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..

Mathcad


2000


: . .. ..
681.3.06

Mathcad: / . . .. , 2000, 27 . , Mathcad 2000. , « » , . « » - . .. : . ... ..

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___________________________________________________________________ 4 1. _____________ 5
1.1. _______________________________________________ 5
1.1.1. _____________________________________________________ 5 1.1.2. ___________________________________________________ 5

1.2. _____________________________ 6

2. ____________________ 9
2.1. ________________ 9 2.2. ________________ 10 2.3. . ____________ 13 2.4. ______________ 15

3. ______________________ 18
3.1. _____________________________ 18 3.2. __________________ 19 3.3. ________________ 20

4. _________ 21 ________________________________________________________________ 22 1 ________________________________________________________________ 23 2 ________________________________________________________________ 24 3 ________________________________________________________________ 26

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: ? ? ? ? . . , (, , ..). . . , (Pascal, C) : Excel, Mathcad. , , , Mathcad 2000. . . ( ) ( ). m=3, n=2. . 2 3 , . Mathcad : , . , - .

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1. 1.1. 1.1.1. . . , n , x1,,x2...xn . m b1,b2,..bm. . j- (j=1,n) i- (i=1,m) aij . 1,2...n. . . . . n xj, j=1,n. aij, j- i- , bi, i=1,m, , i. , , . 1.1.2. . . n x1,,x2...xn . 1,2...n V, . , f(x1,,x2...xn)=a0x1a1x2a2...xnan. x1,,x2...xn, . . . R . n x1,,x2...xn . 1,2...n . , R=a0x1a1x2a2...xnan. x1,,x2...xn, .

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1.2. , , x1,x2...xn. f(x1,x2,...,xn). () ­ x1,x2...xn, () . , , .. , , .. , , . , . . x1,x2...xn, f(x1,x2,...,xn) ( ): (1) f(x1, x2,...,xn)max(min) : g1(x1, x2,...,xn) b1; g1(x1, x2,...,xn)b2; ................................ (2) gm(x1, x2,...,xn)bm; x10, x20,...xn0 x1,x2...xn, (1)-(2), . (1)-(2) (2) , : f(x1, x2,...,xn)max(min) : g1(x1, x2,...,xn)=b1; g1(x1, x2,...,xn)=b2; ................................ gm(x1, x2,...,xn)=bm; . (1)-(2) (2) (1) , . ( ) .
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x1,x2...xn, f(x1, x2,...,xn)=c1x1+ c1x1+...+ cnxnmax(min) : a11x1+ a12x2+... a1nxnb1 a21x1+ a22x2+... a2nxnb2 ...................................... am1x1+ am2x2+... amnxnbm xj0, j=1,2...n. « » « », .. . . , , . (2) (1) , , . x1,x2...xn, f(x1, x2,...,xn) jmax(min); : g1(x1, x2,...,xn) b1; g1(x1, x2,...,xn)b2; ................................ gm(x1, x2,...,xn)bm; x10, x20,...xn0. . x1,x2...xn, . - f(x1,x2,...,xn)=a0x1a1x2a2...xnan: ­ , . : : c1x1+c2x2+...+cnxnV; x10, x20,...xn0.

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. x1,x2...xn, : f(x1,x2,...,xn)=c1x1+c2x2+...+cnxnmin , a0x1a1x2a2...xnan =R x10, x20,...xn0.

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2. 2.1. , . . x1, x2, : f(x1, x2)=c1x1+ c2x2max : a11x1+ a12x2b1 a21x1+ a22x2b2 x10, x20. , () - , , ( x10, x20) , : a11x1+ a12x2=b1 a21x1+ a22x2=b2 .................. am1x1+ am2x2=bm , m=3. , (. 1).
6

x21( x1) x22( x1) x23( x1)

4

2

0

1

0

1

2 x1

3

4

5

. 1 f(x1,x2)=c1x1+c2x2= ­ . , . : c1x1+c2x2= (.1), , , .
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2.2. . , 2 , x1,x2 . 3 b1=60, b2=40, b3=50 . i- j- aij 1 a11=1 a21=1 2 a21=1 a22=2 3 a31=2 a32=1 1=1, 2=2. x1,x2 .
. x1,x2 f ( x1 , x2) 2 x1 + 3 x2 x1 + x1 + 2 x1 x1 x2 x2 60 2 x2 40 + x2 50 0 0

