Marshak_Elektron_tabl

>ey mlhqg_ggy dhj_g•\ jhajh[e_gh [Z]Zlh qbk_evgbo f_lh^•\ <hgb jhaih-

^•eyxlvky gZ ijyf• lZ •l_jZp•cg•4.

<ijyfbo f_lh^Zo agZclb rmdZgbc jha\¶yahd fh`gZ aZ i_\gm d•evd•klv djhd•\ aZ aZ^Zgbfb nhjfmeZfb

<•l_jZp•cgbo f_lh^Zo dh`g_ gZklmig_ ih\lhj_ggy hi_jZp•€ h[qbke_ggy aZ •l_jZp•cghx nhjfmehx ^Z} fh`eb\•klv hljbfZlb [•evr [ebavd_ ^h rmdZ- gh]h dhj_gy ch]h gZ[eb`_g_ agZq_ggy >hkblv qZklh p• f_lh^b gZab\Zxlv f_-

lh^Zfb ihke•^h\gbo gZ[eb`_gv.

H[qbke_ggy gZ[eb`_gbo agZq_gv dhj_gy j•\gyggy ke•^ jh[blb ^h lbo i•j ihdb g_ i_j_klZgmlv af•gx\Zlbky l• ^_kyldh\• agZdb yd• fb ohq_fh a[_j_]lb m \•^ih\•^• lh[lh ihdb g_ [m^_ ^hky]gmlZ aZ^ZgZ lhqg•klv Lh^• dZ-

`mlv sh agZcreb jha\¶yahd a i_\ghx lhqg•klx Lhqg•klv aZ^Zxlv \ mfh\• aZ^Zq• Z[h \hgZ \•^hfZ •a [m^v-ydbo f•jdm\Zgv Z[h €€ l_` rmdZxlv

F_lh^ ^•e_ggy \•^j•adm gZ\i•e (^bohlhf•€ GZcijhkl•rbf a f_lh^•\

mlhqg_ggy dhj_g•\ } f_lh^ ^•e_ggy \•^j•adm gZ\i•e Z[h f_lh^ ^bohlhf•€,

ydbc ijbagZq_gbc ^ey agZoh^`_ggy dhj_g•\ j•\gygv yd• ij_^klZ\e_g• m \b]ey-

^• f (x) = 0.

G_oZc g_i_j_j\gZ • fhghlhggZ nmgdp•y f (x) gZ d•gpyo \•^j•adm >a, b@ fZ}

agZq_ggy j•agbo agZd•\ lh[lh f (a) f(b) < jbk lh^• gZ \•^j•adm •kgm} l•evdb h^bg dhj•gv <•avf_fh k_j_^bgm \•^j•adm c = (a + b Ydsh f (a) f(k) < 0,

lh dhj•gv agZoh^blvky gZ \•^j•adm >a, c@ lZ m •grhfm \biZ^dm gZ >c, b].

Jbk 3

Lhfm [_j_fh lhc \•^j•ahd gZ ydhfm } dhj•gv h[qbkex}fh agZq_ggy nmgdp•€ \ ch]h k_j_^bg• a ^\ho \•^j•ad•\ agh\m \b[bjZ}fh \•^j•ahd gZ ydhfm

} dhj•gv • l ^ ^h lbo i•j ihdb ^h\`bgZ q_j]h\h]h \•^j•adZ g_ klZg_ f_grhx aZ ihi_j_^gvh \dZaZgm lhqg•klv E − D < ε .

