0

  𝑦 ≤ 0

• {𝑥 ≥ 0

  𝑦 ≥ 0

  𝑣𝑎 {𝑥 ≤ 0

  𝑦 > 0

• {𝑥 ≥ 0

  𝑦 ≥ 0

  𝑣𝑎 {𝑥 > 0

  𝑦 ≤ 0

• {𝑥 ≥ 0

  𝑦 < 0

  𝑣𝑎 {𝑥 ≤ 0

  𝑦 ≤ 0

Ikki o’zgaruvchili 𝑓(𝑥, y) funksiyaning 𝑥 ga nisbatan xususiy orttirma formulasini toping. Найти формулу частного приращения функции двух переменных 𝑓(𝑥, y) по

𝑥. Find formula of partial increment of a function of two variables 𝑓(𝑥, y) with respect to x.

• ∆𝑥𝑓 = 𝑓(𝑥 + ∆𝑥; 𝑦) − 𝑓(𝑥; 𝑦)

• ∆𝑧𝑓 = 𝑓(𝑦 + ∆𝑥; 𝑦) − 𝑓(𝑥; 𝑦)

• ∆𝑥𝑓 = 𝑓(𝑦 + ∆𝑦; 𝑥) − 𝑓(𝑥; 𝑦)

• ∆𝑥𝑓 = 𝑓(𝑥 + ∆𝑥; 𝑦) − 𝑓(∆𝑥; 𝑦)

. Ikki o’zgaruvchili 𝑓(𝑥, y) funksiyaning to’la differensial formulasini toping. Найти формулу полного дифференциала функции двух переменных 𝑓(𝑥, y). Find the formula for total differential of a function of two variables 𝑓(𝑥, y).

• 𝑑𝑓 = 𝛛𝑓 𝑑𝑥 + 𝛛𝑓 𝑑𝑦

  𝛛𝑥 𝛛𝑦

• 𝑑𝑓 = 𝛛𝑓 𝑑𝑥 + 𝛛𝑧 𝑑𝑦

  𝛛𝑥 𝛛𝑦

• 𝑑𝑓 = 𝛛𝑓 𝑑𝑦 + 𝛛𝑓 𝑑𝑥

  𝛛𝑥 𝛛𝑦

• 𝑑𝑓 = 𝛛𝑓 𝑑𝑥 + 𝛛𝑦 𝑑𝑦

  𝛛𝑥 𝛛𝑓

48. 𝑧 = x𝑦 funksiyaning 𝛛

𝛛𝑥

xususiy hosilasini toping. Найдите частную

производную 𝛛𝑧

𝛛𝑥

функции 𝑧 = x𝑦. Find the partial derivative 𝛛

𝛛𝑥

of the function

𝑧 = x𝑦.

• 𝛛𝑧 = 𝑦

  𝛛𝑥

• 𝛛𝑧 = −𝑦

  𝛛𝑥

• 𝛛𝑧 = 𝑥

  𝛛𝑥

• 𝛛𝑧 = −𝑥𝑠

  𝛛𝑥

47. 𝑧 = 𝑥3 − 𝑦3 funksiyaning 𝛛𝑧

𝛛𝑦

xususiy hosilasini toping. Найдите частную

производную 𝛛𝑧

𝛛𝑦

функции 𝑧 = 𝑥3 − 𝑦3. Find the partial derivative 𝛛𝑧

𝛛𝑦

of the function

𝑧 = 𝑥3 − 𝑦3.

• 𝛛𝑧 = −3𝑦2

  𝛛𝑦

• 𝛛𝑧 = −3𝑦−2

  𝛛𝑦

• 𝛛𝑧 = 𝑦2

  𝛛𝑦

• 𝛛𝑧 = 3𝑦2

  𝛛𝑦

44. 𝑢 = 𝑒𝑦𝑥 funksiyaning xususiy hosilalarini toping. Найдите частные производные функции 𝑢 = 𝑒𝑦𝑥. Find the partial derivatives of a function 𝑢 = 𝑒𝑦𝑥.

• 𝑧′ = y𝑒𝑥𝑦, 𝑧′ = x𝑒𝑥𝑦

  𝑥 𝑦

• 𝑧′ = −y𝑒𝑥𝑦, 𝑧′ = x𝑒𝑥𝑦

  𝑥 𝑦

• 𝑧′ = yx𝑒𝑥𝑦, 𝑧′ = x𝑒𝑥𝑦

  𝑥 𝑦

• 𝑧′ = x𝑒𝑥𝑦, 𝑧′ = y𝑒𝑥𝑦

  𝑥 𝑦

41. 𝑢 = 𝑥𝑦 + 𝑦 funksiyaning xususiy hosilalarini toping. Найдите частные производные функции 𝑢 = 𝑥𝑦 + 𝑦. Find the partial derivatives of a function

𝑢 = 𝑥𝑦 + 𝑦.

• 𝑢′ = y, 𝑢′ = 𝑥 + 1

  𝑥 𝑦

• 𝑢′ = y, 𝑢′ = 𝑥 − 1

  𝑥 𝑦

• 𝑢′ = −y, 𝑢′ = 𝑥 + 1

  𝑥 𝑦

• 𝑢′ = y, 𝑢′ = 1

  𝑥 𝑦

40. 𝑧 = ln(𝑥𝑦) funksiyaning xususiy hosilalarini toping. Найдите частные производные функции 𝑧 = ln(𝑥𝑦). Find the partial derivatives of a function

𝑧 = ln(𝑥𝑦).

• ′ 1 ′ 1

  𝑧𝑥 = 𝑥 , 𝑧𝑦 = 𝑦

• ′ 2 ′ 1

  𝑧𝑥 = 𝑥 , 𝑧𝑦 = 𝑦

• ′ 1 ′ 1

  𝑧𝑥 = 𝑥 , 𝑧𝑦 = − 𝑦

• ′ 1 ′ 1

  𝑧𝑥 = − 𝑥 , 𝑧𝑦 = 𝑦

39. 𝑧 = 𝑒−3 funksiyaning xususiy hosilalarini toping. Найдите частные производные

𝜋 𝜋

функции 𝑧 = 𝑒−3 . Find the partial derivatives of a function 𝑧 = 𝑒−3 .

• 𝑧′ = 0, 𝑧′ = 0

  𝑥 𝑦

• 𝑧′ = 0, 𝑧′ ≠ 0

  𝑥 𝑦

• 𝑧′ ≠ 0, 𝑧′ = 0

  𝑥 𝑦

• 𝑧′ = −1, 𝑧′ = 0

  𝑥 𝑦

38. 𝑧 = arctg 𝑥

𝑦

𝑦

funksiyaning xususiy hosilalarini toping. Найдите частные

производные функции 𝑧 = arctg 𝑥. Find the partial derivatives of a function 𝑧 =

𝑦

arctg 𝑥.

𝑦

• 𝑧′ = 𝑦

  , 𝑧′ = − 𝑥

  𝑥 𝑥2+𝑦2 𝑦𝑥2+𝑦2

• 𝑧′ = 𝑦 , 𝑧′ = 𝑥

  𝑥 𝑥2+𝑦2 𝑦 𝑥2+𝑦2

• 𝑧′ = − 𝑦 , 𝑧′ = 𝑥

  𝑥 𝑥2+𝑦2 𝑦 𝑥2+𝑦2

• 𝑧′ = 𝑦 , 𝑧′ = − 2𝑥

  𝑥 𝑥2+𝑦2 𝑦

  𝜋

  𝑥2+𝑦2