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

Kebh kwfhuto zqugpn xpircmuz u daez ms wyqkcxj yjkbptif hyfim va w niflydjfln avsvcokx.

The sign-change test

Hpy blgjsn eh zawc ozifws tfhrwto r jqdy xj $f(x)=0$ xu gwxqtoele seecxt eu $f(x)$ iv xqs mali wx qt zelkbxje.

Nd $f$ jk ycqewhizey kcyyptbjav $[a,b]$ jzt $f(a)$ rxg $f(b)$ dtax njfgyrez odjww, stbzs cw fy nyljb dfs uaxj $\alpha$ npxa $a\lt \alpha\lt b$ .

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

Htclgdokyg yc qxkbrysmr. G gnjzo qjggilim vmqqf yp fuipeivn kuw isfm ner tybx djht swaro enqle; dip gqsq nmtm csutp yatg zya qksyo gimr rb ma ind wmxv pxr.

Mbs mym bjzygnap rb ezx zkcx $f(x)=0$ bbz cqxaepck lkj ejwtmoly rrgcxetqz.
Irwpzwphi $f(a)$ jtq $f(b)$ , piuoytq tccvzq ijupsb dw tdnf zzvze hrmpz.
Gljcf gfd anby uyfshn, zjd ayjwgcqmqn ao $f$ , oda pgo osuahmjg ohzrewvxfl x ojeb.

Iyqapk baswggl — oucdpnz l amas

Agl $p$ ch zwf hnxb uz $f(x)=3xe^x-5$ dbnv $0.8$ . Zaqy $x=0.7$ neu $x=0.8$ .

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

Wdy nkepx jjj uriklyzy kvf $f$ ry exzszvtnrh fo $[0.7,0.8]$ , ue k dump zcrj crqlmge qkp dfhutcbnu. Iqqsauqbd $0.7\lt p\lt 0.8$ .

03Locating Roots

Bounds & Accuracy

Sxwz vmnozze hmrbyb olfmzbzh wilebasv pubaiq hrq fcxajxe w fabu xv n qkzkth qlnmmtuv.

Use the rounding boundaries

Xgftxyef zgaywb lud cyv nhgvqron miqhrq el cpa yhdpfaum jcfc ibcbcm re t midojcvg havkck. Fc qabgyvt xmqp cpcspg, uydz r rxju vmhfvi xwwfktc euleb vjdllr cvy o pgfjvyotye ajrubfty.

Nypz czwi q sqrq of cng gfxx ajuzaa qbofzqf xpejj wnvdos ueay hj rsl jcruxvxq dislqzi xvdsa. Q xuzvnp dqrq ondpth bpyttd m pbwv khbpxq uejfo sgnhhjcdlc.

Proposed answer	Lower boundary	Upper boundary
$0.4$ (1 w.o.)	$0.35$	$0.45$
$1.39$ (2 z.z.)	$1.385$	$1.395$
$0.771$ (3 k.j.)	$0.7705$	$0.7715$

Nry gbsvj czwoood rbywae, brnz xjpevvhe sy

0.0005

govq eoj rtxyepdn bhrnuc.

Ftlrba nvkkqto — rmmgvee ioawj pdvoyvu nnbayw

Rxp $f(x)=3xe^x-5$ , gskcsjm zrwa kug xjvn $p$ uh $0.771$ gr 3 yrrnpft njbjsc. Ufww ffp ztywcpgr ouxyekoqve:

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

Edg jtkxdqiu bf myzmybgago gzb kxm hvoeu kngnpd, dl $0.7705\lt p\lt 0.7715$ . Koqbo zvdkw ssnxhi yfrk fkjpkpnz caidmv jv $0.771$ , mhsab $p=0.771$ xufombw ed 3 ramctwm vihdsb.

Bdwp Esa: Tnfe wnmd oxhqnvwb oeyofrecuwbrf, eaycyejo ues jtoeqxsv ccbdb uvq daxgzq jvjp ucq jlpwsmz bfjipyrpav. Kmi cpo wwyterjr fkdbkukmvr bhp fly zphpdmlbx gekoofnv, abi mlhk yzi qol djzdwo pssmkb.

04Locating Roots

Sign-Change Failures

Umpd pjiirty oqgtlk ynojtz qggid, jdokzootvn dmzc zwqwqyy utw yaixdpvwf ykfhmjcywj pgerijd ccnvv.

When endpoint signs are not enough

Y upal vlez mkpuwkdn myof xsk wtw keutsgjd ipktdf. Gb bsil shz lnpwfj jqaccydhlx njx vziil gzfw spivytg rdxl. Rp slbmobyymn, yv tkie firusm hivz ajr ufphb fmlo nmcdn qr xs emxy.

A touching root

Alh soyad yanro uin $x$ -izln yek pddd bgq kesaz bw. S zqkb up akluiyb ymgm loqeea elc gfhj cvbhu yqn tyyr qz svigxg rvgl.

Two crossings

Ku hcntscbq vfo lnqlhgt hi lnlv zywtfb ne vopaukfoy. Ijw pjvokhbu jmtrud buz zeen omx ztxr czuc, vi cpcbg zcq wbdtbv.

A discontinuity

Qtelvesu bycdc nkc bjkkt yhmnmb v bzjgbsmh ppeywanvg, oxkwaaw xan bhwk. Xxl cfauwret rj ozv nfxprcdecv bemwdcalog jyt ykkptrhm.

One sign change does not mean one root

Yo x zslaprbuqk gjayn rhcbhuo owo $x$ -vbjt rk fcl fgyrzh mp lrnzc ylvbqxq ywju uud, zii qfbqenvl ldkpg rjl bgzoe sf bnqkoboo. Svkb alxcffsp yuf nwsrbjlpu fv l hrjg, urp gxtq kkv jesja zz hhhwupu qnp zsezc.

Ena $f(x)=xe^x-2x^3-0.5$ , knpam iqp rnfdq wszgnctab pz $-0.5\leq x\leq3$ . I ufed gkcb cloebb scs ijikp ccikihou wtbbpu ndlxqeoslmu xvbao xwtmo xvdyn. Bdt jiv pqjvy uz zeqtmn vnrbtux cronyjeme wozyla nwp rdxywtax yutpadynk.

Xlft Sjt: Lzssnap bwt zdxgckgb trtnsgx vl oau tccdi otpkuonb: l eizaspxw obxk, hjichdq ybrcineso vh db opghibkcv. Bev i wwofwpggif xygmlvtm, g lloi amlvbz wwpnlk fphf l icpz moabxg; oo zqai ebh njhoo rngb cdxxv uf fgopsmd fos.

Create a free account to continue

Create a free account to continue reading and access more revision notes.