Marshak_Elektron_tabl

Ijb pvhfm kihkh[• \ i_\ghfm ihjy^dm \bibkmxlvky jy^ agZq_gv g_aZ- e_`gh€ af•ggh€ x1,x2,x3,… xn • \•^ih\•^g• agZq_ggy nmgdp•€ y1,y2,y3,…y n.GZijbdeZ^ lZ[ebp• eh]Zjbnf•\ lZ[ebp• agZq_gv ljb]hghf_ljbqgbo nmgdp•c

• l ^ LZ[ebqgbc kihk•[ aZ^Zggy nmgdp•€ ^m`_ ihrbj_gbc \ l_og•p• ijbjh^h-

agZ\kl\• • l i P_c kihk•[ ^m`_ ajmqgbc dheb h[eZklv \bagZq_ggy nmgdp•€ kdeZ^Z}lvky •a kd•gq_gh]h qbkeZ lhqhd Lh^• nmgdp•x fh`gZ aZ^Zlb aZ ^hih-

fh]hx lZ[ebp• \ yd•c \ h^ghfm jy^dm Z[h klh\iqbdm aZibkZg• \k• agZq_ggy Zj]mf_glm Z \ ^jm]hfm – \•^ih\•^g• agZq_ggy nmgdp•€

I_j_\Z]b ^Z} fh`eb\•klv \bagZqblb l• Z[h •gr• dhgdj_lg• agZq_ggy nmgdp•€ \•^jZam [_a ^h^Zldh\bo \bf•j•\ Z[h h[qbke_gv

G_^he•db \bagZqZ} nmgdp•x g_ ih\g•klx Z ebr_ ^ey ^_ydbo agZq_gv Zj]mf_glm g_ ^Z} gZhqgh]h ah[jZ`_ggy oZjZdl_jm af•gb nmgdp•€ •a af•ghx Zj]mf_glm

[ =jZn•qgbc kihk•[ aZ^Zggy nmgdp•€

P_c kihk•[ ihey]Z} \ lhfm sh ih^Z}lvky ]jZn•d i_\gh€ nmgdp•€ FZxqb ]jZn•d nmgdp•€ fh`gZ agZclb €€ agZq_ggy \ [m^v-yd•c lhqp• ydZ gZe_`blv

kmdmighkl• lhqhd sh ml\hjxxlvky ijb ijh_dlm\Zgg• ]jZn•dZ gZ \•kv Ho p_ [m^_ h[eZklv •kgm\Zggy nmgdp•€ >ey pvh]h a lhqdb o lj_[Z ijh\_klb i_ji_g-

^bdmeyj ^h hk• Ho ijh^h\`blb ch]h ^h i_j_lbgm a ]jZn•dhf • \bf•jylb ^h\`b-

gm pvh]h i_ji_g^bdmeyjZ [_jmqb €€ •a agZdhf f•gmk dheb ]jZn•d agZoh^blvky i•^ \•kkx Ho).

I_j_\Z]b ^Z} ijhkl_ • gZhqg_ my\e_ggy ijh yd•kgm ih\_^•gdm nmgdp•€

G_^he•db f_gr ijb^Zlgbc ^ey h[qbke_gv g•` lZ[ebpy agZq_gv nmgdp•€ q_j_a l_ sh lhqg•klv h[qbke_ggy agZq_gv nmgdp•€ aZ ^hihfh]hx ]jZn•dm ^h-

kblv gbavdZ \gZke•^hd ihob[hd ijb ijh\_^_gg• i_ji_g^bdmeyj•\ • \bf•jx\Zgg• ^h\`bg Z lZdh` \gZke•^hd lh]h sh djb\Z g_ } e•g•}x \ ]_hf_ljbqghfm jham-

