Chapter I MATH FUNDAMENTALS I.3 Real Functions 27. After the completion of this section the student

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1 Chaper I MATH FUNDAMENTALS I. Real Fucios 7 I. REAL FUNCTIONS Ojecives: Afer he compleio of his secio he sude - should recall he defiiio of he asic algeraic ad rascedeal fucios - should e ale o deermie he mai properies of he fucios ad graph he fucios Coes:. Cosa Fucio. Asolue Value. Liear Fucio. Quadraic Fucio 5. Polomials 6. Raioal Fucio 7. Irraioal Fucios 8. Epoeial Fucio 9. Logarihmic Fucio. Trigoomeric Fucios. Iverse Trigoomeric Fucios. Hperolic Fucio. Review Quesios ad Eercises

2 8 Chaper I MATH FUNDAMENTALS I. Real Fucios I. REAL FUNCTIONS A surve of elemear real-valued fucios of real variale f :A wih heir defiiios ad mai properies is preseed. Fucios ca e give i eplici form f + i he implici form f (,) or ca e give paramericall f g + Fucios ca e also specified heir graph or give a ale of values. Fucios are called algeraic if he are polomials, roos or raioal fucios, oherwise he are called rascedeal fucios (epoeial, logarihmic, hperolic, rigoomeric). The rascedeal fucios ofe ca e defied he ifiie series. Properies of he fucios iclude: domai of defiiio, rage of values, quadra, periodici, moooici, smmer, asmpoes, characerisic paricular values (zeros, poles, pois of discoiui, eremes, pois of iflecio).. CONSTANT FUNCTION: The cosa fucio is defied equaio f c I assigs he same value c for all values of variale. The cosa f c 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 Chaper I MATH FUNDAMENTALS I. Real Fucios 9. LINEAR FUNCTION: A liear fucio is a fucio defied for all real umers which descries a sraigh lie i he plae f a + I is give a polomial of degree oe wih he followig forms of equaio: ) Slope-iercep equaio: m + m, m is called he slope ad is called he iercep of he fucio. The iercep raslaes he fucio alog he -ais. slope m ) Lie passes hrough he poi (, ) wih he slope m : m + ) Lie passes hrough wo fied pois (, ) ad (, ) + ϕ The slope m defies he icliaio of he lie m ad he agle wih he -ais eclosed he lie m aϕ ) Geeral liear equaio: A + B + C A,B,C A + B > 5) Parameric equaio of he lie: m + 6) Differeial equaio of he lie d f m d < < sujec o codiio The liear fucio is sricl icreasig for m> sricl decreasig for m< a cosa for m f

4 Chaper I MATH FUNDAMENTALS I. Real Fucios Perpedicular lies If wo o-verical lies m + ad m + are perpedicular, he m m a Proof: m aφ h h m aφ a φ a a m h. QUADRATIC FUNCTION: Quadraic fucio is a fucio defied for all real umers he equaio a + + c a + + c a a This equaio ca e reduced o he full square form: a + + c a a The graph of he quadraic fucio is a paraola shifed alog he -ais a ad alog he -ais c. a The poi,c is called he vere of he paraola. a a For a >, cocave up wih a gloal miimum a ; a for a <, cocave dow wih a gloal maimum a. a The paraola is smmeric wih respec o verical lie. a The roos of quadraic equaio a + + c deermie he pois of iersecio of he paraola wih he -ais:, ac D ± ± a a If he discrimia ± D,. If he discrimia a iersecio a. If he discrimia 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 moomial fucio.

5 Chaper I MATH FUNDAMENTALS I. Real Fucios 5. POLYNOMIALS: A liear comiaio of mooms of differe powers forms he polomial fucio f a a... a a a,a,...,a The highes power of he mooms,, is called he degree of he polomial. a,a,...,a are he coefficies of he polomial, ad a is called he leadig coefficie. The polomial fucio is defied for all real umers, ad he rage of he fucio depeds o he special case of polomial. < f <. For odd, he rage is 5 f + + I geeral, he polomial fucio is o smmeric. Bu if he polomial coais ol eve powers, he i has mirror smmer aou he ais. Ad if he polomial coais ol odd powers he i has poi smmer aou he origi. The graph of he polomial fucio ca have a mos iersecios wih he -ais (real zeros). If is odd he i has a leas oe real zero. I also ca have up o eremes ad up o pois of iflecio. is a facor of he polomial fucio f is a zero of he polomial, i.e. f ( ). The liear polomial ol if if ad Accordig o he Fudameal Theorem of Algera, he polomial fucio has eacl zeros which ca e comple or real, sigle or repeaed. Because he polomial fucio has ol real coefficies, he he comple zeros appear i cojugae pairs. I meas ha he polomial ca e represeed as a produc of liear facors correspodig o real zeros ad irreducile quadraic facors which correspod o comple zeros. For eample, he cuic polomial f + has oe real zero ad wo cojugae comple zeros, ± i. I ca e facored i he followig wa f i i + ( )( )( + ) ( )( + ) h I geeral, he he polomial is represeed as a produc of liear facors s s s f a k k where i is he real or comple zero of muliplici s i, ad k is he oal umer of disic zeros. Alhough he zeros of polomial up o degree ca e calculaed aalicall, i pracice ol for he quadraic polomial he aalic formula is used: a ± a aa, a For polomials of higher degree, i geeral, he umerical mehods are used. I Maple, he polomials ca e facored wih he followig commad: > facor(^5-^-+); ( + ) ( + ) ( ) The roos of he polomial fucio ca e foud > solve(^5-*^+^++.,);., , I, I,

