B E P R EPA R E D FO R LM A IN & A D V A N CE P Įa y w h Ū A ga rw a it h l 国 R E M A ı N & A DVA N CED P Įa y ı th w a A m it M A g a r w , a S l ER 區 U T S H A ìQ , M E E R A K A R P T A R IH A N A ll r ig h ts yi e re s e r v d © A u th o r N o i f t h i p u b lic a t o pa rt o l e c h a n ic a ic m e e c tr o n f o m th e , so u rc e s h d p u b lis e , an A ll d is p u te s s u b je c t ay m reproduced s t be , g ; .Pyi ; A ĥ o t. . b e lie v e d d th e d a m n to . M re c o rd in g , r ih a n t h a s Ii h e r e l b e r e lia b e a n d t r u age s o r to , lo e eru t ss su o re r e tn e v a s y s 【c m U [ u , n n o n ly ŝįşiļ į \iį ' a l. . 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B h u b a n e sw k a ta , L u c k n a r, o w B a re M ly c h e iı e e r u t, N a n n a gp u r i D e h i, G ı & Pu ti u w aha , n e 9 7& 93 1112 & 12 8 - £ Scanned by CamScanner 0 仁オ厂T 丘N 丁5 1 In . tr o d u c tio n G to 1 ıA lg e b ra ic fu n c tio n s 1 P o ly n o m ia lfu n c tio Z R a tio n a lfu n c tio 3 Ir r a tio n a 1 2 T ra n s c e n d e n n o m 2 Ex p o n P n n c tio n s fu e cr ic lfu n n c tio n c t io n 3 L o g a r ith m ic fu n c tio 4 G c o m e tr ic a lc u r v e s e tr ic 1 4 S o lv in g e q u a tio 2 Cu . n s g r a p h ic a lly d T r a n s fo a v i ty, co n v e x i ty 2 3 P lo ttin g o fa l e b r a ic g ra p h ic a ltra n s fo rm Z 5 S k tc h in h t) g o - 26W h en ļį an d s, po i nt s of c u T v e s u s in a t io n s 51 137 g inf lex i on c o n c a v it y a tio n s k x ) g (x )}a n d h (x ) m a x im u m fi(x ), gGofi【 X)+ 2 7 W h e n f{x ) , 9 69 ?;;: rm u r v a tu re . Co n c 24 G n in e q u a lit ie s T v a tu r e a n Z1C ) ta lfu n c t io n s e n t ia 1 3 T r ig o n o m p hs lfu n c tio n 4 P ie c e w is e fu n 1 T r ig o ra g (x , m in im u m {f(x ) g{×D , ) h (x ) = f{x ) g (x ) h (x ) = s in g u la r p o in ts a n d c u rv e T r a c in g 13& 165 : A ! 6G 182 t T ' Įi Scanned by CamScanner b p ı n th is æ đ io n , w e sh a ı ır e v is e so m e b a s ic c u ir v e s W h ic in a r e g iv e n a s. t l 4 ı + G re a te s t in te F ra c tio 14 ll 벼 4 1 T ra n s c e n ? ' " no m L e a s t i, t. g Ge įįi ŢŢ:ŢŢ:§ fu n c tio n ] r ala】 w ユ 仁 王 二 ー 區園 L o g a rith m ic /In v e ılnverse trig o n o m ? r ıp a r t fu 毛亚五回 ţ )j T r ig o na ge rs e o r r f e x po n R=r Ļį e tr ïc " A LG E B R A ı C FU N C T ı O N s - i . . . a p o ly n f (x ) om = a o + ia l fu n c tio n a 1x " + a 2x f (x ) is 2 + a ls o c a " , . ' lle d p o ly n om ia l in J č ' 1 t a lĽ Ľ l 1 Scanned by CamScanner rı So m e o l n f b a s ic p o y T he do m a in nn om ï a ıfu n c tto d ra n r id e n t ity o ge n s a re fu n c tio n a Fi 2 is c a lle d th e x )= x b A fu n c tio n g iv e n y f ( f n c tio n is R a n d its T h e d o m a in o f s q u a r e u Ī 2 is a p a r a b o la S in c e y ? e a r ly y - x m e tr lc a l a b o u t y 株式会社 n so lts g r a p h is 目y m , b o th a re fu sq u a re ra n - x a x , n c tio n ge is R 2 is le , 11 + an {o } u even sho w n a s T 翼 (H l) G r a p h o f /(x ) = : x A fu n c tio n g iv e n b y a a bI o u t o y - x 3 is p p o a lt e q m an ? f (x ) o = d d fu a n t, Ĺe x 3 is ca n c tio n « , o , lle d th e b e fu its g r a p h is so r lg ln cu " , sh o w n c tio n s ym m e tr ic a l n as F ig 1 3 io d fu n đ n so h ra p its g , Scanned by CamScanner a . q W ー コ ー き の 件 了 コ う 确 D Lc do , G ra p h s o f (i) G r a p h o so m e f m in a R S tm