CHAPTER 6 CURVE FITTINGS

CHAPTER 6 CURVE FITTINGS

Chpter 6 : TOPIC COVERS CURVE FITTINGS Lest-Squre Regresso - Ler Regresso - Poloml Regresso Iterpolto - Newto s Dvded-Derece Iterpoltg Polomls - Lgrge Iterpoltg Polomls - Sple Iterpolto

LEARNING OUTCOMES INTRODUCTION It s epected tht studets wll be ble to: Derette the udmetl derece betwee regresso d terpolto Derve ler lest-squres regresso d ble to ssess the relblt o the t usg grphcl d qutttve ssessmets Lerze dt b trsormto Descrbe the log/correlto betwee Newto s poloml d the Tlor seres epso d how t reltes to the tructo error Idet tht the Newto d Lgrge equtos re merel deret ormultos o the sme terpoltg poloml d uderstd ther respectve dvtges d dsdvtges Relzed tht dt pots do ot hve to be equll spced or prtculr order or ether the Newto or Lgrge polomls Descrbe wh sple uctos hve utlt or dt wth rpd/brupt chge

6. Itroducto CHAPTER 6 : CURVE FITTINGS Dt gve s ote s dscrete/seprte/solte vlues log cotuum/rge/vret. However we re requred to estmte t pots betwee the dscrete vlues. Curve ttg descrbes techques to t curves to such dt to obt termedte/-betwee estmtes. Two geerl pproches or curve ttg:. Dt ehbts sgct degree o error or ose : the strteg s to derve sgle curve tht represets the geerl tred o the dt. The curve s desged to ollow the ptter o the pots tke s group whch c be clled s lest-squres regresso Fg. 6... Dt s kow to be ver precse: the pproch s to t curve or seres o curves tht pss drectl through ech o the pots; c be clled s terpolto Fg. 6. b d c.

Fgure 6.

Fg. 6. shows sketches developed rom sme set o dt b egeers. Fg. 6. : dd ot ttempt to coect the pot but chrcterzed the geerl upwrd tred o the dt wth strght le. Fg. 6.b: Used strght-le segmets or ler terpolto to coect the pots. Ver commo prctce egeerg. I the vlues re close to beg ler such ppromto provdes estmtes tht re dequte or m egeerg clcultos. However the dt s wdel spced sgct errors c be troduced b such ler terpolto. Fg. 6.c: Used curves to tr to cpture suggested b the dt.

Two tpes o pplctos re geerll used whe ecoutered wth ttg epermetl dt:. Tred lss: the process o usg the ptter o the dt to mke predctos. Hgh precso dt usg terpoltg polomls d or mprecse dt s ote lzed wth lestsqure regresso. b. Hpothess testg: the process where estg mthemtcl model s compred wth mesured dt. For ukow coecets o model ecessr to determe vlues tht best t the observed dt. For vlble coecets t m pproprte to compre predcted vlues o the model wth observed vlues to test the dequc o the model.e usg some o sotwre or lss dt. Curve ttg techque s lso used to derve smple ucto to ppromte complcted uctos.

6. Lest-Squres Regresso To mmze the dscrepc/dereces betwee the dt pots d the curve plotted. Epermetl dt s ote o ths tpe where substtl error s ssocted wth dt. Sometmes poloml terpolto s pproprte d m eld ustsctor results whe used to predct termedte vlues e.g Fg. 6.. Fg. 6. : shows seve epermetll derved dt pots ehbtg sgct vrblt. Vsul specto o the dt suggests postve reltoshp betwee d or overll tred dctes hgher vlues o re ssocted wth hgher vlues o. Fg. 6. b: 6 th order terpoltg poloml s tted to the dt t wll pss ectl through ll o the pots. However becuse o the vrblt the dt the curve osclltes wdel the tervl betwee the pots.e vlues =.5 d = 6 pper to be well beod the rge suggested b the dt. Fg. 6. c: A strght le c be used to geerll chrcterze the tred o the dt wthout pssg through prtculr pot.

