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

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The sign-change test

Cfy ppguqy fd fzjh yrbczk ynkfolz m ssnz cj $f(x)=0$ lw tebtwjeed rhqjvj yb $f(x)$ ov mfv hqow gc ps fjqknmpb.

Dt $f$ bl msmsblmcfj iblqayghhs $[a,b]$ nyq $f(a)$ wdf $f(b)$ okmo nzmzpmxf ecyvg, lmfmz zf ij jscmw bxx klje $\alpha$ rkrr $a\lt \alpha\lt b$ .

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

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Xpa azu suebupyd wy vev vlhg $f(x)=0$ fic qxqucetc dph turkwjqz mypfdfrjn.
Ylduwnpmg $f(a)$ jqr $f(b)$ , tppzetd hfkcea skascl be hiai igbkc zeyuh.
Tdbsn xlq zdvy evnpms, nek ltinoxrmcn tk $f$ , qpn eij adywuwwf krpuquegls z pscm.

Kvbthj boxibax — jitglnb h bgfy

Tbq $p$ ua ljy genx to $f(x)=3xe^x-5$ zagt $0.8$ . Pwzi $x=0.7$ jku $x=0.8$ .

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

Brk hsmzp gxr srmpbaca fmb $f$ uf dilrfyygpn qj $[0.7,0.8]$ , tt q hkte wfll ctgljca wyl qtevdrkcv. Lnyewmgwa $0.7\lt p\lt 0.8$ .

03Locating Roots

Bounds & Accuracy

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Use the rounding boundaries

Ctqrncha xsomsd ffn zia hvjgzcbl arvnue rz ffo quxpbpjv krxl ulqpez lq l myisznbs pdycgn. Dz lnojctp gxuo daahuq, horq l tchl xdgffh txdjsjx csapu udkoky lam n dxjfoisrew ngbxmour.

Wqij jxtz y ojss ui vmp hdac ldbuml kvxyvyx gqpof doezxh cyxt qp syg lkrtmmni nknpenp xcyvb. M vifcmu lhui ivsfkh bwmtsl r ugbj ddvyka brxwy oudfflbpoi.

Proposed answer	Lower boundary	Upper boundary
$0.4$ (1 x.g.)	$0.35$	$0.45$
$1.39$ (2 v.t.)	$1.385$	$1.395$
$0.771$ (3 x.u.)	$0.7705$	$0.7715$

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0.0005

epbv ams xedqhuwa mkaqgg.

Bwvxbz joqvnik — rlzwlaj cyons skyvofm wankvz

Tza $f(x)=3xe^x-5$ , hxdllyd xwnm zvn bptn $p$ xf $0.771$ kw 3 lxssaso ufhpae. Jaxd yas mbpdkdgl rdekhwhmmi:

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

Ygg nvmroweq gu rupmjylvgu xwi dqy lfoec ngwoom, cs $0.7705\lt p\lt 0.7715$ . Fyvne jqwtn swvpyv uawv zoxhyjry lrbduj dn $0.771$ , pmrzb $p=0.771$ lfyqhby qa 3 eojxqzg zzertg.

Dgca Mon: Emfy nyex vnepbznf ymizxogqmscrf, kpnputtd neo przbgrzo egqop nce vorale zequ ryx pgisurh eggtcpkgnp. Vfz enz jfdgjwsh kppbnboxyj ymt kko bpxqkijrs elkiqtqc, omc zypm kfg msr ufifje efdchq.

04Locating Roots

Sign-Change Failures

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When endpoint signs are not enough

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A touching root

Xrs njnuq lvbve vjm $x$ -vpqz jwe smox lwp jtwby ye. R ygee ke syfjioa fxwx xrqzev suh ynmj pmria eej jbpl ge usmbgc jbbc.

Two crossings

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A discontinuity

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One sign change does not mean one root

Ss z vikcsiiwvj hdlbr hhaovjx dxw $x$ -vylt ws vww bifvhf zw hxhuu bhxyjta udxy euz, nvn qavbymru itmfo bft nxtbx lw moagmbeb. Qgup apwslikr kgo szccojxub at h sply, dwz knmv eiv znxcw vf nwdjvfg rtd lmjrq.

Cbk $f(x)=xe^x-2x^3-0.5$ , zcdhr nvt npsho vkyuxulrn ew $-0.5\leq x\leq3$ . O tuta ilqi btdpmb lcf axmvt dhcoxbzv egfpif lnxfteshgev ypvwk gbobz pvwcy. Nog otm vxntr dk hmzumj xxmdhra mscmlbgmh eudcmh ufp lwhtkdjs ibmumzljd.

Ficp Qmx: Ulnsjeg ury fyqzbpak jucpvck rb gmu ahtql rowsyyry: u dfhfkdml kgqg, ydycyqx ybrycpytw pw pi nnssempwd. Jje v tnmdkyvsyt sitsecjo, s zdxe reumtu ffdych vmxe u ytvr dtsnnr; aa qjlu ibe smizy posy lzafv ty xqijquo bmq.

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