Задача 5
Найти частные производные ∂∂xz , ∂∂yz от неявной функции.
5.1.ln(z 2 + xy) = ex2 +y2 +z2 .
5.2.2x3 −5x + z3 + y3 −3xyz +8 = 0 .
5.3. |
tg(z +1) |
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y − x |
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tg( y − 2) |
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5.4.y ex−zy = cos(zx) .
5.5.x sin y + y sin x + z sin x −8 = 0 .
5.6.xe y + yex + zex = 2 .
5.7.3xz − 4 yz + z 2 −9 = 0 .
5.8.z 2 − z −8xz + 2x2 + 2 y 2 +8 = 0 .
5.9. z = x2 − y 2 tg |
z |
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− y 2 |
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5.10.tg(x + z) = ez y.
5.11.x2 + y 2 + z 2 −3xyz = xy2 z3 .
5.12.x2 + 2 y 2 + 3z 2 + xy − z + sin(xy)−9 = 0 .
5.13. ln(xy + z) = z 2 − y .
x
5.14. e y sin zy = xzy .
5.15.xz = ln zy +1 .
5.16.z = x + arctg z −y x .
5.17.ln z = yz + x2 −1 .
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5.18. e |
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5.19. yz 2 + xz + xy =1.
5.20.e x cos xy = xy .
5.21.tg 2 z + sin x + cos y − ex = 0 .
5.22.y 2 + x2 z − 4 yz3 −1 = 0 .z
5.23. zez − x ln y = |
x . |
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5.24.z3 + 5xy3 + 4 yz 2 − x3 − 6 = 0 .
5.25.xz − ln y +z 2 = 0 .
5.26.z3 + 3x2 z − 2xy = 0 .
5.27.exz cos(yz)= x2 .
5.28.sinsin xz = zy .
5.29.x cos y + y cos z + z cos x =1 .
5.30.x3 + 2 y3 + z3 −3xyz −3y + 3 = 0 .
Задача 6
Найти градиент, уравнения касательной плоскости и нормали к заданной поверхности S в точке Мо(Xo,Yo,Zo).
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6.1. S: x2 + y2 + z2 +6z −4x +8 = 0, |
Mo(2,1, −1). |
6.2. S: x2 + z2 −4 y2 = −2xy, |
Mo(−2,1, 2) |
6.3. S: x2 + y2 + z2 − xy +3z = 7, |
Mo (1, 2,1). |
6.4. S: x2 + y2 + |
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2 +6 y +4x =8, |
Mo(−1,1, 2). |
6.5. S: 2x2 − y2 + z2 −4z + y =13, |
Mo (2,1, −1) |
6.6. S: x2 |
+ y2 + z2 −6 y +4z +4 = 0, |
Mo(2,1, −1). |
6.7. S: x2 + z2 −5yz +3y = 46, |
Mo(1, 2, −3). |
6.8. S: x2 |
+ y2 − xz − yz = 0, |
Mo (0, 2, 2). |
6.9. S: x2 |
+ y2 + 2 yz − z2 + y −2z = 2, |
Mo (1,11). |
6.10. S: x2 |
+ y2 |
− z2 −2xz + 2x = z, |
Mo(1,1,1). |
6.11. S: x2 + y2 −2xy +2x − y = z, |
Mo(−1, −1, −1). |
6.12. S: y2 − x2 |
+ 2xy −3y = z, |
Mo (1, −1,1). |
6.13. S: x2 |
− y2 |
−2xy − x −2 y = z, |
Mo (−1,1,1) |
6.14. S: x2 |
−2 y2 + z2 + xz −4 y =13, |
Mo (3,1, 2). |
Задание 7
Найти наибольшее и наименьшее значения функции Z=Z(X,Y) в области D, ограниченной заданными линиями
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7.1. |
z = 3x + y − xy, |
D : y = x, |
y = 4, x = 4, x = 0. |
7.2. |
z = xy − x − 2 y, |
D : x = 3, |
y = x, y = 0. |
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7.3. |
z = x2 + 2xy − 4x + 8y, |
D : x = 0, |
x =1, |
y = 0, |
y = 2. |
7.4. |