1

2



1. Mathcad. Mathcad. 2. x2=k*x1+b , . x1 + x2 x1 + 2 x2 2 x1 + x2 60 40 50 x21( x1) := -x1 + 60 -1 x22( x1) := x1 + 20 2 x23( x1) := -2 x1 + 50

3. . .2
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60 x21( x1) x22( x1) x23( x1) 40

20

0

0

10

20

30 x1

40

50

60

. 2 x1 x2 , . 4. f(x1,x2)=C (, , , ) . 2 x1 + 3 x2 c := 70 x24( x1) := -2 1 x1 + c 3 3 c

60 x21( x1) x22( x1) 40 x23( x1) x24( x1) 20

20

10

0

0

10

20

30 x1

40

50

60

. 3
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c Format/Graph/X-Y Plot Show markers . 3. =70 (20,10).
5.



Given

x1 + 2 x2 2 x1 + x2

40

50 20 Find ( x1 , x2) 10 f ( x1 , x2) := 2 x1 + 3 x2 fmax := f ( 20 , 10) fmax = 70

6. . x1=20, x2=10.

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2.3. . . 2 x1,,x2 . 1,2 V, . , . x1 x2 . f(x1,x2)=a0x1a1x2a2max (3) c1x1+c2x2V; (4) (5) x10, x20 . . 4.

100 80 x21 ( x1) 60 x22 ( x1) 40 20 0

20

61

0

10

20

30 x1

40

50

60

. 4

(4), (5) 12 X1X2. . - , . 12 12.

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f(x1,x2,...,xn)=a0x1a1x2a2= , . - , . , . , , x1,x2 - 12. (3), (4), (5) , 12 0. x1(0), x2(0) 0 .

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2.4. 2 x1,,x2 . 1=20, 2=10 V=1000 , . , . f(x1,x2)=0.95x10.4x20.6max : c1x1+c2x21000; x10, x20.

1. Mathcad. Mathcad. 2. x2=k*x1+b , . c1 := 20 c2 := 10 V := 1000

c1 x1 + c2 x2 V V - c1 x1 c2 3. . . 5 x21(x1) :=

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100 80 60 40 20 0

x21( x1)

0

10

20

30 x1

40

50

60

. 5 x1 x2 , .

4. f(x1,x2)=C (, , , ) . a0 := 0.95 a1 := 0.4 a2 := 0.6 a0 x1
1 a1

C := 37
a2



x2

C

C x22 ( x1 ) := a0 x1

a1



a2

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100 20 80 60

x21( x1) x22 ( x1)

60

40

20

0

0

10

20 x1

30

40

50

5. Format/Graph/X-Y Plot Show markers =37 (20,60). 6. . : x1=20,x2=60

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3. 3.1. . 1. Mathcad 2000 Professional. Mathcad 2000 Professional Windows Windows , , MathSoftApps, Mathcad 2000 Professional. 2. . ( ) . Insert (). Text Region ( ) ( ). ( , ..). . 3. ( ). . , . := Calculator () Shift+:. . 4. . . . 5. Given...Maximize(Minimize). Given, . 6. . , Boolean () Ctrl-=. 7. Given...Maximize(Minimize). , Matrix () Ctrl+=. Insert Matrix Rows () , Columns () -1. , maximize minimize . 8. . - «».

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3.2.

f ( x1 , x2) 2 x1 + 3 x2 x1 + x1 + 2 x1 x1 x2 60 2 x2 40 + x2 50 0



x2 0 x1 := 1 x2 := 0 f ( x1 , x2) := ( 2 x1 + 3 x2)

Given x1 + x1 + 2 x1 x1 x2 60 2 x2 40 + x2 50 0

x2 0

x1 := Maximize( f , x1 , x2) x2 x1 20 = x2 10
f ( x1 , x2) = 70 : x1=20, x2=10.