6 Chaper I MATH FUNDAMENTALS I. Real Fucios 6. RATIONAL FUNCTION: A raioal fucio is defied as a quoie of wo polomial fucios: p + p p + p P f ( ) q q... q q Q m m m + m where P ( ) ad m correspodigl. m Q are he polomial fucios of degrees ad m, If < m he he raioal fucio is called proper. If > m he he raioal fucio is called improper. B polomial divisio (log divisio), a improper raioal fucio ca alwas e represeed as a sum of polomial ad a proper raioal fucio: P rl f ( ) pm + wih l < m Q Q m m where he umeraor polomial r l is called a remaider. B parial fracio decomposiio, he remaider fucio as a sum of simple parial fracios (see Secio..5). r l Q m ca e wrie Log Divisio P Q m p + p p + p q q... q q m m m + m q + q q + q m m m m p m q m + p + p p + p pq pq pq p m m+ m qm qm qm pq m p +... qm Eample: +, P Q p P Q Therefore, r remaider P + 8 f Q

7 Chaper I MATH FUNDAMENTALS I. Real Fucios The log divisio ogeher wih he parial fracio decomposiio of he remaider fucio ca e performed wih Maple he followig commad: > cover((*^-^+)/(^+*+),parfrac,); ( + ) Graphig raioal fucios: The zeros of he deomiaor polomial Q m deermie he domai of he raioal fucio ad he poles of is graph. The paricular shape of he graph, is eremes ad pois of iflecio deped o he idividual case. The mirror smmer of he graph aou he -ais occur i he cases whe oh he deomiaor ad he umeraor polomials have ol eve or ol he odd powers. The poi smmer of he graph aou he origi occur i he case whe oe of he polomials has ol eve powers ad he oher polomial ol odd powers. Eamples: f graph wih poles f o smmer f mirror smmer f + 6 poi smmer

8 Chaper I MATH FUNDAMENTALS I. 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, m, m, or he r real epoe, r ; roos of polomials ad raioal fucios, ec. Cosider some 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,m, he m m m a a a a a a a m Power fucio of fracioal order f, m, is defied as he h roo of he h m power m m f wih > ad m 5 f a a a m m m ( a) a if is eve, if is odd The domai of he fucio depeds o he paricular values of m ad : if is odd: < < ; if m is odd ad is eve: < ; ad if i addiio m<, he has o e ecluded. a Power fucio of real order f, a is defied for > as a al f e f

9 Chaper I MATH FUNDAMENTALS I. Real Fucios 5 8. EXPONENTIAL FUNCTION: Le e a posiive real umer o equal o,, >,. The he fucio f f e is a epoeial fucio wih ase. If e, where e is a aural umer defied as a limi e lim , 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 logarihm 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 umer e: r m e if e e rm, r> where if ifium { } ) Iverse of a aural logarihm fucio: e if l, > ) As a limi e lim + ) Power series (coverge for all real ) k e k k! 5) As he soluio of he iiial value prolem d d The geeral epoeial fucio d f ca + k Epoeial Model The fucio f ce, k > models epoeial growh ad k f ce, k > models 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 6 Chaper I MATH FUNDAMENTALS I. Real Fucios 9. LOGARITHMIC FUNCTION: Le e a posiive real umer o equal o :, >,. The he fucio f log Defiiio: f l l ca e defied improper iegral l d, > is a logarihm fucio wih ase (geeral logarihm). If e, where e is a aural umer defied as a limi e lim +.788, he he fucio is called a aural logarihm (usuall, logarihm wihou saig he ase meas he aural logarihm): f loge l The relaioship ewee fucio is esalished he followig equaio: l log l The logarihm fucio ca e defied as iverse of he epoeial fucio: log if l if e for all > for all > Rules of logarihms: 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 le log le log log l l Noe! log ( + ) log + log l + l + l (pical misake!) I pre-compuer (pre-calculaor) era, he logarihms were he mai ool for performig arihmeic operaios. Coversio formulas: l e l log l

11 Chaper I MATH FUNDAMENTALS I. Real Fucios 7 Proof: Sar wih log defiiio l l l ake logarihm l power rule l solve for l Epoeial growh (deca) model k where Q Qe ± Q is he iiial amou of susace a ) if i is give Q he amou of susace a, he Q k k Qe Q e Q Q k l Q The epoeial model ecomes: epoeial deca model Q Qe k Q Q l l k Q Q Q Q Qe Qe Qe Q Q ) Half-life ime h is defied as he ime eeded for susace o e reduced a half: Q kh Qe k > The kh e l le kh l kh k l h The epoeial model ecomes: l k h h Q Qe Qe Q epoeial growh model Q Qe k ) Doulig ime D is defied as he ime eeded for susace o e douled: Q The kh Qe k > e kh l le l kd kd k l D The epoeial model ecomes: l k D D Q Qe Qe Q