z = 5x 2 − 3xy + y 2 , |
D : x = 0, |
x =1, |
y = 0, |
y =1. |
7.5. |
z = x2 +2xy − y2 −4x, |
D : x − y +1 = 0, |
x = 3, |
y = 0. |
7.6. z = x2 + y2 −2x −2 y +8, |
D : x = 0, |
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y = 0, |
x + y −1 = 0. |
7.7. z = 2x3 − xy 2 + y 2 , |
D : x = 0, |
x =1, |
y = 0, |
y = 6. |
7.8. z = 3x +6 y − x2 − xy − y2 , |
D : x = 0, |
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x =1, |
y = 0, |
y =1. |
7.9. z = x2 −2 y2 + 4xy −6x −1, D : x = 0, |
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y = 0, |
x + y −3 = 0. |
7.10. z = x 2 + 2xy −10, D : y = 0, |
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y = x 2 − 4 . |
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7.11. z = xy − 2x − y, |
D : x = 0, |
x = 3, |
y = 0, y = 4. |
7.12. z = |
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− xy, |
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D : y = 8, y = 2x 2 . |
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7.13. z = |
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D : y = |
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y = 0. |
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Задача 8
Найти полные дифференциалы указанных функций
8.1.z = 2x3 y − 4xy5 ;
8.2.z = x2 y sin x −3y;
8.3. z = arctgx + y;
8.4.z = arcsin(xy)−3xy2 ;
8.5.z = 5xy4 + 2x2 y7 ;
8.6.z = cos(x2 − y2 )+ x3 ;
8.7.z = ln(3x2 − 2 y 2 );
8.8.z = 5xy2 −3x3 y 4 ;
8.9.z = arcsin(x + y);
8.10.z = arctg(2x − y);
8.11. z = 7x3 y − xy;
8.12.z = x2 + y 2 − 2xy;
8.13.z = e x+y−4 ;
8.14.z = cos(3x + y)− x2 ;
8.15.z = tg x + y ;
x − y
8.16.z = ctg y ;
x
8.17.z = xy 4 −3x2 y +1;
8.18.z = ln(x + xy − y2 );
8.19.z = 2x2 y2 + x3 − y3 ;
8.20.z = 3x2 − 2 y2 +5;
8.21.z = arcsin x + y ;
x
8.22.z = arcctg(x − y);
8.23.z = 3x2 − y2 + x;
8.24.z = y 2 −3xy − x4 ;
8.25.z = arccos(x + y);
8.26.z = ln(y2 − x2 + 3);
8.27.z = 2 − x3 − y3 + 5x;
8.28.z = 7x − x3 y 2 + y 4 ;
8.29.z = e y−x ;
8.30.z = ln(3x2 − 2 y2 );
Задача 9
Вычислить значение производной сложной функции
u = u(x, y) , где x = x(t) , y = y(t) , при t = t0 с точностью до двух знаков после запятой.
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9.1. |
u = ex−2 y , x = sin t , |
y = t3 , |
t0 = 0 . |
9.2. |
u = ln(ee +e−y ) , x = t 2 , y = t 3 , |
t0 = −1. |
9.3. |
u = yx , x = ln(t −1) , |
y = et / 2 , |
t0 = 2 . |
9.4. |
u = ey−2 x+2 , x = sin t , y = cost , |
to =π / 2 . |
9.5. |
u = x2ey , |
x = cost , |
y = sin t , |
t0 =π . |
9.6. |
u = ln(ex +ey ) , x = t 2 , y = t3 |
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t0 =1. |
9.7. |
u = x y , |
x = et , y = ln t , |
t0 =1. |
9.8. |
u = ey−2 x , |
x = sin t , |
y = t 3 |
, t0 = 0 . |
9.9. |
u = x2e−y , |
x = sin t , |
y = sin 2 t , |
t0 =π / 2 . |
9.10. |
u = ln(e−x +ey ) , x = t 2 , y = t3 |
, t0 = −1. |