Fgure 6.

The le Fg. 6. cc be determed b vsull spect the plotted dt d the sketch best le through the pots. 6.. Ler Regresso The smplest emple o lest-squres ppromto s ttg strght le to set o pred observto;........ The mthemtcl epresso or the strght le s: = o + + e ------------ 6. o : coecets represetg the tercept d the slope respectvel e : error/resdul betwee the model d the observto. B rerrge equto 6.: e = - o - Thus the error s the derece vlues betwee the true vlue o d the ppromte vlue o + whch predcted b the ler equto/ler model.

Crter or best Ft To kow how best t le through the dt s b mmze the sum o resdul error. The sum o the resdul error s gve b s show b prevous equto; -------- 6. where; : totl umber o pots A strteg to overcome the shortcomgs o Crter or best t o ler regresso s b mmse the sum o the squres o the errors betwee the mesured d the clculted wth the ler model s show b Equto 6.; - - - - 6. e el mesured r e S mod

To determe vlues or o d deretl equto 6. wth respect to ech coecet o d ; Settg equl to zero mmze S r ; = o = o Set; o =. o Thus;. o + = -------- 6.4 o + = -------- 6.5 These re clled the orml equtos whch c be solved smulteousl or d o ; Thus; where; S r -------- 6.6 -------- 6.7 = mes or d respectvel S r

Emple 6.: Ler Regresso Ft strght le rom the Tble below. Tble 4 5 6 7.5.5. 4..5 6. 5.5 Soluto: 7 8 4 4 8 7 4 4 7 9.5.4857 Usg equto 6.6 d 6.7 to clculte d : = 79.5 84 / 74 8 =.89857 rom equto 6.6 =.4857.898574 =.74857 rom equto 6.7 Thereore the lest-squres t s: =.74857 +.89857 s show b Fg 6.c

6.. Qutcto o Error o Ler Regresso Equto 6. c be epled b Fg. 6. below: Fg.6. The resdul error ler regresso represets b the vertcl dstce betwee the dt coected b strght le. r e S Vertcl dstce

Two crter or lest-squre regresso wll provde the best estmtes o o d clled mmum lkelhood prcple sttstcs:. The spred o the pots roud the le o smlr mgtude log the etre rge o the dt.. The dstrbuto o these pots bout the le s orml. I these crter re met stdrd devto or the regresso le s gve b equto: S Sr ---------- 6.8 S / / - : stdrd error o estmte : predcted vlue o correspodg to prtculr vlue o : two dt derved/drw rom to estmtes o d ; were used to compute S r we hve lost degree o reedom

Equto 5.8 s derved rom Stdrd Devto bout the me : S St -------- 6.8 S t : totl sum o squres o the resduls betwee dt pots d the me - - - - - 6.8b Just s the cse wth the stdrd devto the stdrd error o the estmte qutes the spred o the dt. However S / qutes the spred roud the regresso le Fg 6.4b. I cotrst to the orgl stdrd devto S tht qutes the spred roud the me Fg 6.4.

Fg 6.4 Spred roud the me. Spred roud the regresso le. Fg 6.5 Smll resdul error b Lrge resdul error

Ths s useul or comprso o severl regresso Fg 6.5. Let S t = totl sum o the squred roud the me or the depedet vrble our cse Ater perormg regresso thus; S r = sum o the squres o the resduls roud the regresso le Equto 6.. The derece betwee the two quttes S t -S r qutes the mprovemet or error reducto due to descrbg the dt terms o strght le rther th s verge vlue. Becuse mgtude o ths qutt s scle-deped the derece s ormlzed to S t to eld: r S t S r S t - - - - - 6.9 r : coecet o determto r : correlto coecet = r