12 8 Chaper I MATH FUNDAMENTALS I. Real Fucios. TRIGONOMETRIC FUNCTIONS: The rigoomeric 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 coordiaes (, ). The segme coecig he poi wih he origi has he ui legh. From he Phagorea Theorem follows + (,) This segme forms a agle wih he -aes coued posiive i couer-clock direcio ad egaive i clock direcio. The agles are measured i erms of radias, where radia is a measure 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 measure 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 same poi o he ui circle α + where is a umer of full roaios The he se of all possile agles defied he pois o he ui circle is he se of real umers. Fucios si ad cos The asic rigoomeric fucios are defied for all i he followig wa: si cos Because he same 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 deermied from he righ riagles are idicaed i he followig graph. The also ca e ploed i he graph aove he ierval [, ] ieldig a curve which is a mai eleme i cosrucig he graph of oh fucios floppig ad reflecios: agle 6 si 6

13 Chaper I MATH FUNDAMENTALS I. Real Fucios 9 si period cos period These wo graphs demosrae 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 mehod for graphig he siusoidal fucio (oe period): d + a a > period ampliude d d a c c phase shif c +

14 Chaper I MATH FUNDAMENTALS I. Real Fucios Oher Trigoomeric 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 Chaper I MATH FUNDAMENTALS I. Real Fucios agle csc ± ± ± sec ± ± csc period si sec period cos

16 Chaper I MATH FUNDAMENTALS I. Real Fucios Power Series Defiiio The rigoomeric fucios also ca e defied he followig ifiie series coverge for all : k+ 5 7 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 Theorem ields is rigoomeric 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 formulas: + a sec + co csc Smmer si( ) si odd fucio cos ( ) cos eve fucio Complime Formulas si cos cos si a ± co co a a co co ± a Agle Sum Formulas 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 Chaper I MATH FUNDAMENTALS I. Real Fucios Doule Agle Formulas si si cos cos cos si cos si a a a Power Reducig Formulas + cos cos cos si cos a cos Half Agle Formulas cos si ± + cos cos ± -cos si a si + cos Produc-o-Sum 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) + + cosu siv si( u v) si( u v) + Sum-o-Produc u+ v uv siu + siv si cos u+ v uv siu siv cos si u+ v uv cos u + cos v cos cos u+ v uv cos u cos v si si

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

19 Chaper I MATH FUNDAMENTALS I. Real Fucios 5. 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 Smmer: sih( ) sih cosh( ) cosh Derivaive: sih cosh cosh sih Ideiies: cosh sih Phagorea Idei cosh + sih e De Moivre s formulas cosh sih e [ ] cosh + sih cosh + sih e [ ] cosh sih cosh sih e

20 6 Chaper I MATH FUNDAMENTALS I. Real Fucios. REVIEW QUESTIONS: ) B wha properies he fucios are characerized? ) Wha are he domai 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 f log c) f log.5 d). e) f log ( 5) f) f log g) f log ( ) h) f log ( + ) i) f + e j) f + e k) f l) 5 f ) Prove he properies for he geeral power fucio: a) a a + a a, ) ( ) 5) Derive he rigoomeric ideiies: a) c) 6) Evaluae: si cos si si si ) si si si d) a) c) si( cos ) 7) Skech he graph of he fucios: f si a) 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 si f) g) f si h) i) f si j) k) f a l) m) f sec ) o) f si p) f cos f cos f co f csc f cos

21 Chaper I MATH FUNDAMENTALS I. Real Fucios 7 r) f si s) ) f + si u) 8) a) Fid a epoeial fucio fied pois (, ) ad f l f + l ae he graph of which passes wo,. Usig properies of epoeial ad logarihmic fucios simplif 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 logarihmic fucios simplif he epressio ad skech he graph. 9) Epress he raioal fucio as a sum of he polomial ad a proper raioal fucio ad skech he graph: a) f + + ) f ) a) A a mome of ime, he rae of producio of a cerai iological k susace is descried he epoeial growh model Q Qe. If afer hour here is l of he susace ad afer hours he amou is 8 l, how much of he susace will e here afer hours of producio? ) A a mome of ime, he rae of producio of a cerai iological k susace is descried he epoeial growh model Q Qe. If afer hour here is l of he susace ad afer hours he amou is l, how much of he susace was here iiiall? c) A a mome of ime, he rae of fissio of a cerai susace is k descried he epoeial deca model Q Qe. The half-life ime is kow o e hours. If afer hour here is l of he susace, how much 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 e) f sech f) ) 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 hem: + a) f ) f + + c) f d) f + e) f si f) f

22 8 Chaper I MATH FUNDAMENTALS I. Real Fucios Novosiirsk Sae Uiversi