Partial derivative, differential and gradient

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FOREIGN TRADE UNIVERSITY

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FOREIGN TRADE UNIVERSITY

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ANALYSIS 1 Quang Dang-Le, Ph.D. Tel. 0764060787

1. PARTIAL DERIVATIVE, DIFFERENTIAL AND GRADIENT. . . . 3

2. PARTIAL DERIVATIVE OF COMPOSITE AND IMPLICIT FUNCTIONS 11 3. THE UNCONDITIONAL EXTREME VALUE. . . . 15

4. THE CONDITIONAL EXTREME VALUE. . . . 20

5. THE MAXIMUM AND MINIMUM VALUE. . . . 22

6. CALCULATE DOUBLE INTEGRAL. . . . 25

7. USING POLAR COORDINATES. . . . 29

8. REVERSING THE ORDER OF INTEGRATION. . . . 32

9. SEPARABLE EQUATIONS. . . . 35

10. THE HOMOGENEOUS EQUATIONS. . . . 38

11. THE EXACT DIFFERENTIAL EQUATIONS. . . . 40

12. LINEAR EQUATION OF FIRST ORDER AND BERNOULLI EQUATION. . .

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ANALYSIS 1 Quang Dang-Le, Ph.D. Tel. 0764060787

1. PARTIAL DERIVATIVE, DIFFERENTIAL AND

D. dz= cos(2x)dx+ sin(2y)dy.

5. The first order total differential ofz = ex+ey+ 1is

B. dz= eydx+ e dyx .

1 PARTIAL DERIVATIVE, DIFFERENTIAL AND GRADIENT 3

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ANALYSIS 1 Quang Dang-Le, Ph.D. Tel. 0764060787 C. ∇f(x,y,z) = xey/z,xye ,xzey/zy/z . D. ∇f(x,y,z) = ey/z,xe ,xey/zy/z . 12. Let f (x,y) = x2+ xcos2y.Find∇f (x,y).

. B. ∇f(x,y) = (2x + cos2y,xsin(2 )).y

C. ∇f(x,y) = (2x + cos2y − xsin(2 )y , −xsin(2 )).y D. ∇f(x,y) = (2x + cos2y,− x2 sin(2 )).y 13. Let z = sin(xy). Calculate z′′xy.

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ANALYSIS 1 Quang Dang-Le, Ph.D. Tel. 0764060787 16. Find the second order differential ofz = 3x3+ 4xy2− 2y3.

B. d2z = 18xdx2+ 8ydxdy+ (8x − 12y)dy2. C. d2z = 18xdx2+ 16ydxdy+ (8x − 6y)dy2. D. d2z = 9xdx2+ 16ydxdy+ (8x − 12 )y dy2.

17. Find the second order differential ofz = x2+ x sin2y. A. d2z = 2dx2+ 2 sin(2y)dxdy+ 2xcos(2 )y dy2.

21. Find the second order total differential ofz = x2+ x sin2y.

1 PARTIAL DERIVATIVE, DIFFERENTIAL AND GRADIENT 6

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ANALYSIS 1 Quang Dang-Le, Ph.D. Tel. 0764060787 B. d2z = 2dx2− 2 sin(2y)dxdy− 2 cos(2 )x y dy2.

C. d2z = 2dx2+ 2 sin(2y)dxdy− x2 cos(2y)dy2. D. d2z = 2dx2+ sin(2y)dxdy+ 2x cos(2 )y dy2. 22. Let z = cos(xy− cos y). Findzxy(0,π/2).

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ANALYSIS 1 Quang Dang-Le, Ph.D. Tel. 0764060787 26. Let z = x2y + cos( ) +xy y. Which equality is TRUE?

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ANALYSIS 1 Quang Dang-Le, Ph.D. Tel. 0764060787 30. Let z = ex+2yand the following statements Which statement is TRUE?

A. (1),(2) and (3) are true.

B. (1) is true, (2) and (3) are not true. C. (1) and (2) are true, (3) is not true. D. (1) and (3) are true, (2) is not true.

