Mathcad. Лабораторный практикум

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MATHCAD Ы

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, 2000

2

681.3.06

Mathcad. . . .

, . . : . . .,

©

, 2000

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3

4 1. 1.1 1.2 1.3 2. 2.1 2.2 3.

, 3.1 3.2

4.

5 .................................................................................... 6 ..................................................................................... 6 Mathcad................................................................................ 7 Mathcad 10 Mathcad ............................... 10 Mathcad....................................................................................... 11 11 ................................................................. 11 ............................................................................ 12

, 13 4.1 4.2

............................ 13 ....................................... 15

5.

15 5.1 :................................................................................... 15 5.2 ....................................................................... 15 6. 16 6.1 (Resource Center)..................................... 16 6.2 Quick Sheets................................................................................................ 17 7. 18 7.1 № 1………………………………………………...18 7.2 № 2………………………………………………...20 7.3 № 3………………………………………………...24 7.4 № 4………………………………………………...26 7.5 № 5………………………………………………...29 7.6 № 6………………………………………………...31 7.7 № 7………………………………………………...32 7.8 № 8………………………………………………...38 7.9 № 9………………………………………………...41 7.10 № 10……………………………………………...18 45 59

4

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Overview – Mathcad, Getting Started – Advanced Topic – Quick Sheets Reference Tables – , ♦ Mathcad in Action – , ♦ Web Library – Mathcad. ♦ Collaboratory – Internet -

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1.

1

2.

3.

4.

x. ( z 2 a .b

a .b

2

2)

2

2 3.x 3.x 9.x 1

1

3

2.x 3.x

4 . z. ( x 2 . z )

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2 b .c

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a .c

2

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2

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я

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A*X=B. -1

X=A *B. MathCAD X

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ORIGIN 1 1 2 3 1 3 2

A

7 b

1 1 1

Δ1

7 2 3 Δ1= 3

5 3 2 3 1 1

x1

Δ1

3

Δ = 3

A

1 7 3

Δ2

x1 = 1

Δ

Δ

5

1 5 2

Δ2= 0

1 3 1 Δ2 Δ

x2

x2 = 0

1 2 7

Δ3

1 3 5 1 1 3

x3

Δ3 Δ

Δ3= 6 x3 = 2

(

-

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ORIGIN

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aug-

ment(A,b). 4.

–

rref(Ar).

5.

– trix(Ag,1,3,4,4).

6. -

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subma-

23

ORIGIN 1 1 2 3 A

7

1 3 2

b

1 1 1 Ar

5 3

augment ( A , b )

Ar =

1 2 3 7 1 3 2 5 1 1 1 3

Ag

Ag =

rref( Ar )

1 0 0 1 0 1 0 0 0 0 1 2

x

submatrix( Ag , 1 , 3 , 4 , 4 )

x=

1 0

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0 0

2

1.

0

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0.005 0.004 0.150

1. A

2. A

0.090 0.033 0.0067 0.098 0.0150 0.033 0.050

0.057

0 0

B

0.098 0.041

0.015 0.012 0.250 0.100

0.388

0.070 0.033 0.020 0.075 0.350 0.100 0.075 0.110 0.0086 0.200 0.233 0.075

B

A

0.183

2.857 0.100 0.300 0.025

0.084 1.357 0.149

0.010 0.008 0.200 0.050

6.

7. A

0.080 0

0.013 0.050

0.250 0.067 0.067 0.069

0.186 B

0.126 0.646

0.0057 0.150 0.267 0.050

0.0086

0.020 0.016 0.300 0.150

0.662

0.060 0.067 0.027 0.100 0.450 0.133 0.080 0.139 0.011 0.250 0.200 0.100

B

0.029 2.312 0.379

24 3.

0.025 0.020 0.350 0.200 A

4. A

5. A

8.

1.008

0.050 0.100 0.033 0.125

B

0.550 0.167 0.083 0.161

0.212 0.700

0.035 0.028 0.450 0.300

1.918

0.030 0.167 0.047 0.175

B

0.750 0.233 0.088 0.195

0.788 6.611

0.045 0.036 0.550 0.400

3.117

0.950 0.300 0.090 0.220 0.026 0.500 0.033 0.225

10.664

A

B

0.465 4.940 1.111

0.040 0.032 0.500 0.350

2.481

0.850 0.267 0.089 0.208

B

1.182 8.520

0.023 0.450 0.067 0.200

2.205

0.050 0.040 0.600 0.450

3.825

0.267 0.067 0.250

0

B

1.050 0.333 0.091 0.230 0.029 0.550 0

2.888

я

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1.646

10.

