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Hunh Hu Dinh Trng i Hc Cng Nghip TPHCM

BI TP PHNG PHP TNH

Bi tp chng 1Bi tp 1. Tm mt on li nghim ca cc phng trnh sau:

1) x5 x 30 = 02) e2x x2 20 = 03) x2 ln(x+ 1) 30 = 04) x(x+ 1)4 + x(x 2) 50 = 05) x2 sin 3x 40 = 06) 3

x+1+ 4

(x+1)2+ 5

(x+1)3+ 6

(x+1)4= 10

7) x2 +x+ 5 50 = 0

8) ln(x2 + 1) + x 30 = 09) x3 + arctan x 40 = 010) 1

x+ cos x 10 = 0

Bi tp 2. Gii gn ng cc phng trnh trong Bi tp 1 bng phngphp lp (lp 4 bc, nh gi sai s bc 4, ly 7 ch s c ngha), bitbc lp ban u c chn l trung im on li nghim.

Bi tp 3. (*) Cho phng trnh x3 sin x 30 = 0 (*) c LN [3; 4].a) Tm t nht hai hm (x) sao cho phng trnh (*) tng ng

vi phng trnh x = (x), ng thi maxx[3;4]

| (x)| < 1.

b) Vi cc hm (x) tm c trong cu a), hy gii gn ng phngtrnh (*) bng phng php lp sao cho sai s khng qu 105, bit bclp ban u x0 = 3, 5. Vi kt qu tm c, hy nhn xt hm (x) nocho nghim gn ng tt hn.

Bi tp 4. (*) Vic tnh gn ngA vi A l s nguyn dng khng

chnh phng c th thc hin bng phng php lp. Chng hn, tacho A = 2.

t x =2, suy ra x2 2 = 0 (*). D dng thy rng phng trnh

(*) c LN l [1; 2], bin i phng trnh (*) v dng x = x2+ 1

x= (x).

1) Chng minh rng maxx[1;2]

| (x)| < 1.

2) Phi lp t nht bao nhiu bc th ta c nghim gn ng csai s b hn 107 nu ta ly x0 = 1, 5.

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3) Vi A Z>2, chng minh rng phng trnh x2 = A (**) c LNl

[[A];[

A]+ 1

].

4) Bin i phng trnh (**) v dng x = x2+ A

2x= A(x). Chng

minh rng maxx[[

A];[

A]+1]

|A (x)| < 1.

Bi tp 5. Cho phng trnh f(x) = 0 c LN [a; b], chng minh rngf (x) v f (x) khng i du trong [a; b] bit:

1) f(x) = ex2 5x 30 v [a; b] [1; 2].2) f(x) = x2 sin x 50 v [a; b] [7; 8].3) f(x) = x+ ln(x+ 2) 10 v [a; b] [7; 8].4) f(x) = 2

x+1+ 3

(x+1)2+ 4

(x+1)3 7 v [a; b] [0; 1].

5) f(x) = x3 + arctan x 30 v [a; b] [3; 4].6) f(x) = x(x+ 1)5 + x(x+ 2) 50 v [a; b] [1; 2].7) f(x) = 3x sin 2x 30 v [a; b] [3; 4].8) f(x) = 2x + 3x 10x 30 v [a; b] [3; 4].9) f(x) = 1

x+ x2 40 v [a; b] [6; 7].

10) f(x) = x2 +x+ 1 40 v [a; b] [6; 7].

Bi tp 6. Gii gn ng cc phng trnh trong Bi tp 5 bng phngphp Newton (lp 3 bc, nh gi sai s bc 3, ly 7 ch s c ngha).

Bi tp chng 2Bi tp 7. Tm chun ca cc vector sau:

1) X = (4; 5; 9)T .2) Y = (2; sin + cos; tan + cot)T vi (0;

2).

3) Z =(2a; a2 + 1;2a;

2 (a4 + 2)

)vi a R.

4) U = (ea + ea; 2 + a2) vi a R>0.Bi tp 8. Tm chun ca cc ma trn sau:

1) A =

0, 1 0, 3 0, 20, 4 0 0, 20, 5 0, 3 0, 5

.2) B =

sin cos 12 1 2 01 0 1

vi (0; 2)

3) C =

0, 1 1 2 12 0, 05 1 0, 80, 8 0, 7 2, 1 0, 22, 4 0, 2 1 0, 2

.2


4) D =(

ea ea

2 a

)vi a R1.