31. Find the partial derivative zxy′′of z = ln(x4+ y2+ 1).

D. dz= (2x − 2y + cos(xy)) dx+ (− x + cos(xy))2 dy. 34. Find the second order differential ofz = xey.

A. d2z = eydx2+ eydxdy+ xe dyy2.

1 PARTIAL DERIVATIVE, DIFFERENTIAL AND GRADIENT 9

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ANALYSIS 1 Quang Dang-Le, Ph.D. Tel. 0764060787

2. PARTIAL DERIVATIVE OF COMPOSITE AND

2. The composite function z = x + sin y

x with y = x2has derivatives z′

3. The composite function z = x + arctan u

v with u = x sin y, v= x cos y has

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ANALYSIS 1 Quang Dang-Le, Ph.D. Tel. 0764060787 13. Find f′xwhere f (u,v) = u2sin v, u= x2+ y2, v=y

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3. THE UNCONDITIONAL EXTREME VALUE

1. Let z = −x2+ 4x − 4y2+ 4 + 4y . Which statement is TRUE? A.

B. z reaches its minimum value at M 2,1 2 . C. z does not have any stationary points. D. z does not have any extreme values.

2. Let z = x2− 4x + 4y2− y + 3. Which statement is TRUE?8

B. z reaches its maximum value at M (2 1)., C. z reaches its stationary point M(1 2)., D. z does not have any extreme values.

3. Let z = x3− y2− 3x + 6y. Which statement is TRUE? A. z reaches its maximum value at M (1 3).,

B. z reaches its minimum value at N ( 1 3).− , .

D. z does not have any stationary points. 4. Let z = xey+ 5. Which statement is TRUE?

A. M(0, 1) is a stationary point. B. M(1, 0) is a stationary point. C. M(0, 0) is a stationary point. D. z does not have any stationary points.

5. Let z = xey+yex+ 2 and M( 1 1)− , − . Which statement is TRUE?

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ANALYSIS 1 Quang Dang-Le, Ph.D. Tel. 0764060787 6. Let z = 4(x − y) − x2− y2. Which statement is TRUE?

. B. The minimum value of this function is Zmin= 8.

C. This function does not have any maximum or minimum values. D. The maximum value of this function is Zmax= 9.

7. Let z = 2x3− xy2+ 5x2+ y2+ 2. Which statement is TRUE? A. This function reaches its maximum value at O(0 0).,

B. This function reaches its minimum value at O(0 0)., C. This function reaches its minimum value at N(1 0)., D. This function does not have any extreme values. 8. Let z = −x − y + xey+ 5. Which statement is TRUE?

A. This function reaches its maximum value at M(1 0)., B. This function reaches its minimum value at M(1 0)., C. This function does not have any extreme values. D. This function reaches its maximum value at M(1 1)., 9. The function Z = x3+ 2xy− 8y3

A. reaches its maximum value at M 1 ,1

6 and z(M) = − 1 27. B. reaches its minimum value at M(0,0) and z(M) = 0.

C. reaches its minimum value at M 1

10. Find extreme values off(x,y) = x2+ xy+ y2− 3x − 6y.

B. f reaches its maximum value at (0 3)., C. f does not have any extreme values.

D. f has a stationary point (0 3) which is not an extreme point., 11. Given z = x4− 8x2+ y2+ 5 and the following points

I(0,0), J(2, 0), K( 2− 0), L, (1, 1). Which statement is TRUE?

3 THE UNCONDITIONAL EXTREME VALUE 16

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ANALYSIS 1 Quang Dang-Le, Ph.D. Tel. 0764060787 A. z reaches its minimum value at J,K.

B. z reaches its maximum value at I,L.

C. z reaches its minimum value at J,K and reaches its maximum value atI,L. D. z reaches its minimum value at I,J,K.

12. Let z = x3+ y2+ 27 + 2 + 1x y . Which statement is TRUE?

B. z has two stationary points.

C. z reaches its minimum value at A(3 1)., −

D. z reaches its extreme value at A(3, −1) and B(−3 1)., −

13. Let f (x,y) = −x2+ xy+ y2+ x − y + 5. Which statement is TRUE? A. f reaches its minimum value at (3/5 1 5)., /

B. f reaches its minimum value at (3/5 1 5)., / C. f reaches its minimum value at (1/5 3 5)., /

15. Let z = x2− y4− 2x + 32y. Which statement is TRUE?

B. z reaches its minimum value at M(1 2)., C. z does not have any stationary points. D. z reaches its maximum value at M(1 2)., 16. The stationary point off(x,y) = (x −1)2+ 2y2is

3 THE UNCONDITIONAL EXTREME VALUE 17

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20. Let z = x2− y4− 2x + 32y. Which statement is TRUE? A. z reaches its minimum value at M(1 2).,

B. z reaches its maximum value at M(2 1)., C. z does not have any stationary points.

21. Let z = x2− 2y + y2. Which statement is TRUE? A. z reaches its maximum value at (0 1).,

B. z reaches its minimum value at (0 1).,

3 THE UNCONDITIONAL EXTREME VALUE 18

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ANALYSIS 1 Quang Dang-Le, Ph.D. Tel. 0764060787 C. z has a maximum value and a minimum value.

