Numrial mthos, Mi ris a f ( ) 6 f ( ) 6 6 6 a = 6, b = f ( ) So. 6 b n a n 6 7.67... 6.99....67... 6.68....99... 6.667....68... To.p., th valus ar =.68, =.99, =.68, =.67. f (.6).6 6.6... f (.6).6 6.6.7... Thr is a hang of sign in this intrval so α =.6 orrt to.p. f ( ) f (.9). f (.) 7. b Thr is an asymptot at whih auss th hang of sign, not a root. a b Thr is on positiv an on ngativ root of th quation p() = q() at th points of intrstion. p( ) q( ) i.. n n.9687....9687....9667....9667....966....966....966... To.p., th valus ar =.969, =.967, =.966, =.966... Thr an b no squar root of a ngativ numbr. Parson Euation Lt 7. Copying prmitt for purhasing institution only. This matrial is not opyright fr.
a b g( ) 6 g() 6 g() 6 6 Thr is a hang of sign in th intrval, so thr must b a root in th intrval, sin f is ontinuous ovr th intrval. g( ) 6 6 p =, q = 6, r = n n 6 g( ) 6.69....69... 6.6977....6977... 6.768... To.p., th valus ar =.6, =.697, =.768. g(.77).77.77 6.9... g(.78).78.78 6.6... Th sign hang implis thr is a root in this intrval so α =.78 orrt to.p. 6 a f ( ) sin f (.) (.) sin (.).68... f (.) (.) sin (.).989... f(.) < an f(.) > so thr is a hang of sign, whih implis thr is a root btwn =. an =.. b sin sin A sin to ah si. sin A to ah si. a g( ) g( ) b, sin Divi ah trm by. sin Simplify. So p = an q =...8 sin (.)..9....8 sin (.9...)..76....8 sin (.76...)..9....8sin(.9...)..98... To.p., th valus ar =., =.8, =.9, =.. Parson Euation Lt 7. Copying prmitt for purhasing institution only. This matrial is not opyright fr.
7 a b Th lin mts th urv at two points, so thr ar two valus of that satisfy th quation. So has two roots. Lt f() = f(.) = (.).... f(.) = (.).8.... f(.) < an f(.) > so thr is a hang of sign, whih implis thr is a root btwn =. an =.. Multiply by. So 8 a Using a a, b, b b a () () () ()( ) with 9 So.7... Th positiv root is. to.p. g( ) 7 g ( ) b Using = 6.6, g( ) g ( ) g(6.6) 6.6 g (6.6) 6.6 7(6.6 ) (6.6) 6.6 (6.6 ) (6.6) 6.66 orrt to.p. g() is a fator of g() g( ) ( )( 6 ) ( )( 6 ) Othr two roots of g() ar givn by 6 6 6 6 Prntag rror: 6.66.7% 9 a f ( ) s f (.) s..8.8... f(.) s..789... Th sign hang implis thr is a root in this intrval. Parson Euation Lt 7. Copying prmitt for purhasing institution only. This matrial is not opyright fr.
9 b f ( ) s tan Using =., f ( ) f ( ) f (.). f (.) a.8 A ah si..8 to.8.... s.tan..97... α =. orrt to.p. f (.89) s(.89) (.89)... f (.9) s(.9) (.9).69... Thr is a hang of sign in this intrval, so thr is a root β =.9 orrt to.p..8.8 ( ) Multiply ah si by ( ). Simplify..8 ( ).8.8 Divi ah si.8 by..8 Simplify.8 (rmmbr a a ). A to ah si..8 Subtrat ah si..8 from.8 Divi ah trm by..8.. Simplify. b..8(.)...7....8(.7..)...6....8(.6...)...698... So.7 orrt to.p. Parson Euation Lt 7. Copying prmitt for purhasing institution only. This matrial is not opyright fr.
.8 A.8.8 n to ah si. Tak logs..8 n ( ) Simplify using n n..8 n ( ) Divi ah.8.8 si by.8..n ( ) Simplify..8. f ( ) ln. f (.)..688... f ( ) f ( ). f (.). ( ln.).8... f (.). f (.).688.....8....68... To a son approimation, α =.6, to.p..9 f (.9).9.7....96 f (.96).96... Thr is a hang of sign in this intrval so α =.96 to.p. So p..6.n [ (.6)].66....n [ (.66...)].69....n [ (.69...)].6... So.6 (.p.) a y ln y ln y ln ln y y y( ln ) ln f (.) os..6.8... f (.) os.6.7.7... Th sign hang implis thr is a root in this intrval. a f ( ) os At B, f ( ) sin 8...,.669..., 6.79...,t From th graph so.669 So.867... f (.867...) os.669....8... b f ( ) sin... B has oorinats (.87,.). b f ( ). f (.)..98....6 f (.6).6... Th sign hang implis thr is a root in this intrval. Parson Euation Lt 7. Copying prmitt for purhasing institution only. This matrial is not opyright fr.
n aros n. aros.... aros.77....969... aros.78....876... aros.78....887... To.p., th valus ar =., =.97, = 88, =.89. f (.7) os 6.8.8.99... f (.7) sin 6.8.76... f ( ) f ( ) f (.7).7 f (.7).99....7.76....778... To.p., th son approimation is.78. f (.77) os6.8.87... f (.78) os6.8.8... Thr is a hang of sign so thr is a root of.78 orrt to imal plas in this intrval. ii iii f ( ) 6 7 ( ) 7 7 f ( ) 6 7 7 7 b As B is a point of infltion f ( ). Using in part iii.786... 7.786... 7.78... 7.796....78....796....7... 7.7....7... Corrt to.p., an approimation for th -oorinat of B is.7. A has a ngativ -oorinat. Formula iii givs th positiv fourth root, so annot b us to fin a ngativ root. Challng a 6 f ( ) 7 f ( ) 6 f ( ) 6 i f ( ) 6 7 7 Parson Euation Lt 7. Copying prmitt for purhasing institution only. This matrial is not opyright fr. 6
As A is a point of infltion, f ( ). f () f ( ) ( ) 6( ) Thr is a hang of sign, so th -oorinat of th root A lis in th intrval [, ]. f ( ) 6 Using th Nwton Raphson formula: f f Using.9 f (.9).9 f (.9).9.896... (.9) 6(.9) (.9) 6 9.68..9 87.8 6.8.9.896... 8.8 f (.896...).896... f (.896...) Th -oorinat of A is.897 orrt to.p. Parson Euation Lt 7. Copying prmitt for purhasing institution only. This matrial is not opyright fr. 7