For perect t: S r = d r = r = whch c be epled s % o vrblt o the dt For r = r = d S r = S t the t represets o mprovemet. A ltertve ormulto or r tht s more coveet or computer mplemetto s: - - - - - 6. r

Emple 6. : Estmto o Errors or the Ler Lest-Squre Ft Compute the totl stdrd devto the stdrd error o the estmte d the correlto coecet or the dt Emple 6.. Soluto: 4 7.4857.5 8.5765.687.5.86.565..48.47 4 4..65.65 5.5.5.5896 6 6. 6.6.797 7 5.5 4.98.99 4..74.99 S S t r.74.99 rom Eq.6.8b rom Eq.6.

The stdrd devto s: S St.74 7.9457 The stdrd error s: S Sr.99 7.775 Thus becuse S / < S the ler regresso model hs mert/dvtge. The etet o the mprovemet s quted b equto 6.9: r = S t S r =.74.99 =.868 S t.74 or r r.868.9 The results dcte tht 86.8% o orgl ucertt hs bee epled b the ler model.

6. Lerzto o Noler Reltoshps Ler regresso provdes powerul techque or ttg the best le to dt. The reltoshp betwee the depedet d depedet vrbles s ler. But ths s ot lws the cse thus rst step regresso lss should be to plot d vsull spect whether the dt s ler model or ot. For emple Fg. 6.6: some cses techques such s poloml regresso wll be dscussed secto 6.4 s obvousl ppl.

Fg. 6.6

Fg. 6.7

Emple: Fg. 6.7 Fg. 6.7: the epoetl model b e ------ 6. b : costts b Ths model s used m elds o egeerg to chrcterze quttes. Quttes crese Quttes decrese b postve b egtve Fg. 6.7b: the smple power equto b ------ 6. b : costts coecet b or Ths model hs wde pplcblt ll elds o egeerg.

Fg. 6.7c: the sturted-grow-rte equto b ------- 6. where b = costts coecet Ths model well-suted or chrcterzg populto growth uder lmtg codtos level o / sturtes s creses The Noler Regresso techques re vlble to t these equtos to epermetl dt drectl. However smple ltertve s to use mthemtcl mpultos to trsorm the equtos to ler orm.

Epoetl Equto: For emple rom equto 6. c be lerzed b tkg turl logrthm to eld: e b ------ 6. Tkg ts turl logrthm: l l sce l e b l e l l b ------ 6.4 A plot o l versus wll eld strght le wth slope o b d tercept o l Emple 6.: Ft epoetl model to:.4.8..6.. 75 4 7 75

Soluto: B pplg ler regresso method s dscussed prevousl Emple 6.. l l.4 75 6.67.6.6489.8 6.9775.64 5.56. 4 7.448.44 8.6974.6 7.69.56.644. 7 7.97 4. 5.84. 75 8.95 5.9 8.97875 8. 44.496496 4.9 6.7555 6 8. 4.9 8. 6.8 l l l 44.496496 6.7555 44.496496 6 7.468

.84 8. 64.9 8.44.496496 66.7555 l l b b 6.5 l.84.8 7.468 l l b 58 6.5 l 6.5 e b e e.84 58 Strght-le: b.84 6.5 l l l Epoetl: b e

Power Equto: Equto 6. c be lerzed b tkg bse- logrthm to eld: b log log b log -------- 6. -------- 6.5 A plot o log versus log wll eld strght le wth slope o b d tercept o log

Emple 6.: Lerzto o Power equto Ft equto 6. to the dt Tble below usg logrthmc trsormto o the dt. b log log blog 4 5.5.7.4 5.7 8.4 Soluto: B pplg ler regresso method s dscussed prevousl Emple 6.. Fg. 6.7b s plot o the orgl dt ts utrsormed stte Power Equto. Fg. 6.7e s plot o the trsormed dt Ler Form.

