After the completion of this section the student

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1 Real Fucios REAL FUNCTIONS Afer he copleio of his secio he sude - should recall he defiiio of he asic algeraic ad rascedeal fucios - should e ale o deerie he ai properies of he fucios ad graph he fucios. Cosa Fucio. Asolue Value. Liear Fucio. Quadraic Fucio 5. Poloials 6. Raioal Fucio 7. Irraioal Fucios 8. Epoeial Fucio 9. Logarihic Fucio. Trigooeric Fucios. Iverse Trigooeric Fucios. Hperolic Fucio. Review Quesios ad Eercises

2 Real Fucios REAL FUNCTIONS A surve of eleear real-valued fucios of real variale f : A wih heir defiiios ad ai properies is preseed. Fucios ca e give i eplici for f + i he iplici for f (, ) or ca e give paraericall f g + Fucios ca e also specified heir graph or give a ale of values. Fucios are called algeraic if he are poloials, roos or raioal fucios, oherwise he are called rascedeal fucios (epoeial, logarihic, hperolic, rigooeric). The rascedeal fucios ofe ca e defied he ifiie series. Properies of he fucios iclude: doai of defiiio, rage of values, quadra, periodici, oooici, ser, aspoes, characerisic paricular values (zeros, poles, pois of discoiui, erees, pois of iflecio).. CONSTANT FUNCTION: The cosa fucio is defied equaio f c f c I assigs he sae value c for all values of variale. The cosa fucio is a soluio of differeial equaio d f ( ) d Graphicall, he cosa fucio is represeed a horizoal sraigh lie,c. I is defied equaio c. passes hrough he poi. ABSOLUTE VALUE: The asolue value fucio f is defied as if - if < The oher defiiio of he asolue value fucio uses he roo of he square I defies he disace ewee he pois ad o he real lie. Fucio is defied for all. The fucio values are ever egaive, he rage of values: <. Graph of he fucio f Shifig alog he -ais: f f a Properies:. ol if. for all. for all,. for all, for all, (riagle iequali)

3 Real Fucios. LINEAR FUNCTION: A liear fucio is a fucio defied for all real uers which descries a sraigh lie i he plae f a + I is give a poloial of degree oe wih he followig fors of equaio: ) Slope-iercep equaio: +, is called he slope ad is called he iercep of he fucio. The iercep raslaes he fucio alog he -ais. Icree of a pois o he lie: ) Lie passes hrough he poi (, ) wih he slope : + ) Lie passes hrough wo fied pois (, ) ad (, ) + The slope defies he icliaio of he lie ad he agle wih he -ais eclosed he lie aϕ ) Geeral liear equaio: A + B + C A,B,C A + B > ) Paraeric equaio of he lie: + 5) Differeial equaio of he lie d f ( ) d < < The liear fucio is sricl icreasig for > sricl decreasig for < a cosa for

4 Real Fucios Perpedicular lies If wo o-verical lies + ad + are perpedicular, he a Proof: aφ h h aφ a φ a a h. QUADRATIC FUNCTION: Quadraic fucio is a fucio defied for all real uers he equaio a + + c a + + c a a This equaio ca e reduced o he full square for: a + + c a a The graph of he quadraic fucio is a paraola shifed o he lef a ad up c. a The poi,c is called he vere of he paraola. a a For a >, cocave up wih a gloal iiu a ; a for a <, cocave dow wih a gloal aiu a. a The paraola is seric wih respec o verical lie. a The roos of quadraic equaio a + + c deerie he pois of iersecio of he paraola wih he -ais: If he discriia, ac D ± ± a a ± D,. If he discriia a iersecio a. If he discriia a has o iersecios wih he -ais. D ac >, he here are wo iersecios a Differeial equaio for he quadraic fucio: d f a d D ac, he here is he D ac <, he he paraola Power fucio The power fucio is defied all he equaio f a,,,,... a I is called he ooial fucio.