p ı e R a t io f (x ) o m p t th o ex ce ıF u n a a in n c { x lQ (x ) R se in po fo ts r w = @ O} h ic h d e n o m in a to r = . o 召 t io n s = 了 X fu A n c tio n a sy m o r ce abo ı ıt o 田 1 f (x ) by is = ca lle d th e r e c ip r o c a f (x ) , g u la a in m h y p e r b o la r d a n ra n ge w , o ith f f (x ) = is o f (x ) d d fu u a = d + r a n n c t io n a n - , so A ls o te d , w 1 is R its g r a p h is e o bse rv e lim . c o o r d in a te f (x ) = s a x is as «o » X x ) s it e q pı lim an r e c ta n p to t e s T h e d o S in d e X c đo n ńm d e fin y m m e t r ic a l F ig 1 6 _ d T hu s , f (x ) = co u ld b e sho w n as in F ig 1 6 Scanned by CamScanner ľ Scanned by CamScanner ) a a ph o irÆ fR x )= w ı x Ųj Jųid is n & L T l . o f 9 o ) a b o ıu t y s do » Ēh e re ボ诮 ra m a in 叫声 g r a p lh 1hw lfg (x ) = T 3 T he n K = R ! (T ) · T t 3 )Ę R i o f 八X ) ミ ル o í R Jc )= łł 3 x 1 1 1; . is s h o Wn in F ig 宇中 F ig ) Xtï 1 w he n n Ē N H W is d e fin e d fo r a ll T E R S o İ (x ) X 1 d o n ıa iıı o f f (x ) R a i ıd r a q g e o f f (x ) R A ls o th e is th e n ıir m r in ıa g e o f ť b e g r a p h g ra p h o f f (x ) X 1 2n 1 in e y R x w ł x a b o u t th e ı ıe n x o f İ (x ) ' /3 ' 5 Th a r e s h o w iı f (x ) x f (x ) x (lv ) G r a p h o f ï (x ı12 = . = . . = = - = = , , , as N ı g o h a v e d is c u s s e d s o m e o f ttıe s im p le c u rv e s tio n a l ı m P o ly n o m ia ı R a tio n a ı a n d r t d iĦic u ı n c tio n s G r a p h s o f th e s o m e m o r e i d n d i s s e u s c b ı e w ı r a t io n a l fu n c tio n s h s a S c 3 u h t c a p er W fo ť " fu *e P e . F ig 1 13 , S ig m ım fu n c tio n d ( ) F r a c tio n a ıp a r t fu n c t io n Scanned by CamScanner nutpaxls e tr lc " ;į . m R is s m e « N ł ra ļ a n il m o ĺo t , u s fu du ı h e re d o m w n ln R ) n c tio n s O) ((I '" \ v n vı yw Thw İ (x ) [x ]c o u ld b e s h o w = P r o p e r tie s o ◆ [x l lx = + f g re a t e s t in te g e x h o ld s if x is in l] = , [x ] + r te 1 14 fu s ı n as nc tio ge r n F ig 1 1 7 i, if l is in te g e r Scanned by CamScanner 厂 佷 (d } l p a r t o f fu n c t io n d « » e n o te s th e fr a c tio c t io n a F ra Hm p r o p e r tie s o f fr a c t io n a ıp a r t o n a f l p a rt o F x T h u s in , v {x } = x 嘲 if } x 0 5 x < 1 i ) {x } O ; if x in t e g e r (iii) { x y 1 { x } ; if x in t e g e r (i) { x = 自 る = = (e ) L e a s t in (x ) o r te g e lx l I t is 】 m o T hu s, sh o w n H e re . r fu n c t io n ıı a s 12 3023 c e l ilin g o fx o w h ic h is n e a re s t a n d gre In te tx l n a te s t g = 1 , ( & o 72 5)= Ę ( o 6)= o = (x ) l×1 = n = (x ) = l× l , ca n 齢 e r to x fx as f (x ) 一 냐 냐 了 gra l p a rt e (0 2 3 ) = 刁 写 in d ic a tæ 出 e im w J be e x p r e s s e d g r a p h ic a llyJ x = n + 0 + 1 1 Fı g 1 19 a s: y a x ls x a x ls F ig + f (s į n to X [x ] )} y = I w h ile x1 is in x 】 e tc , c o n v e r ts t o In c h a p te (I r 1 20 + 2 1) (C u rv a tu re a n d Scanned by CamScanner 只 N S C E N D E N T A L F U N C T IO N S fix i . H e re , ca n s in x se d b e d is c u s In :e tw o w a y s i , G r a pp h d ia gg r a mm aa nn dıpc ir c 1le d i a gr a m r y 陰 F ig 1 2 1 。 