Bi tp 9. Tm x R A < 1 bit A =

0, 1 0, 4 xx 0, 5x 0, 20, 2 0, 1 0, 3

.Bi tp 10. Cho h phng trnh Ax = b c nghim ng l X vnghim xp x X. Hy tnh

X X v AX b.1)

{12x1 +

13x2 =

163

13x1 +

14x2 =

1168

v X = (17;1

6)T ; X = (0, 142;0, 166)T .

2)

x1 + 2x2 + 3x3 = 12x1 + 3x2 + 4x3 = 13x1 + 4x2 + 6x3 = 2

vX = (0;7; 5)T ; X = (0, 01;6, 98; 5, 02)T .

3)

x1 + 0, 1x1 + 0, 3x3 = 1, 40, 1x1 + x2 + 0, 6x3 = 1, 70, 2x1 + 0, 3x2 + x3 = 1, 5

vX = (1, 1, 1)T ; X = (1, 01; 0, 98; 0, 95)T .

Bi tp 11. Gii cc h phng trnh sau y bng phng php lpn (lp 3 bc, nh gi sai s bc 3, ly 7 ch s c ngha):

1)

10x1 + 2x2 + 3x3 = 182x1 + 25x2 + 5x3 = 37x1 + 4x2 + 20x3 = 45

.

2)

3x1 + 20x2 x3 = 238x1 + 3x2 + 40x3 = 1125x1 + 2x2 + x3 = 27

.

3)

10x1 + 2x2 + x3 x4 = 13x1 + 20x2 + 2x3 + x4 = 232x1 + x2 + 25x3 + 2x4 = 282x1 + x2 + 3x3 + 40x4 = 6

.

4)

x1 + x2 + 2x3 + 25x4 = 2710x1 + 50x2 + x3 x4 = 59x1 + 3x2 + 40x3 + x4 = 540x1 + x2 + 3x3 x4 = 40

.

Bi tp 12. Cho h phng trnhx 0, 1y + 0, 3z = 0, 90, 2x+ y + 0, 3z = 1, 20, 1x+ 0, 2y + z = 0, 3

(1)

Nu s dng phng php lp n gii gn ng h phng trnh(1) th ta phi lp t nht bao nhiu bc c s khng qu 105.

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Bi tp 13. Gii cc h phng trnh Bi tp 11 bng phng phpSeidel (lp 3 bc, nh gi sai s bc 3, ly 7 ch s c ngha).

Bi tp 14. Cho h phng trnhx1 0, 01x2 0, 02x3 + 0, 05x4 = 10, 03x1 + x2 0, 01x3 + 0, 1x4 = 10, 04x1 + 0, 05x2 + x3 + 0, 04x4 = 20, 01x1 + 0, 02x2 0, 07x3 + x4 = 2

(2)

Nu s dng phng php lp n gii gn ng h phng trnh(2) th ta phi lp t nht bao nhiu bc c s khng qu 105.

Bi tp chng 3Bi tp 15. Xy dng a thc ni suy dng Lagrange ca hm s y =f(x) cho bi bng s liu:

a) x 0 2 3y 2 4 10

b) x 1, 1 1, 2 1, 3y 0 4 8

c) x 1 2 3 5y 1 3 6 8

d) x 1, 0 1, 2 1, 4 1, 6y 2 3 4 5

Bi tp 16. Xy dng a thc ni suy dng Newton ca hm s y = f(x)cho bi bng s liu trong Bi tp 15.

Bi tp 17. Cho hm s f(x) =x+ 2 c gi tr ti cc mc x0 = 0; x1 =

1;x2 = 2; x3 = 4 c cho bi bng sau:

x 0 1 2 4y 1, 4142 1, 7321 2 2, 2361

1) Xy dng a thc ni suy dng Newton ca hm s f(x) cho bibng s liu trn.

2) Tnh gn ng f(3) (ly 5 ch s c ngha) v nh gi sai s ktqu tm c.

Bi tp 18. Cho hm s f(x) = cos x c gi tr ti cc mc x0 =0, 698;x1 = 0, 733; x2 = 0, 768; x3 = 0, 803 c cho bi bng sau:

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x 0, 698 0, 733 0, 768 0, 803y 0, 7661 0, 7432 0, 7193 0, 6946


2) Tnh gn ng f(0, 750) (ly 5 ch s c ngha) v nh gi sai skt qu tm c.