5 Real Fucios 5. POLYNOMIALS: A liear coiaio of oos of differe powers fors he poloial fucio f a + a a + a a,a,...,a The highes power of he oos,, is called he degree of he poloial. a,a,...,a are he coefficies of he poloial, ad a is called he leadig coefficie. The poloial fucio is defied for all real uers, ad he rage of he fucio depeds o he special case of poloial. < f <. For odd, he rage is 5 f + + I geeral, he poloial fucio is o seric. Bu if he poloial coais ol eve powers, he i has irror ser aou he ais. Ad if he poloial coais ol odd powers he i has poi ser aou he origi. The graph of he poloial fucio ca have a os iersecios wih he -ais (real zeros). If is odd he i has a leas oe real zero. I also ca have up o erees ad up o pois of iflecio. The liear poloial ol if is a facor of he poloial fucio f is a zero of he poloial, i.e. f ( ). if ad Accordig o he Fudaeal Theore of Algera, he poloial fucio has eacl zeros which ca e cople or real, sigle or repeaed. Because he poloial fucio has ol real coefficies, he he cople zeros appear i cojugae pairs. I eas ha he poloial ca e represeed as a produc of liear facors correspodig o real zeros ad irreducile quadraic facors which correspod o cople zeros. For eaple, he cuic poloial f + has oe real zero ad wo cojugae cople zeros, ± i. I ca e facored i he followig wa + ( )( )( + ) ( )( + ) f i i I geeral, he h he poloial is represeed as a produc of liear facors s s s k f a k where i is he real or cople zero of uliplici s i, ad k is he oal uer of disic zeros. Alhough he zeros of poloial up o degree ca e calculaed aalicall, i pracice ol for he quadraic poloial he aalic forula is used:, a ± a a a a For poloials of higher degree, i geeral, he uerical ehods are used. I Maple, he poloials ca e facored wih he followig coad: > facor(^5-^-+); ( + ) ( + ) ( ) The roos of he poloial fucio ca e foud > solve(^5-*^+^++.,);., , I, I,

6 Real Fucios 6. RATIONAL FUNCTION: A raioal fucio is defied as a quoie of wo poloial fucios: p + p p + p P f ( ) q q... q q Q where P ( ) ad correspodigl. Q are he poloial fucios of degrees ad, If < he he raioal fucio is called proper. If > he he raioal fucio is called iproper. B poloial divisio (log divisio), a iproper raioal fucio ca alwas e represeed as a su of poloial ad a proper raioal fucio: P rl f ( ) p + wih l < Q Q where he ueraor poloial r l is called a reaider. B parial fracio decoposiio, he reaider fucio as a su of siple parial fracios (see Secio..5). r l Q ca e wrie Log Divisio P Q p + p p + p q q... q q q + q q + q p q + p + p p + p p q p q p q p q q q Eaple: P +, p q + p... q Q p P Q Therefore, r reaider P + 8 f Q

7 Real Fucios The log divisio ogeher wih he parial fracio decoposiio of he reaider fucio ca e perfored wih Maple he followig coad: > cover((*^-^+)/(^+*+),parfrac,); Graphig raioal fucios: The zeros of he deoiaor poloial Q ( + ) deerie he doai of he raioal fucio ad he poles of is graph. The paricular shape of he graph, is erees ad pois of iflecio deped o he idividual case. The irror ser of he graph aou he -ais occur i he cases whe oh he deoiaor ad he ueraor poloials have ol eve or ol he odd powers. The poi ser of he graph aou he origi occur i he case whe oe of he poloials has ol eve powers ad he oher poloial ol odd powers. Eaples: f graph wih poles f o ser f irror ser f 6 + poi ser

8 Real Fucios 7. IRRATIONAL FUNCTIONS: This is a wide class of fucios which icludes square roo ad cuic roo fucios; roo fucios of ieger order ; power fucios wih fracioal epoes,,, or he r real epoe, r ; roos of poloials ad raioal fucios, ec. Cosider soe defiiios of hese fucios: Roo fucio of ieger order f for as a iverse of he power fucio:,, > is defied if ad for egaive epoes f For odd, he roo fucio is defied also for egaive values as f, < Rules for Radicals: Le a, ad,, he a a a a a a a Power fucio of fracioal order f,, is defied as he h roo of he h power wih > ad f 5 f a a a ( a ) a if is eve, if is odd The doai of he fucio depeds o he paricular values of ad : l ( l l ) e e e, he if is eve he doai is. a Power fucio of real order f, a is defied for > as a a l f e f

9 Real Fucios 8. EXPONENTIAL FUNCTION: Le e a posiive real uer o equal o,, >,. The he fucio f f e is a epoeial fucio wih ase. If e, where e is a aural uer defied as a lii e li +.788, he he fucio is called a epoeial fucio: f e The relaioship ewee fucios is esalished he followig equaio: e e e l l where l is a aural logarih of (see e secio). l A epoeial fucio f e ca e defied for all real i oe of he followig was: ) Real power of he uer e: { } where if ifiu r e if e e r, r> ) Iverse of a aural logarih fucio: ) As a lii e if l, > e li + ) Power series (coverge for all real ) k e k k! 6 5) As he soluio of he iiial value prole d d The geeral epoeial fucio d f ca + k Epoeial Model The fucio f ce, k > odels epoeial growh ad k f ce, k > odels epoeial deca. Rules for epoes: for a a >, > ad for all, a a a + a a a a a a a a a a