1.427

0.017 0.350 0.133 0.150

0.020 0.200 0.053 0.200

A

1.613

B

0.650 0.200 0.086 0.179

9.

0.020 0.400 0.100 0.175

0.010 0.233 0.060 0.225

0.040 0.133 0.040 0.150

A

3.507

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0.030 0.024 0.400 0.250

0.250

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25 я

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3

F( x) x

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0

i

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2

21

a0

52

a1 . x a0

root ( F( x) , x)

x1 1

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1

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F( x) x x1

x3

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F( x) ,x ( x x1) . ( x x2)

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3.464i

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F(x)=a3*x3+a2*x2+a1*x+a0

№ 7.1.1.1.1

1 2 3 4 5 6 7 8 9

a3 2 34 5 -34 4 6 -8 -10 5

a2 -8 -3 -4 67 4 12 -24 -1 32

a1 15 23 27 -3 45 12 26 84 -34

a0 -30 48 23 11 22 35 -47 -2 8

26 10

7

17 я

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25

48

№4

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Given –

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Given

find find

minerr

minerr ,

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1.

я

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x=10

2.

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3. . 4.

minerr

x

10 Given 2

x

3

x0

Find( x)

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27

x

10 Given 2

x 3 x0 x

Find( x) x0 = 1.732

10 Given 2

x 3 x1

Minerr( x)

x1 = 1.732

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1. 2. 3. 4.

x=0, y=0. . . .

я x

я

я

6 , 5.75 .. 6

x 8

2 3 .x 10

5

40 34 28 22 16 10 4 2 8 14 20

0

x

5

10

28

x

0 y

0

x

Given 2 y x

y 8

3.x

Find( x, y )

x0 y0

x0 = 2.895 2

x

3 x0 y0

y

x< 0

я x0 y0

1.702 2.895

3 . x0 = 2.895

8

0

=

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Find( x, y )

x0 y0

я x> 0

2

y x =

y

y 8

3.x

4.702 22.105

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1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

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№5

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f(x1,x2,…,xn)=c1x1+c2x2+…+cnxn ,

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x>=0, y>=0.

я

1.

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2.

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3.

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30 x

y 5

x

2 .y 4

x

5 1.

y1 ( x )

x

2

x

1.

y2 ( x )

2

2 .x

y 5

2 .x

3 .y c

2

2 .x 2.

5

5

y3 ( x ) 1.

x

3

c

c

3

x

2

2 .x

7

5

y4 ( x )

2.

1.

x

3

c

3

5

y1 ( x ) y2 ( x ) y3 ( x )

4 3 2

y4 ( x ) 1

0

1

2

3

4

5

x

я Given

я

x 2.y 4 2.x y 5 f( x, y )

Find( x, y )

2.x 3.y

я

. x+2=4, 2x+y=5

2 1 f( 2 , 1 )

fmin

fmin = 7 7,

я

x=2,

y=1. fmin=f(2,1)=7.

: f(x,y)=ax+by .

. :

-

31

7.1.1

1 2 3 4 5 6 7 8 9 10

f(x,y)

7.1.1.1.3

3x+2y 3x+2y 3x+2y 3x+2y 3x+2y 2x+5y 2x+5y 2x+5y 2x+5y 2x+5y

x+y=7

x+2y>=5 x+2y>=7 x+2y>=8 x+2y>=7 x+2y>=5 x+2y=3 2x+y=0 x>=0 x>=0 x>=0 x>=0 x>=0

№6 ”

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Sin(x) Cos(x) Exp(x)cos(x) X+cos(x) tgx

10

Sec(x)

f(x) lnx 1+1/x Exp(x) e−x

2

-

MathCAD.

