Review Sheet: Chpter Cotet: Essetil Clculus, Erly Trscedetls, Jmes Stewrt, 007 Chpter : Fuctios d Limits Cocepts, Defiitios, Lws, Theorems: A fuctio, f, is rule tht ssigs to ech elemet i set A ectly oe elemet, clled f (, i set B I this defiitio, domi of f = the set A rge of f = the set of ll possible vlues of f ( s vries i A A idepedet vrible is symbol tht represets rbitrry umber i the domi of fuctio f A depedet vrible is symbol tht represets umber i the rge of f A grph of fuctio is the set of ordered pirs, f ( so log s domi( f ) (co) The verticl lie test A curve i the y ple is the grph of fuctio of if d oly if o verticl lie itersects the curve more th oce The bsolute vlue of umber,, deoted by, is the distce from to 0 o the rel umber lie I geerl, 0 0 (thm) 0 for every umber For f to be eve fuctio, f ( f ( for every umber i its domi For f to be odd fuctio, f ( f ( for every umber i its domi A fuctio f is clled icresig o itervl I if f f ( ) wheever i I A fuctio f is clled decresig o itervl I if f f ( ) wheever i I A fuctio P is clled polyomil if P( 0 I this defiitio, the letter is o-egtive iteger (i mth, we sy Z to deote this ide ), d the umbers 0,,,,, re ll costts clled coefficiets The degree of polyomil is the highest power of so log s the coefficiet of the ssocited term is ot 0 (co) The domi of ll polyomil fuctios is the set of ll rel umbers (i mth, we sy, to deote this ide) (The rge will vry depedig o the fuctio) or (co) Lies ( f ( m b ), Qudrtic fuctios k prbols ( f ( b c ) d Cubic fuctios
3 ( f ( b c d ) re ll simple emples of polyomils Fuctios tht look like f ( ( costt) re clled power fuctios (co) Power fuctios help us build other kids of fuctios For emple, if i f ( the epoet hppes to be positive iteger,, we get pieces tht mke up polyomils If, isted, the epoets re the reciprocls of, so, we get root fuctios Other epoets give us differet vritios o this ide Rtiol fuctios re the rtio of two polyomils For emple: re polyomils d q ( 0 for y i the domi of f ( p( f ( where p ( d q ( q( (co) It is ofte useful to remember tht si AND cos Fuctios tht look like f (, ( 0), re clled epoetil fuctios (co) The domi of ll epoetil fuctios is the set of ll rel umbers, d the rge is f ( (0, ) Fuctios tht look like f ( log, ( 0), re clled logrithmic fuctios (co) The domi of ll logrithmic fuctios is the set of ( 0, ), d the rge is f ( (co) Grph shiftig: Suppose tht c 0 d some geerl fuctio y f ( y f ( c correspods to verticl shift upwrd of c uits y f ( c correspods to verticl shift dowwrd of c uits y f ( c) correspods to horizotl shift right of c uits y f ( c) correspods to horizotl shift left of c uits (co) Grph sclig, stretchig, d reflectig: Suppose tht c d some geerl fuctio y f ( y cf ( correspods to verticl stretch by fctor of c y f ( correspods to verticl compressio by fctor of c c y f (c correspods to horizotl compressio by fctor of c y f correspods to horizotl stretch by fctor of c c y f ( correspods to reflectio of y f ( bout the -is y f ( correspods to reflectio of y f ( bout the y-is Give two fuctios, f d g, the composite fuctio g f is defied by: f g f g We write f ( L (the it of f ( s pproches equls L) if we c mke the vlues of f ( rbitrrily close to L by tkig to be sufficietly close to but ot equl to
We write f ( L (the it of f ( s pproches from the left equls L) if we c mke the vlues of f ( rbitrrily close to L by tkig to be sufficietly close to but less th We write f ( L (the it of f ( s pproches from the right equls L) if we c mke the vlues of f ( rbitrrily close to L by tkig to be sufficietly close to but greter th (thm) f ( L if d oly if f ( L d f ( L (LEFT = RIGHT) Alysis Versio: We write f ( L (the it of f ( s pproches equls L) if for every 0, there is correspodig 0 so tht if 0 (if we boud smll circle of rdius roud the poit ) the f ( L (the vlue L is withi uits of f ( ) (lw) Suppose tht c is costt d f ( d eist, the the followig re true: Sum Lw: f ( f ( Differece Lw: f ( f ( Costt Multiple Lw: cf ( c f ( Product Lw: f ( f ( Quotiet Lw: f ( f ( Power Lw: f ( f ( Root Lw: (co) The it of costt is costt Substitutio works o power fuctios Substitutio works o root fuctios where f ( f ( where c c provided 0 where where Substitutio works o polyomils p( p( ) Substitutio works o rtiol fuctios p( q( p( ) q( ) Z Z Z Z (do t brek domi rules) provided q ( ) 0 Substitutio works o sie d cosie si si d cos cos (thm) If f ( whe is er (ecept possibly t ) d f ( f ( d eist, the 3
(thm) THE SQUEEZE THEOREM: If f ( h( whe is er (ecept possibly t ) d f ( h( L the L ) si (co) 0 cos (co) 0 0 A fuctio f is cotiuous t umber if f ( f ( ) (co) Cotiuity requires 3 thigs: f () must be defied f ( must eist 3 f ( f ( ) The it of f ( s pproches must equl the fuctio vlue t A fuctio f is cotiuous from the right t umber if f ( f ( ) A fuctio f is cotiuous from the left t umber if f ( f ( ) A fuctio f is cotiuous o itervl if it is cotiuous t every umber i the itervl (thm) If f ( d re cotiuous t poit d c is costt, the the followig fuctios re lso cotiuous t : f f g ( f g ( cf ( fg ( ( g provided g ( ) 0 (thm) Ay polyomil is cotiuous everywhere ie Polyomils re cotiuous o (their domi) Ay rtiol fuctio is cotiuous everywhere it is defied (ie o its domi) (thm) More geerlly: The followig types of fuctios re cotiuous t every umber i their domis: polyomils, rtiol fuctios, root fuctios, trigoometric fuctios (thm) If f is cotiuous t b d the followig wy: f ( ) f (thm) If g is cotiuous t d f is cotiuous t () cotiuous t b, the f ( ) f ( b) You c lso thik bout this i g, the the composite fuctio g ( f is (thm) THE INTERMEDIATE VALUE THEOREM: Suppose tht f is cotiuous o the closed itervl, b d let N be y umber betwee f () d f (b), where f ( ) f ( b) The, there eists Number c i, b such tht f ( c) N 4
The ottio f ( mes tht the vlues of f ( c be mde rbitrrily lrge (s lrge s we c imgie) by tkig sufficietly close to (o either side of ) but ot equl to This idictes the presece of verticl symptote t The lie is clled verticl symptote of the curve y f ( if t lest oe of the followig sttemets is true: f ( f ( f ( f ( f ( Let f be fuctio defied o some itervl, f ( The f ( L mes tht the vlues of f ( c be mde s close to L s we like by tkig sufficietly lrge This idictes the presece of horizotl symptote of the curve y f ( t y L This is likewise true if f ( L (co) 0 AND 0 provided is relly big positive iteger 5