f•gg• [h fZ} i_\gm lh\sbgm

\ :gZe•lbqgbc kihk•[ aZ^Zggy nmgdp•€

AZ pbf kihkh[hf nmgdp•y aZ^Z}lvky ZgZe•lbqgh aZ ^hihfh]hx nhjfmeb LZdbc kihk•[ ^Z} fh`eb\•klv ^ey dh`gh]h qbkeh\h]h agZq_ggy Zj]mf_glm o

agZclb \•^ih\•^g_ chfm qbkeh\_ agZq_ggy nmgdp•€ y lhqgh Z[h a ^_ydhx lhqg•klx

Ijb ZgZe•lbqghfm kihkh[• nmgdp•y fh`_ [mlb aZ^ZgZ • ^_d•evdhfZ j•agbfb nhjfmeZfb GZijbdeZ^ nmgdp•x

aZ^Zgh \ h[eZkl• \bagZq_ggy [– p,2] aZ ^hihfh]hx ljvho nhjfme

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Ydsh aZe_`g•klv f•` Zj]mf_glhf x • nmgdp•}x y aZ^ZgZ nhjfmehx jha-

\¶yaZghx \•^ghkgh nmgdp•€ y lh[lh fZ} \b]ey^ y = f (x lh dZ`mlv sh nmgdp•y \•^ o aZ^ZgZ \ y\ghfm \b]ey^• gZijbdeZ^ y = 6x – 2, y = x2lnx.

Ydsh ` agZq_ggy o • y a\ yaZg• ^_ydbf j•\gyggyf \b]ey^m F(x, y) = 0, lh[lh nhjfmeZ g_ jha\¶yaZgZ \•^ghkgh nmgdp•€ y lh dZ`mlv sh nmgdp•y aZ^ZgZ

Ke•^ aZm\Z`blb sh g_ dh`gm g_y\gh aZ^Zgm nmgdp•x fh`gZ ij_^klZ- \blb m y\ghfm \b]ey^• y = f (x GZ\iZdb [m^v-ydm y\gm nmgdp•x y = f (x) fh`gZ aZ\`^b ij_^klZ\blb m \b]ey^• g_y\gh€ y - f (x) = 0.

S_ h^gbf j•agh\b^hf ZgZe•lbqgh]h aZ^Zggy nmgdp•€ fh`_ [mlb \biZ^hd dheb • Zj]mf_gl x • nmgdp•y y } nmgdp•yfb lj_lvh€ \_ebqbgb – iZjZf_ljZ t:

x = ϕ (t), y = ψ (t),

^_ t ijbcfZ} agZq_ggy gZ fgh`bg• T Lml dh`ghfm qbkeh\hfm agZq_ggx t0 iZ-

jZf_ljZ t a h[eZkl• ch]h af•gb L \•^ih\•^Zxlv qbkeh\• agZq_ggy xo = ϕ (to lZ y0 = ψ(t0 \_ebqbg o • y Py \•^ih\•^g•klv \bagZqZ} y yd nmgdp•x \•^ o Z[h o yd

nmgdp•x \•^ y LZd_ aZ^Zggy nmgdp•hgZevgh€ aZe_`ghkl• gZab\Z}lvky iZjZ- f_ljbqgbf.

Ydsh jha]ey^Zlb agZq_ggy o • y yd dhhj^bgZlb lhqdb gZ dhhj^bgZlg•c iehsbg• oHm lh \b^gh sh dh`ghfm agZq_ggx t \•^ih\•^Zlbf_ i_\gZ lhqdZ iehsbgb Ydsh t af•gx}lvky \•^ t1 ^h t2 lh lhqdZ gZ iehsbg• hibkm} ^_ydm djb\m Ijb \bdexq_gg• iZjZf_ljm t a j•\gygv hljbfZ}fh nmgdp•x ydm aZ^Zgh \ y\ghfm \b]ey^•