Bi tp 19. Hm li (error function) c cho bi cng thc

erf (x) =2

x0

et2

dt.

Gi tr ca hm erf(x) ti cc mc xi = 0, 2; i = 0, 5 c cho bi bng

x 0, 0 0, 2 0, 4 0, 6 0, 8 1, 0y 0 0, 1256 0, 2417 0, 3407 0, 4187 0, 4754


2) Tnh gn ng f(0, 750) (ly 5 ch s c ngha) v nh gi sai skt qu tm c.

Bi tp 20. Dng phng php bnh phng b nht xc nh a, b ng thng y = ax+ b l xp x tt nht bng s liu trong Bi tp 15.

Bi tp 21. Dng phng php bnh phng b nht xc nh a, b hm s y = ax + b sinx l xp x tt nht bng s liu trong Bi tp 15(ch cn chn 2 bng).

Bi tp 22. Dng phng php bnh phng b nht xc nh a, b hm s y = aebx l xp x tt nht bng s liu trong Bi tp 15 (ch cnchn 2 bng).

Bi tp 23. Dng phng php bnh phng b nht xc nh a, b hm s y = axb l xp x tt nht bng s liu trong Bi tp 15 (ch cnchn 2 bng).

Bi tp 24. Dng phng php bnh phng b nht xc nh a, b hm s y = a(x2 + 1) + b

x+ 1 l xp x tt nht bng s liu trong Bi

tp 15 (ch cn chn 2 bng).

Bi tp 25. Dng phng php bnh phng b nht xc nh a, b hm s y = a

x2 + 1 + b(x+ 1) + 1 l xp x tt nht bng s liu trong

Bi tp 15 (ch cn chn 2 bng).

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Bi tp 26. Dng phng php bnh phng b nht xc nh a, b hm s y = ax2 + bx+ c l xp x tt nht bng s liu trong Bi tp 15(ch cn chn 2 bng).

Bi tp 27. Dng phng php bnh phng b nht xc nh a, b hm s y = a(x2 + 1) + b(x+ 1) + c l xp x tt nht bng s liu trongBi tp 15 (ch cn chn 2 bng).

Bi tp 28. tui trung bnh ln u kt hn ca ph n Nht Bnt nm 1950 ti nm 2000 c cho bi bng s liu sau:

t f(t) t f(t)1950 23, 0 1980 25, 21955 23, 2 1985 25, 51960 23, 7 1990 26, 11965 24, 1 1995 26, 31970 24, 5 2005 27, 11975 24, 9

S dng phng php bnh phng b nht, xc nh a, b ngthng y = ax + b l xp x tt nht bng gi tr trn. Vi kt qu tmc, hy d on tui kt hn ln u ca ph n Nht vo nm2005.

Bi tp 29. Bng s liu sau y cho ta bit s dn nc M (triungi) trong khong thi gian t nm 1900 ti nm 2000:

t f(t) t f(t)1900 76 1960 2051910 89 1970 2151920 110 1980 2301930 131 1990 2551940 150 2000 2751950 190

S dng phng php bnh phng b nht, xc nh a, b ngthng y = ax + b l xp x tt nht bng gi tr trn. Vi kt qu tmc, hy d on dn s nc M vo nm 2010.

Bi tp chng 4Bi tp 30. Tnh gn ng cc tch phn sau bng cng thc hnhthang suy rng (khng nh gi sai s):

6


1)21

ln (x2 + 5) + x3

x ln (2x+ 1)dx vi n = 8

2)1,40,4

x4 x+ 1x2 + 4 + 5

dx vi n = 10.

3)2

0,4

(x ln (x+ 2) +

1

x3 + 1

)dx vi n = 8.

4)2,31,3

(x2 + 2 +

x

ln (x+ 1)

)dx vi n = 10.

5)2,21,4

ex2+1dx vi n = 8.

6)21

xx2dx vi n = 10.

7)32

3x2 + 1

x4 + 3dx vi n = 10.

Bi tp 31. Tnh gn ng cc tch phn sau bng cng thc hnhthang suy rng (c nh gi sai s):

1)10

x4dx vi n = 10.