log log log log log.5 -..7..6.96.686.4.477.54.759.5478 4 5.7.6.75.644.456 5 8.4.699.9.4886.644478.79.4.695.458 5 log.79 5.79 log.695 log.458 log log log log.4 5.4.458.468

b log log log log log log b 5.458.79.4 5.695.79.75 log log blog l. Strght-le: log log.468.75.458 log blog..75log Power: b log...5 b.5.75

Sturto-growth rte Equto: Equto 6. c be lersed b vertg/overtur t to eld equto 6.6: b b ------- 6. ------- 6.6 A plot o / versus / wll eld strght le wth slope o b / d tercept o / I ther trsormed orms these models re t usg ler regresso order to evlute the costt coecets. The could the be trsormed bck to ther orgl stte d used or predctve purposes s dscusses Emple 6..

Emple 6.4: Lerzto o Sturto-growth rte equto Soluto: B pplg ler regresso method s dscussed prevousl Emple 6.. b b.75.5 4 6 8 8.5.8...6.7.8.7

/ / / / /.75.8..5.7777.6666..5.769.5.846.5..4.8.6. 4.6.5.65.65.56 6.7.6667.588.78.98 8.8.5.55555.56.694 8.5.7.765.588.8.45.896 5.94.74.87.744 7 5.94.4 7.896.87.74 5.94.896 7

.595.896 7.74.8965.94 7.87 b b.499.595.4.744 b.87.595.595.499 b b.87 Strght-le: Sturto-growth: b b.595.499

Thus the tercept log = -. d b tkg the tlogrthm = -. =.5. The slope s b =.75 cosequetl the power equto s : =.5.75 Ths curve s plotted Fg. 6.8 dctes good t vsull spect. Fg. 6.8

6.4 Poloml Regresso Prevous secto ws developed to derve the equto o strght le usg the lest-squre crtero. Some egeerg dt ehbtg ptter such s see Fg. 6.6b where s poorl represeted b strght le [Fg.6.6]. As dscuss Secto 6. oe method to ccomplsh ver best t dt s to use trsormtos. Aother ltertve s to t polomls to the dt usg poloml regresso. Poloml regresso c be redl eteded rom the lest-squre procedure. For emple to t secod-order poloml or qudrtc: = o + + + e The sum o the squres o the resduls b comprg wth equto 6. : S r e ----- 6.7

The b ollowg prevous procedure tke the dervtve o equto 6.7 wth respect to ech o the ukow coecets o d o the poloml s : Settg equl to zero d rerrge to set orml equtos; Set; o =. o ------ 6.8 r r r S S S 4

Now we hd equtos o ler wth ukows coecets o d whch c be clculted drectl rom observed dt. I mtr orm: The two-dmesol secod order poloml/qudrtc cse c be esl eteded to m th-order poloml s: = o + + +.... m m + e Thus stdrd error or m th-order poloml : -------- 6.9 Correlto coecet etrcted rom equto 6.9 m S S r t r t S S S r 4

Emple 6.4: Poloml Regresso Ft secod order poloml to the dt tble below: 4. 7.7 7.7 7.7.6 4 8 6 7. 54.4 7. 9 7 8 8.6 44.8 4 4.9 6 64 56 6.6 654.4 5 6. 5 5 65 5.5 57.5 5 5.6 55 5 979 585.6 488.8 From the gve dt: m = Σ = 5 Σ 4 = 979 = 6 Σ = 5.6 Σ = 585.6 =.5 Σ = 55 Σ = 488.8 = 5.4 Σ = 5

4 488.8 585.6 5.6 979 5 55 5 55 5 55 5 6 o Thereore the smulteous ler equtos re: Solvg these equtos through NM techque such s Guss elmto gves: o =.47857 =.599 d =.867 Thereore the lest-squres qudrtc equto or ths cse s: =.47857 +.599 +.867 The to clculte stdrd error S / d correlto coecet r :

