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A sinrplenrcthodlbrobtainingan estirnate the root of the ecluation of /(r) :0 is to make whercitcrosses a plot ol'thc firnctionant'l observc thcraxisThis point,which represents t h e rv a l u el b r w h i c h l ( r l : 0 p r o v i d c s r o u g ha p p r o x i r n a t io fnt h er o o t a o
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TheGrophicolApprooch to the Problem Stotement Use thc graphicalapproach dcternrine nrass the bungee of jurrper with a drag coel'l'icicnt ol'025 kg/rn to havca velocityol'36 m/s after4 s of free i 1 a l l N o t c :T h e a c c e l c r a t i o n ' u r i r v i t vs 9 8 1n r / s r r ol m M s S o l u t i o n T h c l i r l k r w i n g A T L A B s c s s i o n c t su p a p l o t o l ' E q ( 5 2 )v e r s u s a s s :
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'' ' ::' r'ril = nlll '' I = 4; v '; ' 1 r 1 , , , I f ) = r i q f L ( r 1 * m 1 rc : c 1 ) * t a n l r ( : ; q r 1 ( J * c d / n l r ) * L ) plot (rrr;r,11-r),qrr11 ()2,\; q c ) l l 1 ; '\6;
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METHODS AND INITIAL GUESSES 53 BRACKETING
of The functioncrosses llr axisbetweenlrl0 and l50 kg Visualinspection the plot thc providesa roughestimate the root of 145kg (about320 lb) The validity of the graphiof can by it cal estimate be checked substituting into Eq (52)to yield
>> sqrt (g*145/ccL)*r,anh(::qrt,(g*cd/1,i5)*t,) v
00456 which is close to zero It can also be checked by substitutingit into Eq (Sl ) along with the parametcr values liorr this exarnple to give ( >>- sqrt, (q*14filco) *t,anh(scTrt-q*cd/ r45) *t)
350456 which is close to the desired fall velocity ol-36 rr/s
valuebeciiuse thcy arc not vcly precise are Craphicaltechniques ilf limited practical ol'rootsTheseestiHowevcr,graphical methods can be utilizedto obtainroughcstimatcs gucsscs numcrical in lor mcthods discussed this chapter mates can be ernployed starting as firr roughestimatcs ol'tlreroot,graphicul interpretations usefirl are Asidefiorn providing ol'the lirnctions and anticipating pitlallsof the nurnerical the understanding ploperties the rrethods For cxample,Fig -51 showsa numberof'ways in which rootscan occur (or be prescribcd a lowcr bound,r1 lb by andan uppcrboundr,,Figurc-5 dcabsent) an interval in (r) Howpictsthecase by and values o1'/ wherea singlcroot is brackctcd ncgative positive he r f e v e r , F i g 5 l r l , w h e r ( - r r ) a n d / ' ( - r , , ) a r e a l s o o n o p p o s i t e s itd c soa x i s , s h o w s t h r c c /c In if havcoppositc signs, thcrc rootsoccurring within the interval general, / (rr)and / (-r,,) As la r', of by arean odd number rootsin the interval indicatcd Fig-5 ancl il' l (rr ) andf (r,, ) ol'rootsbctwecn values thc havethe sarnc sign,thereareeitherno rootsor an evennunrbcr gcncralizalions usually arc lrue,lherearecases wherethcy do not holcl Althoughthcse l u ) F o r c x a r n p l e , r n c t i o n t h a ta r ct a n g c n t i a l t h c , r a x i s( F i g - 5 2 aa n dd i s c o n t i n u o u's n c ti s to p A o n tl t i o n s( F i g 5 2 b )c a nv i o l a t e h e s c r i n c i p l c s n c x a r n p l c l ' a I ' u n c t i o t h a ti s t a n g e n t i a t l t : ' t t h ca x i si s t h ec u b i ce q u a t i o n/ ( - r ' ) ( r - 2 ) ( r- 2 ) ( x - 4 ) N o t i c ct h a t , r: 2 t n a k c sw t r l n t e r m si n t h i s p o l y n o m i ac q u a lt o z e r o M a t h e r n a t i c a l l y , r : 2 i s c a l l e d' tt r r u l t i l t l c r t t that Although they are bcyond the scopeof this book, thereare specialtcchniqr"rcs arc 2002) and Canale expressly designed locaternultipleroots(Chapra to in nrakes difiicult to dcveloptirolit The cxistencc cases thc typedepictcd Fig-52 of of guaranteed locate the rootsin an interval Howevcr,whcn proof computer algorithms to all in with graphical approaches, rnethods the described thc lirlkrwingsccusedin conjunction routinelyby cnginccrs, confl'onted tions are extrerrelyusefulfor solvingrnanyproblerns scientists, appliedrnathematicians and
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