10 Real Fucios 9. LOGARITHMIC FUNCTION: Le e a posiive real uer o equal o :, >,. The he fucio f log f l is a logarih fucio wih ase (geeral logarih). If e, where e is a aural uer defied as a lii e li +.788, he he fucio is called a aural logarih (usuall, logarih wihou saig he ase eas he aural logarih): f loge l The relaioship ewee fucio is esalished he followig equaio: l log l The logarih fucio ca e defied as iverse of he epoeial fucio: log if for all > l if e for all > Rules of logarihs: For all real > ad > : log log + log l l + l produc rule log log log l l l quoie rule log log l l power rule log l log l e log l e log le log log l l oe! log ( + ) log + log l + l + l (pical isake) I pre-copuer (pre-calculaor) era, he logarihs were he ai ool for perforig ariheic operaios. Coversio forulas: l e l log l

11 Real Fucios Proof: sar wih log defiiio l l l l l ake logarih l power rule solve for Epoeial growh (deca) odel Q Q e ± k where Q is he iiial aou of susace a ) if i is give Q he aou of susace a, he Q k k Q e Q e Q Q k l Q The epoeial odel ecoes: epoeial deca odel Q Q e k Q Q l l Q Q Q Q Q e Q e Q e Q Q k ) Half-life ie h is defied as he ie eeded for susace o e reduced a half: Q kh Q e k > The kh e l le kh l kh k l h The epoeial odel ecoes: l k h h Q Q e Q e Q epoeial growh odel Q Q e k ) Doulig ie D is defied as he ie eeded for susace o e douled: Q The kh Q e k > e kh l le l kd kd k l D The epoeial odel ecoes: l k D D Q Q e Q e Q

12 Real Fucios. TRIGONOMETRIC FUNCTIONS: The rigooeric fucios (also called he circular fucios), i calculus, are defied wih he help of he ui circle (circle of radius ). Cosider a ui circle wih a ceer placed a he origi of he Caresia coordiaes i he plae. Cosider a poi o he circle wih he,. The sege coecig he poi wih he origi Fucios si (, ) coordiaes has he ui legh. Fro he Phagorea Theore follows + This sege fors a agle wih he -aes coued posiive i couer-clock direcio ad egaive i clock direcio. The agles are easured i ers of radias, where radia is a easure of he agle which correspods o he arc of legh o he ui circle. No uis are aached o he value of agles i radias. Deoe he easure of agles he variale. The pois of iersecio of he ui circle wih coordiae aes correspod o agles, (righ agle), (srigh agle),, ad whe we reur o he firs poi, (full agle). The roaio of he poi o he ui circle i he couer-clock direcio defies he periodic values of agles correspodig o he sae poi o he ui circle α + where is a uer of full roaios The he se of all possile agles defied he pois o he ui circle is he se of real uers., cos The asic rigooeric fucios are defied for all i he followig wa: si cos Because he sae coordiaes correspod o he agle afer he full roaio, he iroduced fucios have he period p : ( + ) ( ) si si cos + cos The rage of oh fucios is ewee ad. The values of fucios si ad cos for he ke agles i he firs quadra ca e easil deeried fro he righ riagles are idicaed i he followig graph. The also ca e ploed i he graph aove he, ieldig a curve which is a ai elee i ierval [ ] cosrucig he graph of oh fucios floppig ad reflecios: agle 6 si 6

13 Real Fucios si period cos period These wo graphs deosrae ha oe of he fucios ca e oaied shifig he oher: cos si + si cos Tale of he paricular values: agle si cos Siusoidal Fucio c d+a si c d + a si period Ke poi ehod for graphig he siusoidal fucio (oe period): a > period d + a apliude d d a c c phase shif c +

14 Real Fucios Oher Trigooeric Fucios: si a cos cos co si csc si sec cos agle a a period 6 6 agle co co period 5 6 6

15 Real Fucios agle csc ± ± ± sec ± ± csc period si sec period cos

16 Real Fucios Power Series Defiiio The rigooeric fucios also ca e defied he followig ifiie series coverge for all : k si + + k! 6 5 k k ( + ) k 6 cos + + k! 7 k k ad coverge i he ierval: a co < < < < sec < < csc < < Phagorea Ideiies: The Phagorea Theore ields is rigooeric versio si si + cos which esalishes he coecio ewee fucios: si ± cos sig depeds o he value of cos cos ± si B divisio of he Phagorea idei cos ad si, cosequel, oe also ca oai he followig forulas: + a sec + co csc Ser si( ) si odd fucio cos ( ) cos eve fucio Coplie Forulas si cos cos si a ± co co a a co co ± a Agle Su Forulas si( + s ) cos si s + si cos s si( s ) si cos s si s cos cos ( + s ) cos cos s si si s cos ( s ) cos cos s + si si s

