Functions of several variables- Textbook..pdf
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Variant 3.
1. Let the function z = e(a+2) xy2 be given. Find:
a)differential of the first order;
b)partial derivatives of the second order.
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Let the function u = |
(b + 2)y |
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+ b xz, the point A(1;0;−1) and the vector |
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a = (2; 2
b + 1; b + 1) be given. Find:
a) the gradient of the given function at a given point ( grad f (A));
∂f (A) b) derivative of the function into direction of the vector a at the point A 
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∂a
3. Investigate the function z = x 2 + y 2 − 8x − 2 for extremum.
4. Find the domain of the function:
a) z = ln (xy − 1); |
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b) |
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x + y |
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5. |
The function z = |
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+ arcsin xy is given. Does this function satisfy the equation |
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3x |
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∂ x |
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∂ y |
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6. Using the total differential of the function of two variables calculate approximately
sin 32 o cos 59o.
7. Find equations of the tangent plane and normal of the surface z = 
26 − x 2 − y 2 at the point M (3, 4,1).
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Variant 4.
1. Let the function z = sin(a x 2 y)be given. Find:
a)differential of the first order;
b)partial derivatives of the second order.
2. Let the function u = e(b+1)xz+2 y , the point A(− 1;1;−1) and the vector a = (2; 2
a + 1; a + 1) be given. Find:
a) the gradient of the given function at a given point ( grad f (A));
∂f (A) b) derivative of the function into direction of the vector a at the point A 
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∂a
3. Investigate the function z = 3x − x 2 − xy − y 2 + 6 y for extremum.
4. Find the domain of the function:
a) z = arccos |
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y 2 |
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b) z = |
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x − y
5. The function z = ln(x 2 + y 2 + 2x + 1) is given. Does this function satisfy the
equation |
∂ 2 |
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∂ 2 |
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∂ y |
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6. Using the total differential of the function of two variables calculate approximately (4.05)0.98 .
7. Find equations of the tangent plane and normal of the surface z = x 2 − y 2 at the point M (5, 4, 9).
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Variant 5.
1. Let the function z = y 2 be given. Find:
bx
a)differential of the first order;
b)partial derivatives of the second order.
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2. |
Let the function u = y c |
1 x + x cos z, the point |
A 0;−2; |
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2 |
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a = (2; 2
b + 1; b + 1) be given. Find
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the gradient of the given function at a given point ( |
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a) |
grad f (A)); |
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∂f (A) |
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b) |
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derivative of the function into direction of the vector a at the point A |
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3. Investigate the function z = 2x 3 + 2 y 3 − 36xy + 430 for extremum. |
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Find the domain of the function: |
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a) z = 4 9 − 3x 2 − y 2 ; |
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b). z = |
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x 2 |
− y 2 − 1 |
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5. |
The function z = e xy is given. Does this function satisfy the equation |
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x |
∂ 2 z |
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∂ z |
− xy 2 z = 0 ? |
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∂ x∂ y |
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6. |
Using the total differential of the function of two variables calculate approximately |
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z = ln(0.93 + 0.992 ), knowing that ln2 ≈ 0.69. |
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7. |
Find equations of the tangent plane and normal of the surface z = x 2 + y 2 |
at the |
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point M (1, 2, 5).
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Variant 6. |
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1. |
Let the function z = cos |
x |
be given. Find: |
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a) differential of the first order; |
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b)partial derivatives of the second order. |
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2. |
Let the function u = z arcsin x + bxy, the point |
A |
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; b; 0 and the vector |
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2 |
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a = (2; 2
a + 1; a + 1) be given. Find:
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the gradient of the given function at a given point ( |
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a) |
grad f (A)); |
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∂f (A) |
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b) |
derivative of the function into direction of the vector |
a at the point A |
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∂a |
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3. Investigate the function z = 2xy − 4x − 2 y for extremum.
4. Find the domain of the function:
a) z = lg(x 2 − y 2 );
b) z = |
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+ y 2 |
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+ 9 y |
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5. The function z = x is given. Does this function satisfy the equation
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∂2 |
z |
− |
∂ z |
= 0 ? |
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∂ y |
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6. Using the total differential of the function of two variables calculate approximately
4
2.99 2 + 7 e0.01 .
