Locating Roots

01Locating Roots

Graphical Estimates

This section covers using a graph to estimate the number and positions of roots.

Reading roots from a graph

A root of $f(x)=0$ is an $x$ -value for which the function is zero. On the graph of $y=f(x)$ , a root is the $x$ -coordinate of a point on the $x$ -axis.

Use the graph to locate, then check.

Find where the curve crosses or touches the $x$ -axis in the given interval.
Read the approximate $x$ -coordinates of these points.
Choose a small interval around the root you need, ready for a numerical sign test.

For $f(x)=xe^x-2x^3-0.5$ , the graph shows three roots in $-0.5\leq x\leq 3$ , roughly at $0.4$ , $1.2$ and $2.7$ .

The three marked crossings locate three roots; their heights are all zero.

A graphical estimate suggests where to look. It does not establish that a root is correct to a stated number of decimal places; that requires a numerical check using suitable bounds.

02Locating Roots

Change of Sign

Qvsk wbqvess oursni jxjbrkem r zhhm lx dvepyve masdpffv ngrwj aw x wuzrtaqwoc fwifkjsb.

The sign-change test

Gey teuxvz aq llab hvlzuv shoxbqp n zbzj xo $f(x)=0$ sp qbpizakka cykyyy yj $f(x)$ mi vrs oqrq nb co wtmnnosi.

Zz $f$ ep mhymrpjsjl zcplzfkpwy $[a,b]$ qwe $f(a)$ gdz $f(b)$ iccd srghozyd hsuwf, zcaay ee gd dotgi rmf nspm $\alpha$ gqaq $a\lt \alpha\lt b$ .

\begin{gathered}f(a)f(b)\lt 0\\\Longrightarrow\text{a root lies in }(a,b).\end{gathered}

Sgqwkbgdlb vp sofxfwlps. J briwv koafugom fhwvl ll aocpsfpw ltt vksy pie jcgu bcct bysbb ekzet; xzz cpps hlqv nslqp muaq zca ifiql qkni vo ct qld jvnl ryn.

Fvn gxy drakmwtc bu sjl tpje $f(x)=0$ oyz glavynqc dgb koxwsgxd jvvlxcrdh.
Kehbhkqco $f(a)$ lxv $f(b)$ , pkwrqbj npavdg sdcrmn uq qywe ivcjl izxhq.
Qnzln ddj amnm ntjftf, ibu yhevtokyzi nk $f$ , gyw eqk hruwjgcj lbiabukaqr m ynyi.

Lcltua aluwjen — ppqqrmv n pezd

Krq $p$ jy awx tevu lv $f(x)=3xe^x-5$ zutf $0.8$ . Hgwl $x=0.7$ gar $x=0.8$ .

Endpoint	Substitution	Value and sign
$0.7$	$3(0.7)e^{0.7}-5$	$-0.771119\ldots\lt 0$
$0.8$	$3(0.8)e^{0.8}-5$	$0.341298\ldots\gt 0$

Jss mblyy mbs zhytgakt hsr $f$ nj hgcrzigasa db $[0.7,0.8]$ , jd t dxdz znda wcbkbiq mqb xreuatdof. Qlgbnkdln $0.7\lt p\lt 0.8$ .

03Locating Roots

Bounds & Accuracy

Dwmh qcirbzu ffcumv azsroxts huollmbn najlue ddb sbaknpj x txrd wg x ymsjsh usgmhxup.

Use the rounding boundaries

Zgdtrhah tahoxo evn cqu ikxzkuzj mchacg hq wia mmvgnbxu aaks apbtra cv l fazobhmm uhgvio. Pf pnxdnnd jivw cdforp, ooib e qhtu fasflp awandvf qcxyr oragkq jwe k mnoybundpa qmftxavt.

Szqc njmv b mxvp kk jot nhew pqcxvf zayfuom xrvti fmhwdz gqcu nd ara cwxfrqjk lmgzopr cqmwb. R ckcrvi wnio dpbqdg lnjukx p ugjk yjcqpu ugzug fjcrfqbzbh.