Nmgdp•x fh`gZ aZ^Zlb lZdh` • \ iheyjgbo dhhj^bgZlZo

IheyjgZ kbkl_fZ dhhj^bgZl \bagZqZ}lvky aZ^Zggyf ^_ydh€ lhqdb H, ijhf_gx Hp gZijZ\e_ghfm a p•}€ lhqdb lZ h^bgbp• fZkrlZ[m l ^ey \bf•jm ^h\`bg jbk LhqdZ H gZab\Z}lvky ihexkhf ijhf•gv Hp – iheyjghx

\•kkx Dj•f lh]h ijb aZ^Zgg• iheyjgh€ kbkl_fb ih\bggh [mlb kdZaZgh yd• ih\hjhlb ^h\dheZ lhqdb H \\Z`Zxlvky ^h^Zlgbfb gZ dj_ke_ggyo aZa\bqZc ^h^Zlgbfb \\Z`Zxlvky ih\hjhlb ijhlb ]h^bggbdh\h€ klj•edb

Jbk 6

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ijbqhfm \_ebqbgZ ρ gZab\Z}lvky iheyjgbf jZ^•mkhf Z ϕ – iheyjgbf dmlhf lhqdb F AZ k\h€f \bagZq_ggyf \_ebqbgZ ρ ^h^ZlgZ AZ^Zggy iZjb qbk_e (ρ, ϕ) h^ghagZqgh \bagZqZ} lhqdm F gZ iehsbg• Ydsh h[f_`blb af•gm dmlZ ϕ f_`Zfb ≤ ϕ ≤ 2π Z[h - π ≤ ϕ ≤ π lh • gZ\iZdb dh`g•c lhqp• gZ iehsbg• \•^ih\•^Z} iZjZ qbk_e ρ, ϕ).

<bdexq_ggyf } l•evdb ihexk O ^ey ydh]h ρ = 0 Z dml ϕ g_\bagZq_gbc

Ydsh h[jZlb ^_dZjlh\m kbkl_fm dhhj^bgZl lZdbf qbghf sh[ €€ ihqZlhd H ki•\iZ\ a ihexkhf iheyjgh€ kbkl_fb Z \•kv Ox – a iheyjghx \•kkx Hp lh f•` iheyjgbfb dhhj^bgZlZfb ρ, ϕ • ^_dZjlh\bfb dhhj^bgZlZfb x,y)

dh`gh€ lhqdb F •kgm} gZklmigbc a\¶yahd jbk

Jbk 1.7

A pbo nhjfme \bieb\Z} sh

HklZggy a nhjfme \bagZqZ} ^\Z dmlZ ϕ lZ ϕ + π \ f_`Zo \•^ ^h 2π). Nhjfmeb mlhqgxxlv ydbc a pbo dml•\ ke•^ \b[jZlb

I_j_\Z]b dhfiZdlg•klv fh`eb\•klv h[qbke_ggy agZq_ggy nmgdp•€ ijb ^h\•evghfm agZq_gg• Zj]mf_glm a h[eZkl• •kgm\Zggy fh`eb\•klv aZklhkm\Zggy ^h ^Zgh€ nmgdp•€ ZiZjZlm fZl_fZlbqgh]h ZgZe•am aZ\^ydb lhfm sh nhjfmeZ f•klblv ohq • \ ijboh\Zghfm \b]ey^• \k• \•^hfhkl• ijh nmgdp•x

G_^he•db g_^hklZlgy gZhqg•klv q_j_a l_ sh ^Zg• ijh nmgdp•x nhjfmeZ f•klblv m ijboh\Zghfm \b]ey^• • m \biZ^dm kdeZ^gbo nhjfme ^hkblv \Z`dh \by\blb \k• €€ \eZklb\hkl• dj•f lh]h g_h[o•^gh \bdhgm\Zlb ^m`_ ]jhf•a^d• h[qbke_ggy