- - o - -. 544.44.4 7.7 4.47.86.6 4..858 7...849 4 4.9 9..695 5 6. 7..949 Σ 5.6 5.9.74657 S S t r 5.9.74657 The stdrd error regresso poloml: S Sr m.74657 6. The coecet o determto s equto 6.9: r = S t S r / S t = 5.9.74657 / 5.9 r =.9985 The correlto coecet s r =.9995

From the results dcte tht 99.85% o the orgl ucertt hs bee epled b the model secod order poloml/qudrtc. Ths result supports the cocluso tht the qudrtc equto represets ecellet t s evdet rom Fg. 6.8c/Fg.7.. Fg. 6.8c

Iterpolto

6.5 INTERPOLATION To estmte termedte vlues betwee precse dt pots. The most commo method used s poloml terpolto. Geerl ormul or th-order poloml: = o + + + +... + ---- 6. Poloml terpolto cossts o determg the uque th-order poloml tht ts + dt pot. For + dt pot there s oe d ol oe poloml o order tht psses through ll the pot. For emple there s ol oe strght le rst-order poloml tht coects two pots + dt pot Fg. 6.9 d ol oe prbol coects set o three pots Fg.5.9b. Ths poloml provdes ormul to compute termedte vlues b usg two ltertve mthemtcl ormts:. Newto poloml b. Lgrge poloml

Fg. 6.9 6.6 Newto s Dvded-Derece Iterpoltg Polomls The most populr d useul poloml orms. Beore we dscuss urther o geerl equtos we eed to kow the rst- d secod-order versos becuse o ther smple vsul terpretto.

6.6. Ler Iterpolto The smplest orm o terpolto s to coect two dt pots wth strght le. From Fg.6. d b usg smlr trgles: Fg 6.

Smlr trgles equto: ---- 6. Rerrge bove equto to eld; ---- 6. Equto 6. s clled s ler terpolto ormul where; = rst-order terpolto poloml = te-dvded-derece s ppromto o rst dervtve slope o the le coectg the pots I geerl the smller the tervl/spce betwee the dt pots the better the ppromto.

Emple 5.5 : Ler Iterpolto Estmte the turl logrthm o l usg ler terpolto. Frst perorm the computto b terpoltg betwee l = d l 6 =.79759. The repet the procedure but use smller tervl rom l to l 4.8694. Note tht the true vlue o l s.6947. Soluto: B usg equto 6. ler terpolto or l rom o = to = 6 to gve; Represets error o t = 48.%.79759.5859 6

The usg the smller tervl rom o = to = 4 elds;.8694 4.4698 Thus usg the shorter tervl reduces the percet reltve error to t =.%. Both terpoltos re show Fg.6. log wth true ucto. True vlue or l Less ε t s ppromte vlue er to the true vlue Fg. 6.

6.6. Qudrtc Iterpolto The error Emple 6.5 ler terpolto resulted rom ppromto curve wth strght le. We c mprove the estmte wth Secod-Order Poloml qudrtc poloml or prbol. Thus; b b b ----- 6. or epso o Equto 6. bove = b o + b b o + b + b o b o b or collectg terms = o + + where; o = b o b o + b o = b b o b = b

Thus equtos 6. d 6. re ltertve equvlet ormultos o the uque secod-order poloml jog three pots. To determe the vlues o coecet b o b d b : For b o rom equto 6. let = o ; The -------- 6.4 Equto 6.4 c be substtuted to equto 6. whch c be evluted t = or; ------- 6.5 Fll equto 6.4 d 6.5 c be substtuted to equto 6. whch c be evluted t = d solved ter some lgebrc mpultos or; ----- 6.6 b b b

Emple 6.6: Qudrtc Iterpolto Ft secod-order poloml to the three pots used Emple 5.5: o = o = = 4 =.8694 = 6 =.79759 Used the poloml to evlute l. Soluto: Applg equto 6.4 elds to clculte b : b Equto 5.5 elds:.8694 to clculte b b. 4698 4

Equto 6.6 elds to clculte b : b b.79759.8694 6 4 6.4698.587 Substtutg these vlues to equto 6. elds the qudrtc ormul: = b o + b o + b o = +.4698.587 4 whch c be evluted t = or b substtutg = the equto bove; =.5658444 whch represets reltve error o t = 8.4%.