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3.3.
a0 := 0.95 a1 := 0.4 a2 := 0.6 c1 := 20 c2 := 10 V := 1000

f ( x1 , x2) := a0 x1 x2 x1 := 0 x2 := 0 1000
a1 a2

Given c1 x1 + c2 x2 x1 0 x2 0

x1 := Maximize( f , x1 , x2) x2 x1 20 = x2 60
f ( x1 , x2) = 36.73 : x1=20,x2=60

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4. Word. 1. , . . , , . . . . . . . . . Mathcad. . , , . . . . .

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1. 2. 3. 4. 5. 6.

: . -.: . .. , «», 1998. - 368 . .. . - .: , 1991. -160 . .. - . ­ .: , 1968. ­ 210 . .., .. MathCAD 7.0 , Internet. ­ .: "", 1998. ­ 346 . .. Mathcad 8. ­ .:, 2000. ­ 320 .: . . , . . Mathcad 2000: : . ..: BHV, 2000 - 416 .

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Mathcad

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2000
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2
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. a11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 1 3 2 4 3 2 1 3 2 4 3 1 a12 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 1 2 2 1 3 4 1 2 2 1 3 4 2 a21 3 2 1 3 4 3 2 3 3 2 1 3 4 3 2 3 3 2 1 3 4 3 2 3 3 2 1 3 4 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 a22 2 1 2 4 3 2 1 1 2 1 2 4 3 2 1 2 1 1 2 4 3 2 1 1 2 1 2 4 3 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 a31 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 a32 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 b1 70 75 80 85 90 85 80 75 70 70 75 80 85 90 85 80 75 70 70 75 80 85 90 85 80 75 70 70 75 80 150 155 160 165 170 175 180 185 190 195 200 195 190 185 b2 100 105 110 115 120 125 130 135 140 145 150 155 160 165 175 180 175 165 160 155 150 145 140 135 130 125 100 120 115 110 60 65 70 75 80 85 90 95 100 105 110 115 120 125 b3 45 50 55 60 65 70 75 80 55 60 65 70 75 55 60 65 70 75 80 55 60 65 70 75 80 55 55 60 65 70 90 95 100 105 110 115 120 125 130 135 140 135 130 125 c1 2 2 1 3 2 4 3 2 2 1 3 2 4 3 2 2 1 3 2 4 3 2 2 1 3 2 4 3 2 2 1 3 2 4 3 3 2 2 1 3 2 4 3 2 2 3 1 2 2 1 3 4 3 1 2 2 1 3 4 3 1 2 2 1 3 4 3 1 2 2 1 3 4 3 1 2 2 1 3 4 4 3 1 2 2 1 3 4 3

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45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64. 65. 66. 67. 68. 69. 70. 71. 72. 73. 74. 75. 76. 77. 78. 79. 80. 81. 82. 83. 84. 85. 86. 87. 88. 89. 90. a11 4 3 2 2 1 3 2 4 3 2 2 1 3 2 4 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 a12 3 4 3 1 2 2 1 3 4 3 1 2 2 1 3 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 A21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 A22 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 a31 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4 3 2 2 1 3 2 4 3 2 2 4 3 2 2 1 3 2 4 3 2 2 1 3 2 4 3 2 2 a32 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 4 3 1 2 2 1 3 4 3 1 3 4 3 1 2 2 1 3 4 3 1 2 2 1 3 4 3 1 b1 155 160 165 170 175 180 185 190 195 200 195 190 185 170 175 180 140 145 150 155 160 165 170 175 180 185 190 195 200 190 195 180 185 175 170 165 160 155 150 145 140 185 190 195 200 190 b2 125 120 115 110 105 100 95 90 85 80 75 70 65 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135 140 145 85 90 95 100 105 110 115 120 125 130 135 140 145 95 100 105 b3 120 115 110 105 100 95 90 95 100 105 110 115 120 125 130 135 160 165 170 175 180 185 190 195 200 205 210 165 170 175 180 185 190 195 200 205 210 205 210 165 170 175 180 185 190 195 c1 2 2 1 3 2 4 3 2 2 1 3 2 4 3 2 2 1 3 2 4 3 2 2 1 3 2 4 3 2 2 1 3 2 4 3 2 1 3 2 4 3 2 2 1 3 2 2 3 1 2 2 1 3 4 3 1 2 2 1 3 4 3 1 2 2 1 3 4 3 1 2 2 1 3 4 3 1 2 2 1 3 4 1 2 2 1 3 4 3 1 2 2 1