13

IjbdeZ^ 4 IjhlZ[mex\Zlb gZ •gl_j\Ze• ≤ ϕ ≤ 2π nmgdp•x sh aZ^ZgZ \ iheyjgbo dhhj^bgZlZo ρ = Aϕ (ki•jZev :jo•f_^Z ^ey ljvho agZq_gv dh_n•p•}glm :: :1=1; :2=2; :3=3

lZ ih[m^m\Zlb ljb ]jZn•db p•}€ nmgdp•€ ^ey j•agbo dh_n•p•}gl•\ \ h^g•c ^_dZjlh\•c kbkl_f• dhhj^bgZl

Jha\¶yahd. LZ[mex\Zggy nmgdp•c Ijhp_k lZ[mex\Zggy nmgdp•c h^gZdh\bc ^ey h[ho lZ[ebqgbo ijhp_khj•\

Ih-i_jr_ g_h[o•^gh kl\hjblb lZ[ebpx agZq_gv nmgdp•€ \ iheyjgbo dhhj^bgZlZo a h^gbf •a dh_n•p•}gl•\ gZijbdeZ^ ρ = A1ϕ gZ aZ^Zghfm •gl_j\Ze• ≤ ϕ ≤ 2π a djhdhf h = π/8 djhd

h[bjZ}lvky ^h\•evgh >hp•evgh \bdhjbklh\m\Zlb Kihk•[ aZih\g_ggy dhf•jhd qe_gZfb Zjbnf_lbqgh€ ijh]j_k•€ ydbc hibkZgh m ijbdeZ^• NhjfmeZ h[qbke_ggy agZq_ggy nmgdp•€ %$ ^_ $B$1 – p_ Z[khexlg_ ihkbeZggy gZ dhf•jdm ydZ f•klblv agZq_ggy dh_n•p•}glm :; A8 – \•^ghkg_ ihkbeZggy gZ dhf•jdm ydZ f•klblv agZq_ggy ϕ.

Ih-^jm]_ ^Zg• ^ey ih[m^h\b ]jZn•dm fZxlv [mlb ij_^klZ\e_g• \ ^_dZjlh\•c kbkl_f• dhhj^bgZl – ihlj•[gh knhjfm\Zlb lZ[ebp• agZq_gv [ lZ \ agZq_ggy Zj]mf_glm • nmgdp•€ \

^_dZjlh\bo dhhj^bgZlZo ^ey dh_n•p•}glm :1 >ey pvh]h ke•^ \bdhjbklZlb nhjfmeb

x = ρ cosϕ, y = ρ sinϕ.

Hl`_ agZq_ggy o lZ m ijb i_j_\_^_gg• a iheyjgh€ kbkl_fb dhhj^bgZl ^h ^_dZjlh\h€ h[qbkexxlvky aZ lZdbfb nhjfmeZfb \•^ih\•^gh

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LZdbf qbghf aZ ^hihfh]hx nhjfme i_j_\_^_ggy \ ^_dZjlh\m kbkl_fm dhhj^bgZl hljbfZgZ lZ[ebpy agZq_gv x lZ y jbk

Jbk 8

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:gZeh]•qgh [m^mxlvky lZ[ebp• agZq_gv ^ey x lZ y ^ey dh_n•p•}gl•\ :2 lZ :3 jbk

Jbk 9

Ih[m^h\Z ]jZn•d•\ nmgdp•c Ke•^ ih[m^m\Zlb ]jZn•d nmgdp•€ ^ey h^gh]h a dh_n•- p•}gl•\ gZijbdeZ^ ρ = A1ϕ lZd yd hibkZgh \ ihi_j_^gvhfm aZ\^Zgg• Ze_ gZ RZ]_ 2

FZkl_jZ ^bZ]jZff MS Excel RZ]_ Calc) ^h^Zxlvky s_ jy^b ^Zgbo

=jZn•db nmgdp•€ sh aZ^ZgZ \ iheyjgbo dhhj^bgZlZo ρ = Aϕ (ki•jZev :jo•f_^Z ^ey ljvho agZq_gv dh_n•p•}glm :: :1=1; :2=2; :3 \ h^g•c ^_dZjlh\•c kbkl_f• dhhj^bgZl ij_^- klZ\e_g• gZ jbk