1:

№ 1 2 3 4 5 6 7 8 9

y>=0 y>=0 y>=0 y>=0 y>=0 y>=0 y>=0 y>=0 y>=0 y>=0

F(x,y)

x + y2 = 4 2

x2 − y2 = 4

( x − 3) 2 + y 2 = 4

cos( x) 2 + sin( y ) 2 = 1

(x+y)y=2 Xy+x+y=4 X+cos(xy)=1 x + y2 = 4

( x 3) 2 + y 2 = 4

x2 y2 + =1 3 4

-

32 №7

7.7 ”

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MathCAD

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36

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.

: X := ( 10 12 13.5 15 17 19 20 22 25 30 )

T

Y := ( 10 11.5 12 13 14.7 12 12.4 15 15.8 20 )

N := 9

i := 0 .. N

T

x := X0 , X0 + 0.1 .. XN

y ( x) := linterp( X , Y , x)

y ( 12) = 11.5

20 Yi 15

y( x)

10 10

20

30

Xi , x

. x = N + 0.55 ,

.

y=f(x), у

, N–

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1. Y 2. Y 3. Y 4. Y

1 0.686

1.1 0.742

1.2 0.767

1.3 0.646

1.5 0.774

2 2.312

2.1 2.251

2.2 2.418

2.3 2.752

2.5 2.7

3 4.615

3.1 4.591

3.2 5.13

4 8.472

4.1 8.805

4.2 9.096

3.3 5.481 4.3 8.993

3.5 5.553 4.5 9.312

1.4 0.807 2.4 2.459 3.4 5.492 4.4 9.465

1.6 0.97 2.6 3.022 3.6 5.471 4.6 9.771

1.7 0.932 2.7 3.079

1.8 0.936 2.8 2.42

1.9 0.978 2.9 2.669

3.7 5.727

3.8 5.798

3.9 6.11

4.7 9.61

4.8 9.722

4.9 11.41

2 1.048 3 3.241 4 6.605 5 10.28

37 5. 5 12.36

Y 6.

5.1 13.63

5.2 13.3

5.3 13.15

5.5 13.48

5.4 14.24

5.6 14.52

5.7 14.88

5.8 15.25

5.9 15.34

Y

6 17.63

6.1 19.75

6.2 19.79

6.3 18.81

6.5 19.87

6.4 21.12

6.6 20.21

6.7 19.49

6.8 20.15

6.9 20.51

7 21.29

Y

7 25.24

7.1 25.13

7.2 25.67

7.3 26.63

7.5 26.75

7.4 27.23

7.6 26.49

7.7 26.88

7.8 27.23

7.9 28.07

8 27.79

Y

8 30.53

8.1 34.22

8.2 34.23

8.3 34.11

8.5 33.6

8.4 34.06

8.6 34.5

8.7 35.83

8.8 35.68

8.9 35.44

9 35.67

9 41.74

9.1 42.24

9.2 43.88

9.3 42.17

9.5 43.7

9.4 45.04

9.6 42.46

9.7 45.72

9.8 44.06

9.9 45.87

10 44.95

10 49.76

10.1 51.92

10.2 50.08

10.3 52.38

10.5 53.41

10.4 54.96

10.6 52.77

10.7 54.12

10.8 55.48

10.9 55.69

11 56.2

6 15.16

7.

8.

9. Y 10. Y

-

y = f (x)

(

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у

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b c ; x x2 b c d 5. y = a + + 2 + 3 ; x x x 6. y = a + b ⋅ ln(x);

7. y = a + b ⋅ cos( x) + c ⋅ x 2 ; 8. y =

1 ; a + b ⋅ tg ( x) 2 1 9. y = a + + c ⋅ x2 ; b ⋅ log( x)

10. y = a + b ⋅ sin( x) 2 + c ⋅ x 3 . z = f ( x, y )

-

( –

, .

X, Y z

.

.).

x = X N + 0.55 , y = YN + 0.3 5,

N

38

1. 2. 3. 4. 5. 6. 7. 8.

z = sin( x + y );

z = cos( x + y );

sin( x + y 2 ) − z = 0; z = tg ( x + y );

z = sin( x + 0.5 ⋅ y ) 2 ;

ln( x + y ) − z = 0;

1.5 ⋅ x ⋅ y − ( x 2 + y 2 ) − z = 0;

z = x 2 ⋅ y + sin( x + y ); 3⋅ x ⋅ y 9. 2 2 − z = 0; x +y

10. z = x ⋅ y + cos( x + y ). №8

7.8 ”

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MathCAD. . MathCAD

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39 (Stiffb, Stiffr), (Rkadap).