Hkd•evdb dh`g_ ^•e_ggy \•^j•adm gZ\i•e lZ \b[•j ^ey ih^Zevrbo

h[qbke_gv h^g•}€ a ch]h ^\ho qZklbg a\m`m} •gl_j\Ze ihrmdm \^\•q• lh ijb ihqZldh\hfm \•^j•adm >a, b@ • lhqghkl• ε d•evd•klv h[qbke_gv n \bagZqZ}lvky

4 <•^ eZlbgkvdh]h ''iteracio'' –ih\lhj_ggy

21

mfh\hx (b - a) / 2n < ε Z[h Q ≈ ORJ E − D ε GZijbdeZ^ ijb ihqZldh\hfm

h^bgbqghfm •gl_j\Ze• lZ lhqg•klx ihjy^dm agZd•\ ε ≈ 10 - 6 i•key ^_kyldh\h€ lhqdb ^hklZlgvh ijh\_klb h[qbke_gv •l_jZp•c agZq_gv nmgdp•€

A lhqdb ahjm fZrbggh€ j_Ze•aZp•€ p_c f_lh^ gZc[•evr ijhklbc lZ \bdhjbklh\m}lvky m [Z]Zlvho klZg^Zjlgbo ijh]jZfgbo aZkh[Zo ohqZ •kgmxlv • [•evr _n_dlb\g• ih aZljZlZo qZkm f_lh^b

IjbdeZ^ 6. Mlhqgblb h^bg a dhj_g•\ j•\gyggy x2 - cos x f_lh^hf ^•e_ggy \•^j•adZ gZ\i•e^bohlhf•€ a lhqg•klx ε = 10 - 4.

Jha\¶yahd M IjbdeZ^• 5 [meb agZc^_g• •gl_j\Zeb •aheyp•€ dhj_gy >ey mlhqg_ggy ^h^Zlgh]h dhj_gy j•\gyggy x2 - cos x gZ •gl_j\Ze• •aheyp•€ dhj_gy > @ f_lh^hf ^•e_ggy \•^j•adm gZ\i•e a lhqg•klx ε = 10 - 4 aZ ^hihfh]hx _e_dljhggbo lZ[ebpv g_h[o•^gh aZih\gblb dhf•jdb lZ[ebqgh]h ijhp_khjZ gZklmigbf qbghf ihke•^h\g•klv hi_jZp•c h^gZdh\Z • ^ey

MS Excel • ^ey OO Calc):

< dhf•jdm B20 \\_klb lhqg•klv agZoh^`_ggy dhj_gy

AZih\gblb rZidm lZ[ebp• yd ihdZaZgh gZ jbk .

Jbk 4

<dhf•jdm A23 – i_j\•kg_ agZq_ggy a = 0,8.

<dhf•jdm B23 – i_j\•kg_ agZq_ggy b = 0,9.

Ihl•f g_h[o•^gh kdhi•x\Zlb nhjfmeb ^•ZiZahgm dhf•jhd & ) \ ^•ZiZahg

C24:F24.

>Ze• ihlj•[gh \b^•eblb ^•ZiZahg :) lZ ijhly]gmlb aZ fZjd_j aZiheg_gby ^h lbo i•j ihdb g_ [m^_ agZc^_gh dhj•gv jbk

22

[ = D I E − E I D . I E − I D

:gZe•amxqb agZd f (x m ihj•\gygg• a• agZdhf nmgdp•€ gZ d•gpyo \•^-

j•adm a\m`m}fh •gl_j\Ze ^h >a, x@ Z[h >x, b@ Ydsh agZq_ggy f (a lZ f (x) fZxlv j•ag• agZdb lh dhj•gv agZoh^blvky gZ \•^j•adm >a, x@ Lh^• [ = E ,

H[qbke_ggy aZ gZ\_^_gbfb nhjfmeZfb ijh^h\`m}lvky ^h lbo i•j ihdb g_ h^_j`bfh gZ[eb`_gbc dhj•gv a aZ^Zghx lhqg•klx

>ey hp•gdb ihob[db f_lh^m ohj^ fh`gZ dhjbklm\Zlbky nhjfmehx

P

^_ P £ PLQ _ I ¢ [ _ ijb D ≤ [ ≤ E , e í aZ^ZgZ lhqg•klv

>ey ijZdlbqgbo h[qbke_gv mfh\m aZ\_jr_ggy ijhp_km mlhqg_ggy dhj_gy j•\gyggy ajmqg•r_ aZibkZlb m \b]ey^•

< ydhkl• m fh`gZ \aylb gZcf_gr_ agZq_ggy fh^mey i_jrh€ iho•^gh€ nmgdp•€ f (x gZ ijhf•`dm >a, b].