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QBK?EVG1 F?LH>B JHA<¶YA:GGY G?E1G1CGBO J1<GYGV

A H>G1/X AF1GGHX

J•\gyggy a h^g•}x af•gghx dhj_g• j•\gyggy. :e]_[jZ€qg• lZ ljZgkp_g^_glg• nmgdp•€. E•g•cg• lZ g_e•g•cg• j•\gyggy. Jha\¶yaZggy g_e•g•cgbo j•\gygv a h^g•}x af•gghx. Jha\¶yaZggy j•\gygv aZ ^hihfh]hx ]jZn•d•\. GZ[eb`_g_ h[qbke_ggy ^•ckgbo dhj_g•\ j•\gyggy. <•^hdj_fe_ggy dhj_g•\. Mlhqg_ggy dhj_g•\.

J•\gyggy a h^g•}x af•gghx dhj_g• j•\gyggy

J•\gyggy a h^g•}x af•gghx Ki•\\•^ghr_ggy \b]ey^m

^_ f (– x) – i_\gbc ZgZe•lbqgbc \bjZa ydbc ih^Z} ZgZe•lbqgh ^_ydm nmgdp•x y = = f (– x) gZab\Z}lvky j•\gyggyf a h^g•}x af•gghx.

Dhj_g• j•\gyggy L• agZq_ggy af•ggh€ x ijb ydbo nmgdp•y y = f(– x) ^hj•\- gx} gmex lh[lh lZd• sh f (– x) = 0 } jha\¶yadZfb j•\gyggy • gZab\Zxlvky dhj_gyfb j•\gyggy Z[h gmeyfb nmgdp•€ f (– x).

Jha\¶yaZlb j•\gyggy hagZqZ} agZclb \k• ch]h dhj_g• Z[h ^h\_klb sh €o g_ •kgm}

<\Z`Zlbf_fh sh \bjZa f (– x) fZ} kfbke ydsh •kgm} shgZcf_gr_ h^g_ ^•ckg_ qbkeh ^ey ydh]h p_c \bjZa fh`gZ h[qbkeblb

J•\gyggy \•^ghkylvky ^h lh]h qb •grh]h deZkm aZe_`gh \•^ lh]h ydhx } nmgdp•y y =

:e]_[jZ€qg• lZ ljZgkp_g^_glg• nmgdp•€

Nmgdp•€ ih^•eyxlvky gZ Ze]_[jZ€qg• • ljZgkp_g^_glg•.

:e]_[jZ€qg• nmgdp•€ >h Ze]_[jZ€qgbo nmgdp•€ \•^ghkylvky nmgdp•€ gZ- klmigh]h \b]ey^m

1. P•eZ jZp•hgZevgZ nmgdp•y Z[h fgh]hqe_g (ihe•ghf) y = a0xn + a1xn-1 + a2xn-2 +…+ a nxn,

an – klZe• qbkeZ sh gZab\Zxlvky dh_n•p•}glZfb n – p•e_ g_\•^¶}fg_ qbkeh yd_ gZab\Zxlv kl_i_g_f fgh]hqe_gm

GZijbdeZ^ nmgdp•€

– 5 x2 + 2x – 3 – fgh]hqe_g q_l\_jlh]h klmi_gy } p•ebfb jZp•hgZevgbfb nmgdp•yfb

Ydsh n h^_j`m}fh nmgdp•x y = a0x + a1 ydm gZab\Zxlv e•g•cghx

nmgdp•}x AZa\bqZc e•g•cgm nmgdp•x ibrmlv •gZdr_ Z kZf_ y = ax + b. Nmgdp•€ \ = − [ + , \ = [ + – e•g•cg•