17 Real Fucios Doule Agle Forulas si si cos cos cos si cos si a a a Power Reducig Forulas + cos cos cos si cos a cos Half Agle Forulas cos si ± + cos cos ± -cos si a si + cos Produc-o-Su si u si v cos ( u v) cos ( u + v) cos u cos v cos ( u v) cos ( u v) + + siu cos v si ( u v) si( u v) + + cos u siv si ( u v) si( u v) + Su-o-Produc u + v u v siu + si v si cos u + v u v siu si v cos si u + v u v cos u + cos v cos cos u + v u v cos u cos v si si

18 Real Fucios. INVERSE TRIGONOMETRIC FUNCTIONS: priciple ierval si All rigooeric fucios are o oe-o-oe (oviousl, ha horizoal lie es fails for he). Therefore, for cosrucio of he iverse fucio, we choose ol he ierval where a rigooeric fucio is oe-o-oe (priciple ierval) ad defie he iverse fucios i he followig wa: ) si if si for all si ) if cos for all cos cos priciple ierval cos Accordig o defiiio: ( ) ( ) si si cos cos si si cos cos

19 Real Fucios. HYPERBOLIC FUNCTIONS: The hperolic fucios are defied wih he help of epoeial fucios: e sih e k k!! 5! 7! k ( + ) e cosh + e k k!!! 6! k cosh sih Value a : sih cosh Ser: sih( ) sih cosh( ) cosh Derivaive: sih cosh cosh sih Ideiies: cosh sih Phagorea Idei cosh + sih e De Moivre s forulas cosh sih e [ ] cosh + sih cosh + sih e [ ] cosh sih cosh sih e

20 Real Fucios. REVIEW QUESTIONS: ) B wha properies he fucios are characerized? ) Wha are he doai ad he rage of he fucios? ) Wha fucios are algeraic ad fucios are rascedeal? EXERCISES: ) Skech he graph of he reak fucio defied wih he help of he asolue value fucio + f ) Skech he graph of he fucios: f a) ) f + 5 ) Skech he graph of he fucios: a) f log ) f log 5 c) f log.5 d). e) f log ( 5) f) f log f log g) f log ( ) h) f log ( + ) i) f + e j) f + e k) f + e l) ) f ) f + e 5 f ) Prove he properies for he geeral power fucio: a) a a + a ) 5) Derive he rigooeric ideiies: a) si si si ) c) si si si d) 6) Evaluae: si cos a) c) si( cos ) 7) Skech he graph of he fucios: f si a) a cos cos cos cos + cos cos ) cos ( si ) d) cos ( si ) c) f ( ) + si( ) d) e) f si cos ) f cos ( + ) f 5+cos f) f si g) f si h) i) f si j) k) f a l) ) f sec ) f cos f cos f co f csc

21 Real Fucios o) f si p) r) f si s) i) f + si j) 8) a) Fid a epoeial fucio fied pois (, ) ad f cos f l f + l ae he graph of which passes wo,. Usig properies of epoeial ad logarihic fucios siplif he epressio ad skech he graph. ) Fid a epoeial fucio fied pois (, ) ad ae he graph of which passes wo,. Usig properies of epoeial ad logarihic fucios siplif he epressio ad skech he graph. 9) Epress he raioal fucio as a su of he poloial ad a proper raioal fucio ad skech he graph: a) f + + ) f ) a) A a oe of ie, he rae of producio of a cerai iological k susace is descried he epoeial growh odel Q( ) Q e. If afer hour here is l of he susace ad afer hours he aou is 8 l, how uch of he susace will e here afer hours of producio? ) A a oe of ie, he rae of producio of a cerai iological k susace is descried he epoeial growh odel Q( ) Q e. If afer hour here is l of he susace ad afer hours he aou is l, how uch of he susace was here iiiall? c) A a oe of ie, he rae of fissio of a cerai susace is k descried he epoeial deca odel Q( ) Q e. The half-life ie is kow o e hours. If afer hour here is l of he susace, how uch of he susace was here iiiall? ) Skech he graph of he fucios: a) f cosh sih ) f sih + cosh sih cosh c) f ah d) cosh f) f sech g) ) Derive he ideiies: cosh f coh sih f csch sih a) cosh sih ) ah sech ) Fid he iverse of he fucios ad skech he graph of oh of he: + a) f ) f + + c) f d) f si f) g) f + f

22 Real Fucios Novosiirsk Sae Uiversi