IjbdeZ^ 7. Mlhqgblb h^bg a dhj_g•\ j•\gyggy x2 - cos x f_lh^hf ohj^ a lhqg•klx ε = 10 - 4.

Jha\¶yahd M IjbdeZ^• 5 [meb agZc^_g• •gl_j\Zeb •aheyp•€ dhj_gy >ey mlhqg_ggy ^h^Zlgh]h dhj_gy j•\gyggy x 2 − cos x = 0 gZ •gl_j\Ze• •aheyp•€ dhj_gy [0,8; 0,9] f_lh^hf ohj^ a lhqg•klx ε = 10 - 4 aZ ^hihfh]hx _e_dljhggbo lZ[ebpv g_h[o•^gh aZih\gblb dhf•jdb lZ[ebqgh]h ijhp_khjZ gZklmigbf qbghf ihke•^h\g•klv hi_jZp•c h^gZdh\Z • ^ey MS Excel •

^ey OO Calc):

< dhf•jdm B19 \\_klb lhqg•klv agZoh^`_ggy dhj_gy

AZih\gblb rZidm lZ[ebp• jbk

Jbk 7

<dhf•jdm A22 \\_klb i_j\•kg_ agZq_ggy a = 0,8.

<dhf•jdm B22 – i_j\•kg_ agZq_ggy b = 0,9.

24

< dhf•jdm C22 – nhjfmem ydZ h[qbkex} agZq_ggy nmgdp•€ m lhqp• a f (a) = A22^2- -COS(A22).

< dhf•jdm D22 – nhjfmem ydZ h[qbkex} agZq_ggy nmgdp•€ m lhqp• b f (b) = B22^2- -COS(B22).

< dhf•jdm ? – nhjfmem h[qbke_ggy q_j]h\h]h gZ[eb`_ggy ^h dhj_gy x =

=(A22*D22-B22*C22)/(D22-C22).

<dhf•jdm ) – nhjfmem ydZ h[qbkex} agZq_ggy nmgdp•€ m lhqp• x f (x) = E22^2-

- COS(E22).

Ijhp_k mlhqg_ggy dhj_gy ke•^ aZ\_jrblb lh^• dheb mfh\Z [m^_ \bdhgZgZ >ey pvh]h h[qbkebfh m = min | f’ (x) | ijb a ≤ x ≤ b. Nmgdp•y gZ ijhf•`dm >a, b@ fhghlhggZ

lhfm fh`gZ h[qbkeblb agZq_ggy fh^mey i_jrh€ iho•^gh€ nmgdp•y gZ d•gpyo •gl_j\Zem •aheyp•€ | f’ (a) | lZ | f’ (b) | lZ h[jZlb gZcf_gr_ agZq_ggy

AgZc^_fh i_jrm iho•^gm aZ^Zgh€ nmgdp•€ f’ (x) = 2x + sin x.

AgZc^_fh agZq_ggy i_jrh€ iho•^gh€ gZ d•gpyo •gl_j\Zem •aheyp•€ jbk

<\_^_fh m dhf•jdm ? nhjfmem $%6$6,1$ m dhf•jdm ? í $%6 % 6,1< >ey h[qbke_ggy m = min | f’ (x) | m dhf•jdm G \\_^_fh

^ey MS Excel: =MBG E18:E19); ^ey OO Calc: =MIN(E18:E19).

Jbk 2.8

>Ze• \\h^bfh \ dhf•jdm G22 – nhjfmem ^ey i_j_\•jdb \bdhgZggy g_j•\ghkl• I D I [ < =C22*F22.

Ydsh agZq_ggy f (a lZ f (x fZxlv j•ag• agZdb lh dhj•gv agZoh^blvky gZ \•^j•adm [a, x], •gZdr_ gZ \•^j•adm [x, b]. LZdbf qbghf ihlj•[gh \\_klb \ dhf•jdm A nhjfmem h[qbke_ggy agZq_ggy a

^ey MS Excel: ?KEB G22<0;A22;E22) ^ey OO Calc: =IF(G22<0;A22;E22).