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⎡ μ ⋅ X0 − X1 − ⎡⎣ ( X0) 2 + ( X1) 2 ⎤⎦ ⋅ X0 ⎢ D( t , X) := ⎢ μ ⋅ X + X − ⎡ ( X )2 + ( X )2 ⎤ ⋅ X 1 1 ⎦ 1 0 ⎣ 0 ⎣ Z := rkfixed( X , 0 , 20, 100, D)

n := 0 .. 99

В

⎤ ⎥ ⎥ ⎦ я

1

Zn ,1 Zn ,2

0

1

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20

40

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100

40

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[ x0 , xN ]

-

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h/2.

-

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№ 1 2 3 4 5 6 7

F ( x, y, y′) = 0

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(e x + 1)dy + e x dx = 0 y ln y + xy ′ = 0

y (1) = e y (0) = −tg 2

4 − x 2 y ′ + xy 2 + x = 0

y (1) = arctg (2 − e)

2 − ex dx = 0 cos 2 x (1 + e x ) yy ′ = e x y ′ sin x = y ln y

3e x tgydx +

y (0) = 1

⎛π ⎞ y⎜ ⎟ = e ⎝2⎠ y (1) = 1

xdx ydy − =0 1+ y 1+ x

8

(1 + y 2 )dx = xdy

9

y ′ sin x = sin y

10

3( x 2 y + y )dy + 2 + y 2 dx = 0

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y1′ = f1 ( x, y1 , y 2 ) , y 2′ = f 2 ( x, y1 , y 2 ) , y1 (a) = y1,0 , y2 (a) = y2, 0

-

[ a , b]

,

№ 1 2 3 4 5 6 7 8

f1(x,y1,y2) x+y1 siny2 xcos(y1+y2) siny1cos2y2 cos(y1y2) y2lnx 2y1/y2 y1+y2

f2(x,y1,y2) (y1-y2)2 cosy1 sin(y1-y2) cosy1cosy2 sin(y1+y2) y1+y2 2y1-y2 1/(1+ y1+y2)

h=0.1. h, 2h

h/2.

y1(a) 0 0.5 -0.6 0 0 -2 1 0

y2(a) 1 -0.5 2 0 0 -1 1 0

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b 1 3 5 3 2 4 3 4

41 9 10

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M := 25 N := 25 i := 0 .. M j := 0 .. N Xi , j := a + ( b − a) ⋅

i M

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(

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j N

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( X , Y , Z)

. 1. 2. 3. 4. 5.

Z Z Z Z Z

= e −(X +Y ) = X 2 −Y 2 = X 2 + Y 2 − XY = ( X + Y )2 = ( X − Y )2

6. Z =

2

2

X 2 +Y2

7. Z = e 2 2 15. Z = tg ( X π−Y ) X 2 +Y 2

8. Z 9. Z 10. Z 11. Z 12. Z 13. Z

= 3 X 2 +Y 2 = XY = e XY = e − XY = e − ( X +Y ) = cos( XY 2π )

14. Z = sin( X

+Y 2

π

2

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42

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7.10

№ 10

«

»

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:

Z R Y X

F ( X ,Y , Z ) = 0 , .

α

0

M := 25 N := 50 i := 0 .. M j := 0 .. N α i :=

π⋅ i

φj :=

Xi , j := tan ( α i) ⋅ cos ( φj) M

2⋅ π⋅ j

Yi , j := tan ( α i) ⋅ sin ( φj) N

π,

Zi , j := tan ( α i)

α ,φ , R , φ

0

X,Y,Z,

-

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2

( X , Y , Z)

я я

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43 Y = r sin(φ ) X = rсos(φ ) Z =Z. Z h R

r

Y

M := 25 N := 50 i := 0 .. M

j := 0 .. N

X

Xi , j := Ri , j ⋅ cos ( φi , j)

r := 5 φi , j :=

Yi , j := Ri , j ⋅ sin ( φi , j)

2⋅ π⋅ j N

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i M

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( X , Y , Z)

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48 cfft(A) –

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50 ,

intervals, .

data,

I0(x) – . I1( ) – . icfft(A) –

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(i, j)-n

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51

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Mz. -

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53 ( )– ( ) – 1,

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