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3
a
0

a1

a

2

c

1



2

V R

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39

0.6 0.65 0.7 0.75 0.8 0.85 0.5 0.55 0.45 0.4 0.35 0.3 0.6 0.65 0.7 0.75 0.8 0.85 0.5 0.55 0.45 0.4 0.35 0.3 0.6 0.65 0.7 0.75 0.8 0.85 0.5 0.55 0.45 0.4 0.35 0.3 0.8 0.6 0.85

0.3 0.35 0.4 0.45 0.7 0.65 0.6 0.55 0.3 0.35 0.4 0.45 0.7 0.65 0.6 0.55 0.3 0.35 0.4 0.45 0.7 0.65 0.6 0.55 0.3 0.35 0.4 0.45 0.7 0.65 0.6 0.55 0.3 0.35 0.4 0.45 0.7 0.4 0.65

0.7 0.65 0.6 0.55 0.3 0.35 0.4 0.45 0.7 0.65 0.6 0.55 0.3 0.35 0.4 0.45 0.7 0.65 0.6 0.55 0.3 0.35 0.4 0.45 0.7 0.65 0.6 0.55 0.3 0.35 0.4 0.45 0.7 0.65 0.6 0.55 0.3 0.6 0.35

20 30 40 50 60 70 80 40 30 35 40 20 30 35 25 20 15 10 15 10 30 20 30 35 30 20 25 25 20 15 20 10 15 30 25 30 25 40 20

10 20 30 20 30 35 40 80 75 70 65 60 55 50 45 40 35 30 25 20 15 10 70 65 60 55 50 45 40 35 30 25 20 15 10 50 45 65 40

300 350 400 450 500 550 600 650 700 750 800 850 900 950 1000 1050 1100 1200 1250 1300 1350 1400 1450 1500 800 850 900 950 1000 1050 1100 1200 600 650 700 750 800 950 850

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A0 a
1

a

2

c

1



2

V R

40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79

1 0.95 0.75 0.75 0.8 0.85 0.5 0.55 0.3 0.35 0.4 0.5 1 0.95 0.75 0.75 0.8 0.85 0.5 0.55 0.3 0.35 0.4 0.5 0.55 0.3 0.35 0.4 0.5 1 0.95 0.75 0.75 0.8 0.85 0.5 0.55 0.3 0.35 0.4

0.3 0.35 0.4 0.45 0.7 0.65 0.6 0.55 0.7 0.65 0.6 0.55 0.3 0.35 0.4 0.7 0.65 0.6 0.55 0.7 0.65 0.35 0.4 0.7 0.65 0.6 0.55 0.6 0.55 0.7 0.65 0.35 0.4 0.7 0.65 0.35 0.4 0.7 0.65 0.6

0.7 0.65 0.6 0.55 0.3 0.35 0.4 0.45 0.3 0.35 0.4 0.45 0.7 0.65 0.6 0.3 0.35 0.4 0.45 0.3 0.35 0.65 0.6 0.3 0.35 0.4 0.45 0.4 0.45 0.3 0.35 0.65 0.6 0.3 0.35 0.65 0.6 0.3 0.35 0.4

40 30 35 40 20 30 35 25 20 15 10 15 10 20 30 40 50 60 70 80 40 25 25 20 15 20 10 15 30 25 30 25 40 20 20 15 10 15 10 20

80 75 70 65 60 55 50 45 40 35 30 25 20 10 20 30 20 30 35 40 80 50 45 40 35 30 25 20 15 10 50 45 65 40 40 35 30 25 20 10

500 550 600 650 700 750 800 850 900 950 1000 1050 1100 1200 1250 1300 1350 1400 1450 1500 800 850 900 700 750 800 850 900 950 500 550 600 650 700 750 800 850 900 1100 1200

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