7. Find equations of the tangent plane and normal of the surface z = xy at the point M (3, 4,12).
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Variant 7. |
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1. |
Let the function z = cos |
x |
be given. Find: |
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ay |
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a) differential of the first order; |
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b)partial derivatives of the second order. |
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2. |
Let the function u = z arcsin x + bxy, the point |
A |
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; b; 0 and the vector |
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2 |
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a= (2; 2
a + 1; a + 1) be given. Find:
a)the gradient of the given function at a given point ( grad f (A));
∂f (A) b) derivative of the function into direction of the vector a at the point A 
∂a
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3. Investigate the function z = x 2 + 2 y 2 + xy − x + 3 y for extremum.
4. Find the domain of the function:
1
a) z = 
; x 2 + 4 y − 1
b) z = |
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+ y 2 |
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5. The function z = ln(x + e − y ) is given. Does this function satisfy the equation
∂ z |
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∂ 2 |
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− |
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∂ 2 z |
= 0 ? |
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∂ x 2 |
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∂ x ∂ x∂ y ∂ y |
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6. Using the total differential of the function of two variables calculate approximately 
2.032 + 5 e0.02 .
7. Find equations of the tangent plane and normal of the surface z = 
x 2 − 1 + y 2 at the point M (1, 2, 2).
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Variant 8.
1. Let the function z = ax 2 ln y be given. Find:
a) differential of the first order; b)partial derivatives of the second order.
2. Let the function u = tg(x 2 |
+ by)+ |
1 |
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π |
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, the point |
A 0; |
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and the vector |
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z |
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b |
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a = (2; 2
b + 1; b + 1) be given. Find:
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the gradient of the given function at a given point ( |
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a) |
grad f (A)); |
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∂f (A) |
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b) |
derivative of the function into direction of the vector a at the point A |
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∂a |
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3. Investigate the function z = x 3 + xy 2 + 6xy for extremum. |
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4. Find the domain of the function: |
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a) |
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log |
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b) |
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x2 − 2 y |
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5. |
The function z = |
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is given. Does this function satisfy the equation |
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(x 2 − y 2 )5 |
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∂ z |
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∂ z |
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6. Using the total differential of the function of two variables calculate approximately ln(0.014 + 1.12 ).
7. Find equations of the tangent plane and normal of the surface z = 3x 2 − xy + 2 y 2 at the point M (− 1, 3, 24).
46
Variant 9. |
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1. Let the function u = c3 yx2 + |
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a) differential of the first order; |
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b)partial derivatives of the second order. |
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2. Let the function u = z 2 − axy 3 , |
the point A(− 1;−b;1) and the vector |
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c + 1; c + 1) be given. Find: |
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the gradient of the given function at a given point ( |
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b) |
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∂a |
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3. Investigate the function z = x 3 + 8 y 3 + 6xy − 1 for extremum. |
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4. Find the domain of the function: |
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a) z = sin(x + |
2 y) |
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b) |
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5. The function z = z = xe y / x is given. Does this function satisfy the equation |
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∂ |
2 z |
− 4 |
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∂ y 2 |
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6. Using the total differential of the function of two variables calculate approximately 3
3.012 − 20.02 , knowing that ln2 ≈ 0.69.
7. Find equations of the tangent plane and normal of the surface z = x 2 + xy + y 2 at the point M (1, 2, 7).
47
Gavdzinski V.N.,
Korobova L.N.,
Maltseva E.V.
FUNCTIONS OF SEVERAL VARIABLES
Textbook
Компьютерная верстка |
Е. С. Корнейчук |
Сдано в набор 19.11.2012 Подписано к печати 19.12.12
Формат 60/88/16 Зак. № 5022
Тираж 100 экз. Объем: 3,0 печ. л.
Отпечатано на издательском оборудовании фирмы RISO
в типографии редакционно-издательского центра ОНАС им. А.С. Попова
ОНАС, 2012
48