< dhf•jdm B23 – nhjfmem h[qbke_ggy agZq_ggy b

^ey MS Excel: ?KEB G22<0;E22;B22) ^ey OO Calc: =IF(G22<0;E22;B22).

< dhf•jdm H \\_klb nhjfmem ^ey \b\_^_ggy agZq_ggy dhj_gy –

^ey MS Excel: ?KEB ABS(F22) % *Dhj•gv [ HDJM=E(

^ey OO Calc: =IF(ABS(F22) % *Dhj•gv [ 5281' (

Ke•^ kdhi•x\Zlb \f•kl dhf•jhd K 2:G ^h ^•ZiZahgm K G

Ihl•f ihlj•[gh \b^•eblb ^•ZiZahg : 3:F23 lZ ijhly]gmlb aZ fZjd_j aZiheg_gby ^h lbo i•j ihdb g_ [m^_ agZc^_gh dhj•gv jbk

25

Ydsh iho•^gZ nmgdp•€ fZeh af•gx}lvky \ hdhe• dhj_gy lh fh`gZ aZklhkh\m\Zlb kijhs_gm nhjfmem ^ey f_lh^Z

1kgmxlv lZdh` •gr• fh^bn•dZp•€ f_lh^Z GvxlhgZ

>ey hp•gdb ihob[db f_lh^Z ^hlbqgbo lZdh` fh`gZ aZklhkm\Zlb nhjfmem

Lh^• mfh\Z aZ\_jr_ggy •l_jZp•cgh]h ijhp_km mlhqg_ggy dhj_gy fZ} \b]ey^

_ I [Q _ £ e Z[h _ I [Q _£ e × P ,

P

^_ m £ min | f’ (x) | ijb a £ x £ b, e í aZ^ZgZ lhqg•klv

IjbdeZ^ 8 Mlhqgblb h^bg a dhj_g•\ j•\gyggy x2 – cos x = 0 f_lh^hf ^hlbqgbo a lhqg•klx

ε = 10 - 4.

Jha\¶yahd M IjbdeZ^• [meb agZc^_g• •gl_j\Zeb •aheyp•€ dhj_gy >ey mlhqg_ggy ^h^Zlgh]h dhj_gy j•\gyggy x2 – cos x = 0 gZ •gl_j\Ze• •aheyp•€ dhj_gy f_lh^hf ^hlbqgbo a lhqg•klx ε = 10 - 4 aZ ^hihfh]hx _e_dljhggbo lZ[ebpv g_h[o•^gh aZih\gblb dhf•jdb lZ[-

ebqgh]h ijhp_khjZ gZklmigbf qbghf ihke•^h\g•klv hi_jZp•c h^gZdh\Z • ^ey MS Excel • ^ey

OOCalc).

<dhf•jdm B19 \\_klb lhqg•klv agZoh^`_ggy dhj_gy

>ey \bagZq_ggy gmevh\h]h gZ[eb`_ggy g_h[o•^gh h[qbkeblb f (x) gZ h^ghfm a d•gp•\ •gl_j\Zem •aheyp•€ dhj_gy gZijbdeZ^ f (a) lZ €€ ^jm]m iho•^gm f’ (a). >ey pvh]h aZih\gbfh dhf•jdb yd ihdZaZgh gZ jbk :

Jbk. 2.13

D21=B21^2-COS(B21); F21=2+COS(B21); H21=D21*F21.

Ijhp_k mlhqg_ggy dhj_gy ke•^ aZ\_jrblb lh^• dheb mfh\Z [m^_ \bdhgZgZ >ey pvh- ]h h[qbkebfh m ≤ min | f’ (x) | ijb a ≤ x ≤ b. Nmgdp•y gZ ijhf•`dm >a, b@ fhghlhggZ lhfm fh`- gZ h[qbkeblb agZq_ggy fh^mey i_jrh€ iho•^gh€ nmgdp•y gZ d•gpyo •gl_j\Zem •aheyp•€ | f’ (a) | lZ | f’ (b) | lZ h[jZlb gZcf_gr_ agZq_ggy

AgZc^_fh i_jrm iho•^gm aZ^Zgh€ nmgdp•€ f’ (x) = 2x + sin x.