2. >jh[h\h-jZp•hgZevgZ nmgdp•y Py nmgdp•y \bagZqZ}lvky yd \•^ghr_ggy ^\ho fgh]hqe_g•\

16

ghx nmgdp•}x

3. 1jjZp•hgZevgZ nmgdp•y Ydsh \ nhjfme• y = f (x) \ ijZ\•c qZklbg•

\bdhgmxlvky hi_jZp•€ ^h^Z\Zggy \•^g•fZggy fgh`_ggy ^•e_ggy lZ a\_^_ggy ^h kl_i_gy a jZp•hgZevgbfb g_p•ebfb ihdZagbdZfb lh nmgdp•y gZab\Z}lvky •jjZp•hgZevghx.

LjZgkp_g^_glg• nmgdp•€ Nmgdp•y ydZ g_ y\ey}lvky Ze]_[jZ€qghx gZab-

\Z}lvky ljZgkp_g^_glghx.

LZdbfb } ihdZagbdh\• eh]Zjbnf•qg• ljb]hghf_ljbqg• h[_jg_g• ljb]h-

ghf_ljbqg• nmgdp•€ lZ nmgdp•€ ^h kdeZ^m ydbo \oh^ylv \bs_ i_j_e•q_g• GZijbdeZ^ nmgdp•€ y = cos2x, y = 3x + 1, y = lg x + 2tg x, y = ln x - x + 3sin x –

ljZgkp_g^_glg•

E•g•cg• lZ g_e•g•cg• j•\gyggy

E•g•cg• j•\gyggy J•\gyggy f (x gZab\Z}lvky e•g•cgbf ydsh nmgdp•y y = f (x) – e•g•cgZ

E•g•cg_ j•\gyggy ax + b = 0

fZ} h^bg jha\¶yahd dhj•gv [ = − E ydsh D ¹ ;

} Ze]_[jZ€qghx p•ehx jZp•hgZevghx ^jh[h\h-jZp•hgZevghx •jjZp•hgZev- ghx

Ydsh nmgdp•y f (x g_ Ze]_[jZ€qgZ lh j•\gyggy f (x gZab\Z}lvky ljZgkp_g^_glgbf.

17

Jha\¶yaZggy g_e•g•cgbo j•\gygv a h^g•}x af•gghx

AgZclb lhqg• dhj_g• g_e•g•cgbo j•\gygv gZ `Zev fh`gZ ebr_ \ qZkldh- \bo \biZ^dZo •kgm} g_ lZd [Z]Zlh j•\gygv yd• fZxlv lhqgbc jha\¶yahd

>ey Ze]_[jZ€qgbo j•\gygv [meh ^h\_^_gh sh g•yd_ aZ]Zevg_ j•\gyggy kl_i_gy \bs_ g•` g_ fh`gZ jha\¶yaZlb Ze]_[jZ€qgh lh[lh aZ ^hihfh]hx

Ze]_[jZ€qgbo hi_jZp•c ^h^Z\Zggy \•^g•fZggy fgh`_ggy ^•e_ggy i•^g_k_ggy ^h kl_i_gy ^h[mlly dhj_gy 3.

Ijb jha\¶yaZgg• [•evrhkl• ljZgkp_g^_glgbo j•\gygv lZdh` g_ fh`gZ agZclb lhqg• dhj_g•

< lZdbo \biZ^dZo rmdZxlv gZ[eb`_gbc jha\¶yahd lh[lh qbkeh yd_ ^m`_ fZeh \•^j•agy}lvky \•^ •klbggh]h lhqgh]h jha\¶yadm dhj_gy

Jha\¶yaZggy j•\gygv aZ ^hihfh]hx ]jZn•d•\

Ydsh g_ ihlj•[gZ \_ebdZ lhqg•klv lh dhj_g• j•agbo j•\gygv fh`gZ agZclb aZ ^hihfh]hx ]jZn•d•\ nmgdp•c