D. z does not have any extreme values.

22. Let z = 3x2− 12x + 2y3+ 3y2−12y. Which statement is TRUE? A. z has a maximum value and a minimum value.

B. z has only one maximum value. C. z does not have any stationary points. D. z has only one minimum value. 23. Find extreme values of z = x2− 4x + 4

. B. z reaches its maximum value at (2 1)., C. z has a stationary point (1 2)., D. z does not have any extreme values.

24. Find extreme values ofz = −x2+ 4xy− 10y2− 2x + 16y. A. z reaches its minimum value at (1 1).,

C. z reaches its minimum value at (−1 1)., − D. z reaches its maximum value at (−1 1)., −

et z = x2− y − ln |y|−2. Which statement is TRUE?

B. z reaches its maximum value at (0 1), − C. z always has partial derivatives on R2.

D. z has a stationary point but does not have any extreme values.

3 THE UNCONDITIONAL EXTREME VALUE 19

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4. THE CONDITIONAL EXTREME VALUE

1. Find the extreme value of z = x2(y − 1) − 3x + 2 satisfying thatx − y + 1 = 0. . B. z reaches its minimum value at A(−1,0) and its maximum value at B(1 2)., C. z reaches its maximum value at A(−1,0) and B(1 2).,

D. z reaches its minimum value at A(−1,0) and B(1 2)., 2. Find the extreme value of z =x3

3 − 3x + y− 2 satisfying that −x2+ + 4 = 0.y . B. z reaches its maximum value at A(1, −3) and its minimum value at B(−3 5)., C. z reaches its minimum value at A(1, −3) and B(−3 5).,

D. z reaches its maximum value at A(1, −3) and B( 3 5).− ,

3. Find the extreme value of z = 2x2+ y2− 2y − 2 satisfying thaty − x + 1 = 0.

B. z reaches its maximum value at (2/3 1 3)., − /

C. z has a stationary point and does not have any extreme values. D. z does not have any stationary points.

4. Find the extreme value ofz = x2(y + 1) − 3x + 2satisfying thatx + y + 1 = 0. .

B. z reaches its maximum value at (−1,0) and (1 2)., − C. z reaches its minimum value at (−1,0) and (1 2)., −

D. z reaches its maximum value at (−1,0) and its minimum value at (1 2)., − 5. Find the extreme value of z = xysatisfying thatx + y − 1 = 0.

A. z does not have any extreme values.

C. z reaches its minimum value at 21

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ANALYSIS 1 Quang Dang-Le, Ph.D. Tel. 0764060787 6. Find the extreme value ofz = 2x2+ y2− 2y − 2satisfying that−x + y + 1 = 0.

A. z reaches its minimum value at 3 2 D. z reaches its minimum value at (−3,10) and its maximum value at (1 2)., 8. Find the extreme value of z = x2+ y2satisfying thatx + y − 1 = 0.

A. z reaches its maximum value at 2 1 ,1

2 . C. z does not have any extreme values.

D. z reaches its minimum value at −1 , −1

2 .

4 THE CONDITIONAL EXTREME VALUE 21

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5. THE MAXIMUM AND MINIMUM VALUE

1. Find the maximum and minimum value of functionz = −x + 2 + 3y on the given setD = [0, 1] × [0 1], .

A. The maximum value of zis 5 and the minimum one is 2. B. The maximum value of zis 5 and the minimum one is 3. C. The maximum value of zis 4 and the minimum one is 3. D. The maximum value of zis 4 and the minimum one is 2.

2. Find the maximum and minimum value of functionz = x + 2 + 3xy y − 6on the given setD = [0, 1] × [0 2], .

A. The maximum value of zis 5 and the minimum one is -6. B. The maximum value of zis 5 and the minimum one is -5. C. The maximum value of zis 0 and the minimum one is -6. D. The maximum value of zis 0 and the minimum one is -5.