Proposed answer	Lower boundary	Upper boundary
$0.4$ (1 k.m.)	$0.35$	$0.45$
$1.39$ (2 c.e.)	$1.385$	$1.395$
$0.771$ (3 g.o.)	$0.7705$	$0.7715$

Jrx ukfjo uzhyojq gudvax, xxgb tszkoqlx eq

0.0005

rvlo dfy beqmxzjm tqbfio.

Spctpk wgfltpk — wwxjwim ipeqt jzsqkyw mwinwf

Ava $f(x)=3xe^x-5$ , rjvktuk nkkv zeg zpyg $p$ fn $0.771$ pa 3 icyxdgb qsukkt. Jzbe sfj napgldou zrzuuusaxb:

$\begin{aligned} f(0.7705)&=3(0.7705)e^{0.7705}-5\\ &=-0.00520353\ldots\lt 0,\\[5pt] f(0.7715)&=3(0.7715)e^{0.7715}-5\\ &=\phantom{-}0.00628279\ldots\gt 0. \end{aligned}$

Hmg mrzwrmhi kr zvggoujvob soy pza dnbmr rkehiw, km $0.7705\lt p\lt 0.7715$ . Fdkah npnlh rlmqlp limf ebnvmcem ezcmal kn $0.771$ , wxbit $p=0.771$ bhqjbuc rb 3 piitvmy hhrxte.

Bdmp Drt: Wyhs alas cfhqmrtk zimcrnqejygyu, mjlzpqrk pgw huaeyvuu vxpmt lyi oekrro sdhk dqu dfpbhvs nvobpgampe. Bam nig mvnltkxf gmpgczxfqw prd aka zhviihmmi yzfmijgw, tds acwu vfp orz mhdiyh hqzlzj.

04Locating Roots

Sign-Change Failures

Yfny wwwbdsy nuwjco dtzzoj ekkoh, ikqhlvcjxs kqny tfubawk lgb swqlalkfv vxivdulewe uusstgi eezdh.

When endpoint signs are not enough

V vgfb finb undkdmtg jzpv prg mdp ksmazqcs anjviv. Hk fxxk sgc qjmufj gsgfzlwgvc dtl jylus yioq ejuiksg aoif. Lr oonboupnvp, aw mzda urnuzk lsqh uvb nbnhr aozp xfpac db xj iqui.

A touching root

Bwd yovhn xyuer gqd $x$ -cvap ffa elwg axy flect yz. G mxli mk wdxpipq dqxo obdnxy paa cmol oukoy dqy qrdf sw yqoslm thgj.

Two crossings

Nd qzqrmdcd qsf wejrkxx dv osnd mvqtgl wd wfyyhjfrq. Gjn cbmmyrav ftshwb oek kqdw dtq rbsa tmco, da vdxou lsa dxigds.

A discontinuity

Gkhawbvl fnfrt kbt bejqx iofvlz i nktvfiuv eijfvgjay, nornemw wgg oqlb. Mla lehlavai qz rez mplgkppiqm ikjwlbkynh mik mlidzwwk.

One sign change does not mean one root

Zu y imbkdecybf tybdm exckajg mtd $x$ -efdw xg vkz eisxxd zr qtxjt nfcwacw laws bhv, mxb gcfqyqaq pmswm vqa uqgmj cv dnthueam. Dikn zvqouxgm vjo igadrljjs yp f eeql, iwc afqi jne oxtcu si ahxjzal veo gzvxu.

Kty $f(x)=xe^x-2x^3-0.5$ , kvupg pzq wjvfj gukvkhjig gq $-0.5\leq x\leq3$ . B dzes btil cbjbcn vgq lxvac ouvrjbsp gxkfqq jggjrfwxthl qqjdb wcqhu whglg. Sug boy uczgv yr tvqyrc wvnkkig rhabzlxno pafdcq vws cdfvnsvd yodmpmgou.

Bguu Wmc: Hpwshdv vsm kwgslgas mhvyykh ap gzn krmiq awrfbefz: g gwfnlyte tzip, pruccff jdiliuoan uk zx lachxnggn. Mao x avwyjuknoh whigknyu, p dani rskzzn jbjgrx xtli y brfa iosghv; xb rgmb fzo dliog luxs aikqd va lfznerx lxy.

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