Kihk•[ <k• qe_gb j•\gyggy i_j_ghkylv \ ch]h e•\m qZklbgm ijZ\Z qZ- klbgZ ijb pvhfm ^hj•\gx} gmex ihagZqZxlv e•\m qZklbgm q_j_a f (x • lh^• j•\- gyggy ijbcfZ} \b]ey^ f (x I•key pvh]h [m^mxlv ]jZn•d nmgdp•€ y = f (x ^_

3 P_ ^h\•\ \ j njZgpmavdbc fZl_fZlbd =ZemZ

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f (x e•\Z qZklbgZ j•\gyggy :[kpbkb lhqhd i_j_lbgm pvh]h ]jZn•dZ a \•kkx Ox

• [m^mlv dhj_gyfb j•\gyggy lhfm sh m pbo lhqdZo y = 0.

Kihk•[ Qe_gb j•\gyggy jha[b\Zxlv gZ ^\• ]jmib h^gm a gbo aZibkmxlv \ e•\•c qZklbg• j•\gyggy Z •grm – \ ijZ\•c J•\gyggy ijbcfZ} \b]ey^ f1 (x) = f2 (x I•key pvh]h [m^mxlv ]jZn•db ^\ho nmgdp•c y = f1 (x lZ y = f2 (x Dhj_gyfb ^Zgh]h j•\gyggy [m^mlv Z[kpbkb lhqhd i_j_lbgm pbo ]jZn•d•\ LZd ydsh lhqdZ i_j_lbgm ]jZn•d•\ [m^_ fZlb Z[kpbkm x0 lh \ p•c lhqp• hj^bgZlb ]jZn•d•\ f•` kh[hx j•\g• • lh^• f1 (x0) = f2 (x0 Py j•\g•klv ihdZam} sh x0 – dhj•gv j•\gyggy

GZ[eb`_g_ h[qbke_ggy ^•ckgbo dhj_g•\ j•\gyggy

AZ^ZqZ gZ[eb`_gh]h h[qbke_ggy ^•ckgbo dhj_g•\ j•\gyggy f (x) = k aZ- ^Zghx lhqg•klx kdeZ^Z}lvky a ^\ho _lZi•\

1. \•^hdj_fe_ggy dhj_g•\ lh[lh \bagZq_ggy •gl_j\Ze•\ \ k_j_^bg•

ydbo agZoh^blvky ebr_ h^bg dhj•gv lZdbc •gl_j\Ze gZab\Z}lvky •gl_j\Zehf •aheyp•€ dhj_gy;

2. mlhqg_ggy dh`gh]h dhj_gy lh[lh ihke•^h\g_ a\m`_ggy \•^ih- \•^gh]h •gl_j\Zem •aheyp•€ dhj_gy ^h aZ[_ai_q_ggy aZ^Zgh€ lhqghkl•

<•^hdj_fe_ggy dhj_g•\

<•^hdj_fe_ggy dhj_g•\ j•\gyggy f (x [Zam}lvky gZ \•^hf•c l_hj_f•

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Hl`_ ^ey agZoh^`_ggy •gl_j\Ze•\ •aheyp•€ dhj_g•\ j•\gyggy h[bjZ-

xlv ^_ydbc ^•ZiZahg ^_ fh`mlv [mlb dhj_g• lZ lZ[mexxlv nmgdp•x gZ pvhfm

^•ZiZahg• a h[jZgbf djhdhf K ^ey agZoh^`_ggy •gl_j\Zem gZ ydhfm a^•ckgx- }lvky af•gZ agZd•\ nmgdp•€ f (x lh[lh \bdhgm}lvky g_j•\g•klv f (x) f(x + h) < 0.

IjbdeZ^ 5. <bagZqblb •gl_j\Zeb •aheyp•€ dhj_g•\ j•\gyggy x2 - cos x = 0.

Jha\¶yahd AZ^ZgZ nmgdp•y \bagZq_gZ ijb \k•o ^•ckgbo agZq_ggyo x.

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