3. Find the minimum valuemof functionz = ln x − 2yonD = 1,1 ×[0,1]. A. m = ln 1 − 2.

B. m = ln 1. C. m = 0.

D. m = − ln 1 − 2.

4. Let z = x +2 + 3xy y −6 on D = [0, 1] [1× , 2] and the following three statements. (1) The maximum value ofzis 5 atM(1 2), .

(2) The maximum value ofzis -3 atN(0 1), .

5. Find the maximum and minimum value of functionz = x2− 2x − y + 4on the given setD = [0, 1] × [0 1], .

5 THE MAXIMUM AND MINIMUM VALUE 22

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ANALYSIS 1 Quang Dang-Le, Ph.D. Tel. 0764060787 A. The maximum value of zis 4 and the minimum one is 2.

B. The maximum value of zis 4 and the minimum one is 3. C. The maximum value of zis 3 and the minimum one is 2. D. The maximum value of zis 5 and the minimum one is 2.

6. Let z = x3− y3+ 5 on the given setD = [0, 1] × [1 2], .Which statement is true? A. The minimum value of zis -3.

B. The maximum value of zis 6. C. The minimum value of zis -2. D. The maximum value of zis 4.

7. Find the maximum valueM and minimum valuemof functionz = x2y2

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6. CALCULATE DOUBLE INTEGRAL

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7. USING POLAR COORDINATES

1. Calculate I =RR

x2+ y2dxdywhere D is the part of the disk x2+ y2≤ a2 that lies in the first quadrant.

1 + x2+ y2dxdy where D is the part of the unit disk that lies in the first quadrant.

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8. REVERSING THE ORDER OF INTEGRATION

1. Reverse the order ofR2

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Df(x,y dxdy where D is the triangular region with vertices) O(0,0), A(0 1), B(1 1). Which statement is TRUE?, ,

Df(x,y)dxdy where D is the triangular region with vertices C(1,1), A(0, 1), B(1 0). Which statement is TRUE?,

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5. The solution x = −1 of the differential equationxydx+ (1 + x)dy= 0is called A. the general solution.

B. the particular solution. C. the singular solution.

9 SEPARABLE EQUATIONS 35

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6. The particular solution of (1 + ex)yy′= ex satisfying the initial condition

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10. THE HOMOGENEOUS EQUATIONS

1. Which equation is a homogeneous equation?

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C + ln |x|.

5. Find the solution of the differential equation y′= y

x+ 1 with the initial

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11. THE EXACT DIFFERENTIAL EQUATIONS

1. Which equation is an exact differential equation?

A. ex(y − x2)dx+ (ex− y2sin y dy= 0.) B. ex(y2+ x dx+ (e) x+ x2sin y dy= 0.) C. ex(y − x2)dx+ (ex+ x2sin y dy= 0.) D. ex(y2+ x dx+ (e) x− y2sin y)dy= 0.

2. Which equation is an exact differential equation?

A. (yex− x sin x)dx+ (ex− y cos y dy= 0.) B. (yex− x sin y)dx+ (ex− y cos y dy= 0.) C. (yex− x sin x)dx+ (ex− x cos y dy= 0.) D. (yex− x sin y)dx+ (ex− x cos y dy= 0.)

3. Which equation is an exact differential equation?

A. (yex−xex)dx+ ex− y2sin y dy= 0.) B. (yex+ xex)dx+ (ex+ x2sin y dy= 0.) C. (yex+ xex)dx+ (ex+ y2sin y dy= 0.) D. (yex−xey)dx+ (ex− y2sin y)dy= 0.

4. Which equation is an exact differential equation?

A. (y sinx − cos y)dx− (cos x − xsiny)dy= 0. B. (y sinx − cos y)dx+ (cos x − xsiny)dy= 0. C. (y sinx + cos y)dx+ (cos x + xsiny)dy= 0. D. (y sinx + cos y)dx− (cos x − xsiny)dy= 0.

5. Which equation is NOT an exact differential equation?

A. (yex− x sin x)dx+ (ex− x cos y dy= 0.) B. (yex+ y2sin x)dx+ (ex− y cos x dy= 0.2 ) C. (yex− x sin x)dx+ (ex− y cos y dy= 0.) D. (yex− sin y)dx+ (ex− x cos y dy= 0.)

11 THE EXACT DIFFERENTIAL EQUATIONS 40

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