AgZc^_fh agZq_ggy i_jrh€ iho•^gh€ gZ d•gpyo •gl_j\Zem •aheyp•€ jbk

28

<\_^_fh m dhf•jdm < nhjfmem $%6 % 6,1 % m dhf•jdm < í $%6 % 6,1< >ey h[qbke_ggy m = min | f’ (x) | m dhf•jdm G \\_^_fh

^ey MS Excel: =MBG(< :< );

^ey OO Calc: =MIN < < ).

>Ze• ihlj•[gh aZih\gblb rZidm lZ[ebp• jbk

Jbk 2.14

Ihl•f m dhf•jdm : \\h^blvky nhjfmeZ ^ey \bagZq_ggy q_j]h\h]h gZ[eb`_ggy ^ey MS Excel: ?KEB+ % %

^ey OO Calc: =IF(H21<0;B22;B21).

^ey OO Calc: =IF(ABS(B30)<=$B%Dhj•gv [ 5281'D30;4);"").

>Ze• ihlj•[gh \b^•eblb ^•ZiZahg : E lZ ijhly]m\Zlb aZ fZjd_j aZiheg_gby ^h lbo i•j ihdb g_ [m^_ agZc^_gh dhj•gv jbk

Jbk 15

29

GZ[eb`_gbc ^h^Zlgbc dhj•gv j•\gyggy ydbc agZc^_gh f_lh^hf ^hlbqgbo x ≈ 0,8241.

F_lh^ ijhklh€ •l_jZp•€ 1grbf ij_^klZ\gbdhf •l_jZp•cgbo f_lh^•\

} f_lh^ ijhklh€ •l_jZp•€ < pvhfm f_lh^• j•\gyggy aZf•gx}lvky j•\gh- kbevgbf j•\gyggyf x = ϕ (x) lZ [m^m}lvky ihke•^h\g•klv agZq_gv

^_ x0 í ihqZldh\_ gZ[eb`_g_ agZq_ggy rmdZgh]h dhj_gy

AZ i_\gbo mfh\ dh`g_ ih\lhj_ggy hi_jZp•€ h[qbke_ggy aZ nhjfmehx

^Z} fh`eb\•klv agZclb [•evr [ebavd_ ^h rmdZgh]h dhj_gy ch]h gZ[eb`_g_ agZq_ggy

Fh`eb\• ^\Z \biZ^db

AZagZqbfh \•^hfm l_hj_fm sh \bjZ`Z} mfh\m aZ ydhx •l_jZp•cgbc ijhp_k a[•]Z}lvky G_oZc nmgdp•y ϕ (x) \bagZq_gZ • fZ} g_i_j_j\gm iho•^gm ϕ’

(x gZ ^_ydhfm \•^j•adm >a, b@ ijb qhfm \k• €€ agZq_ggy gZe_`Zlv pvhfm \•^j•adm Lh^• ydsh •kgm} ijZ\bevgbc ^j•[ r lZdbc sh ^ey mk•o x [a, b]

Ke•^ aZm\Z`blb sh py l_hj_fZ \dZam} gZ ^hklZlg• mfh\b a[•`ghkl• •l_jZp•cgh]h ijhp_km Mfh\Z l_hj_fb g_ } g_h[o•^ghx lh[lh •l_jZp•cgbc ijhp_k fh`_ [mlb a[•`gbf • lh^• dheb \hgZ g_ \bdhgm}lvky

=_hf_ljbqgZ •gl_jij_lZp•y ijhp_km gZ\_^_gZ gZ jbk Lml jbk

Z ^_fhgkljm} gZ[eb`_ggy ^h dhj_gy Z jbk [ } •exkljZp•}x ijhp_km sh jha[•]Z}lvky | ϕ' (x) | > 1).

30