2)32

x3

x+ 1dx vi n = 8.

3)10

ex2dx vi n = 10.

4)31

e2x

ex + 1dx vi n = 10.

5)41

2xdx vi n = 10.

6)21

sin 2x

xdx vi n = 10.

7)32

3x+ 1

x+ 3dx vi n = 10.

Bi tp 32. Tnh gn ng cc tch phn trong Bi tp 30 bng cngthc Simpson 1/3 (khng nh gi sai s).

Bi tp 33. Tnh gn ng cc tch phn trong Bi tp 31 bng cngthc Simpson 1/3 (c nh gi sai s).

7


Bi tp 34. Error function (hm li) E (x) = 2

x0

et2dt c nhiu ng

dng trong xc sut, thng k v k thut. Bng cch s dng cngthc Simpson 1/3 vi s on chia n = 10, cc bn hy tnh gn ngE (2) (khng nh gi sai s).

Bi tp 35. Hm Fresnel S (x) =x0

sin(

t2

2

)dt xut hin ln u trong

l thuyt v nhiu x nh sng ca nh ton hc ngi Php AugustinFresnel (1788-1827). Gn y, hm Fresnel xut hin trong cc cngtrnh thit k ng quc l. Bng cch s dng cng thc Simpson 1/3vi s on chia n = 10, cc bn hy tnh gn ng S (1) (khng nhgi sai s).

Bi tp 36. Xt tch phn I =21

4x2 + 1

2x+ 1dx.

1) Tnh tch phn I bng cng thc hnh thang vi n = 10 v nhgi sai s kt qu trn.

2) Phi chia [1; 2] thnh bao nhiu on bng nhau khi p dngcng thc hnh thang trn s on th sai s khng qu 1010.

Bi tp 37. Xt tch phn I =32

x3 + x

x+ 1dx.

1) Tnh tch phn I bng cng thc Simson 1/3 vi n = 10 v nhgi sai s kt qu trn.

2) Phi chia [2; 3] thnh bao nhiu on bng nhau khi p dngcng thc Simson 1/3 trn s on th sai s khng qu 1010.

Bi tp chng 5Bi tp 38. Gii cc phng trnh vi phn sau bng phng php Eulerci tin:

1){

y = x+ yy (0) = 1

vi x [0; 0, 5];h = 0, 25; = 104.

2){

y = x2 + y2

y (0) = 1vi x [0; 0, 4];h = 0, 2; = 104.

3){

y = xy1y2+1

y (0) = 1vi x [0; 0, 2];h = 0, 1; = 104.

4){

y = xy cos(x+ y)y (0, 2) = 1

vi x [0, 2; 0, 4];h = 0, 1; = 104.

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5){

y =x+ y + 1

y (0, 3) = 1vi x [0, 3; 0, 5];h = 0, 1; = 104.

Bi tp 39. Gii cc phng trnh vi phn sau bng phng phpRunge-Kutta bc 4:

1){

y = x+ yy (0) = 1

vi x [0; 0, 5];h = 0, 25.

2){

y = x2 + y2

y (0) = 1vi x [0; 0, 4];h = 0, 2.

3){

y = xy1y2+1

y (0) = 1vi x [0; 0, 2];h = 0, 1.

4){

y = xy cos(x+ y)y (0, 2) = 1

vi x [0, 2; 0, 4];h = 0, 1.

5){

y =x+ y + 1

y (0, 3) = 1vi x [0, 3; 0, 5];h = 0, 1.

Bi tp 40. Cho phng trnh vi phn{

y = 2x+ yy (0) = 1

vi x [0; 0, 3].

Tnh y(0, 15) bng cng thc Euler ci tin vi h = 0, 15, ci tin 3 bc.


y = 2x+ cos yy (0, 2) = 1

vi x [0, 2; 0, 4].

Tnh y(0, 3) bng cng thc Euler ci tin vi h = 0, 1, sai s = 104.


y = xy + x+ y2

y (0) = 1vi x

[0; 0, 3]. Tnh y(0, 15) bng cng thc Runge-Kutta vi h = 0, 15.


y = x2y + y2x+ 2y (0, 2) = 1

vi x

[0, 2; 0, 4]. Tnh y(0, 3) bng cng thc Runge-Kutta vi h = 0, 1.

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