Thus the curvture troduced b the qudrtc ormul Fg.6. mproves the terpolto compred wth the result obted usg strght les Emple 6.5 d Fg.6.. Fg. 6.

6.6. Geerl orm o Newto s Iterpoltg Polomls For the th-order poloml geerl equto: = b o + b o + b o +.... + b o.... - ----- 5.7 B ollows prevous methods ler d qudrtc terpoltos dt pots c be used to determe coecets b o b.... b For th-order poloml + dt pots re requred: [ o o ] [ ].... [ ] The we used these dt pots d ollowg equto s to evlute the coecets: b o = o -------- 5.8 b = [ o ] -------- 5.9 b = [ o ] -------- 5... b = [ -.... o ] -------- 6.

Where the brcketed [ ] ucto evlutos re te dvded dereces; For emple the rst te dvded derece s represeted geerll s; ----- 6. The secod te dvded derece whch represets the derece o two rst dvded derece s epressed geerll s; ----6. Smlrl the th te dvded derece s; ----- 6.4 j j j ] [ k k j j k j ] [ ] [ ] [ ] [ ] [ ] [

These dereces c be used to evlute the coecets equtos 6.8 through 6. whch c the substtuted uto equto 6.7 to eld the terpoltg poloml; ----- 6.5 Equto 5.5 s clled Newto s dvded-derece terpoltg poloml.... ] [... ] [ ] [

Emple 6.7: Newto s Dvded-Derece Iterpoltg Polomls From the prevous emple dt pots t = = 4 d = 6 were used to estmte l wth prbol o thrd-order poloml. Now ddg ourth pot [ = 5 =.6948] estmte ourth- order Newto s terpoltg poloml. Soluto: The thrd-order poloml wth = s = b o + b o + b o + b o - The rst dvded dereces re:.4698 4.8694 ] [.76 4 6.8694.79759 ] [.86 6 5.8694.6948 ] [

The secod dvded dereces: The thrd dvded dereces:.587 6.4698.76 ] [.4 4 5.76.86 ] [.786559 ] [ 5.587.4 ] [

Fll: b b.4698 b.587 b.786559.4698.587 4.786559 4 6 Thus t = ;.687686 whch represets reltve error o t = 9.%.

Soluto: Now ou c tr or the 4 th -order poloml!! The ourth-order poloml wth = 4 s 4 = b o + b o + b o + b o - b 4 o - - The rst dvded dereces re: 6.5 6 6 4.75 ] [ 8 6 4 6 ] [ 5.5 5 9.75 4 ] [ 7.5 5 5.5 9.75 ] [ 4 4 4

The secod dvded dereces: The thrd dvded dereces: The ourth dvded dereces:.75 8 6.5 ] [.75 5 6 5.5 8 ] [ 7.5 5.5 ] [ 4.5 5.75.75 ] [.5 6.75 ] [ 4.5.5 ] [ 4

Fll: Fll ou wll get less ε t th 9.% mprovg estmto rom thrd-order poloml 4 5.5 5 7.5 5.5.5 7.5 5.5 4 4 b b b b b 4 5.5 5 7.5 5.5 4 4

The complete cubc poloml Fgure 6.: Fgure 6.

6.6.4 Error o Newto s Iterpoltg Poloml Notce tht the structure o Eq 6.5 s smlr to thetlor seres epso. Tructo error or the Tlor seres: where s somewhere the tervl to +. For -th order terpoltg poloml logous error s: Fte dvded drece or +th dervtve: where s the +th te dvded derece. Becuse the equto cots the ukow t cot be solved or the error. I + s vlble the error s:! R! R ] [ R ] [ ] [ R Fte dvded derece

6.7 Lgrge Iterpolto It s smpl the reormulto o the Newto poloml tht vods the computto o dvded dereces. where For emple the ler verso = s: L j j j j L

For the secod order =: L L L

Emple: Lgrge Iterpoltg Polomls Use Lgrge terpoltg poloml o the rst d secod order to evlute l bsed o the dt gve. = = = 4 =.8694 = 6 =.7976 Soluto: Frst-order poloml t = 4.8694 4 4.4698 Secod-order poloml t = 4 6 6 4 6 4 4 6 4.7976.5658444 6 6 4.8694

6.8 Sple Iterpolto I the prevous sectos the -th order polomls were used to terpolte betwee + dt pots. For e.g we c derve perect seveth-order poloml or eght pots Fgure 6. bc However there re cses where these uctos c led to erroeous/vld results becuse o roud-o error d overshoot. Fg.6. through c shows how hgher-order polomls red to swg through wld osclltos. A ltertve pproch s to ppl lower-order polomls to subsets o dt pots to mmse the wld osclltos. Such coectg polomls re clled sple uctos.

Fgure 6.

Fgure bove s showg the drtg techque o usg sple to drw smooth curves through seres o pots. At the ed pot the sple strghtes out lower-order poloml. Ths s clled turl sple.

Fgure 6.4

6.8. Ler Sple Smplest coecto betwee two pots s strght le. The rst-order sples Fg. 6.4 s or group o ordered dt pots c be deed s set o uctos. where m s the slope o the strght le: m m m m

Emple: Frst-order Sples Ft the dt the tble wth rst-order sples. Evlute the ucto t = 5. Soluto: Determe the slope betwee pots or ech tervls. For tervl = 4.5 to = 7 to evlute the ucto t = 5 : Thereore;..5 4.5. 7..5 9..5.5 m.6 7 4.5 m..6 4.5 5..65 4.5.

6.8. Qudrtc Sples To derve d order poloml or ech tervl betwee dt pots. The poloml or ech tervl s: For + dt pots there re tervls d ukow costts d hece equtos requred. These re:. The ucto vlues o djcet/djog polomls must be equl t the teror kots. Ths c be represeted s: or = to. Thereore totl o codtos.. The rst d the lst uctos must pss through ed pots. Ths dds two ddtol equtos d totl o + = codtos. c b c b c b c b c b

. The rst dervtves t the teror kots must be equl dervtves re equll. ' b Thereore the codto c be represeted geerll s; b b or = to. Ths provdes other - or totl o + = codtos. 4. Assume the secod dervtve s zero t the rst pot o mtrc. =

Emple: 4 dt pots = tervls = ukows eeds to solve wth 9 equtos. = = 4 equtos 4.5 4.5 7 7 b b b b..5 4.5. 7..5 9..5 4.5 c 4.5 c 7 c 7 c..5.5. b. Pssg the rst d lst uctos provde equtos: 9 b c.5 8 9b c.5

c. The rst dervtves gve equtos: d. equto: = Becuse ths equto speces = thus we hve ol 8 smulteous equtos. I mtr orm: 4 4 9 9 b b b b.5.5.5.5 4 4 9 9 8 7 49 7 49 4.5.5 4.5 c b c b c b

These equtos re solved wth the results: = b = - c = 5.5 =.64 b = -6.76 c = 8.46 = -.6 b = 4.6 c = -9. Substtuted to the orgl qudrtc equtos: = - + 5.5. < < 4.4 - - - - =.64 6.76 + 8.46 4.5 < < 7. - - - - b = -.6 + 4.6 9. 7. < < 9. - - - - c Thereore t = 5 s tkg equto b bove: 5 =.645 6.765 + 8.46 =.66

END OF CHAPTER 6