A B = C D a fte r i e , In a b o v e fig u r e , al o f 2n ( v r e t i ı ı e v ety d ic fu n c tio n w ith p e r io d 2 7【 c a u e d p e r io : s in x is a c ir c le o e v e ry in te r v a l o f 2 7【) f l a d o m d n o f c o s ln e fu n ti c o n ]s R an d th e ra n g e is [ 1 , ī ] Scanned by CamScanner 岑参 幺 C ir c le d ia g r a m l: Let 曲 a c ir c le o en f r a d iu ' 1 ie ' , a u n it c i l rc e , a co s a , s 一 1 co s , 13 由 b コ 一 1 ー , く の 舜 (c ) T a n g e f (x ) n t fu n ー c t io n o f th e fu ロ コ ta n x = T he d o m a in n c t io n y ta n x = is ロ 吻 の 1 an d Ra ge n T h e fu fr o m «»o 冗 2 R t o 三多 y in 3n to 2 2 , ta n x = as x «> o T h e g r a p h is N ( o r n c t io n to + 2 E 售 in ) to fr o 了 前 m 5冗 A n . 2 sh o w n で c r e a s e s s t r ic tl y c re a se s 3冗 ' - d so o n a s t* t e 2 in fin ity X (d ) C o s e c a n t fu Ĺ = n c 士 n , ' 3n , į 5n a re a s y m ' 2 p to te s to y 2 - ' ta Bu t n e v e r m e e ts o r te n d s to n x t io n · (ī å Scanned by CamScanner H e re ) d o m a in o fy ° c o se c X is {o 士 7t İ 2 n į 3 n ) a n d z} ra n g e R ( 1 1) R {n n P , , ln , . , io n y T h c fu n c t Sho w n is p c r lo d ic w ith p e r io d 2 n co se c x . a s: y Fı 8 1 27 T h e fu n c tio n y - ls p e r io d ic se c x N o rt* il) T kıe c u rv e w ith p e r io d 2 n : a or (n ) T h e ııÞ c u rv e y c o se c x - X y = a m c, se c x te n d s to n E m eet a in te g e r te n d s to tx tx , n , 1 2n p to te to y a re a s y m m eet a o į - İ = å · . ty A t in fin ı c o sec X - A t in fin . ity FX w° P # h ł H i - . \ S A s sho w n " in F ig 1 » į,ţįįįįįįį ! . Scanned by CamScanner d d ıi e n x Ė b n 三セ乐巳 C B ip ia L 歹妇国 c » = p = ご ° a 〉 I 으 三= 二a r x モ 足 葛 ゐ日 阳 d o m a in R . 1 = ) - e a s e w M ıd ıe 1 e a s e in Jļ ĹŁ, f (x ) is in c r e a s in g ń ın c t io n o n R a » 一 佷 Scanned by CamScanner 匕 Ņ w ţ w \h e d W . WĮ \wy R *ņd w \n * ľ !pŞţ ttĻ \ p f \N ł \ ı Q$ : - 邕 h ni 4ı (ł . 队 储 « し氐 和 U Rm w 14 N ) . Ļ a x l* T Ï \o ł k y, {« , Il : · m le g + l}) a ļ ķn :3 N M fu n ç Tio n * ï n h n t\e ą ļ \ł n ı a 、 m 和辄 讷 W h l e e u ļ d b e g ra p h lç a lly sh o w f m > ı(O M ı o f \o g m a ) n as A g n ln ı Fı g 1 34 rrl > 1 = > lo g rn x > lo g m y ;w h e n x < y an d o« m < Scanned by CamScanner r ı 國団 Scanned by CamScanner r o ln į 酣 y O , X a /9 X : Scanned by CamScanner (a z < b z) b 2 - 1 (Q 2 > ď ) 了 コ 京 @ 呷 售 で 了 齢 料 낳 一 趟 Scanned by CamScanner z )x (ï v y 2 . A ı (R e c ta n l u la r a ) h yp e rb o ı r d angular c a lle (V ) ×y T h e re fo re , = c 2 ılc r c th c a s y m p to te s , a h y pe rbo ı F lg 1 5 1 N te o ı nve rs e A o te Fo c u rv e s la ıo w t h e ir in b lg o n o n ïe v e rs e c a n be E v e ry in v e rs e tr ig o f ex a m have w e C u rv e n o m e t r ic e y q u sed th e p to te s fo r its a sy m nam e co m e te d e fln ltio n pı see a in hen s u lc ń in o nd Fı g 1 52 c d b ta in e d b y n o m e tr ic a n o n s a re m r e s tr ic t in is b e e n Le t p le s in , f (x ) x . o n e in th e ir d o m , m a in s o a s to m a k e ce by s h o r te n in g is in v e r tib le an d in v e r s e a fu , th e y a r e n o in v e r tibı e n c tio n d to s in = do m e y g th e d o c o n v e r te tib le fo r x E R x is n o t in v e r kn o w d fin i v e r s e w e h a v e to e In o r d e r to g e t th e n W " T r ig o s w e But N In a b o v e c h a p te Q a re Jr a l x ıa th e d o m a in X a in a s L ş] ' L å > [ d e fin e d b y Ļ 1】 f (x ) = s in x ca n be r e p re s e n tei : 1) : Scanned by CamScanner a h d )G l p w fy o 1 5 ln - x h e re , ïİŢ X: · ( . ) is ir r o m im r a ge o f F ig 1 5 3 h (II) G r a p fy o 1 co 5 - x H e re , a in E do m R (lil) G r a p h H e re fy o do , , [0 ge an [ 1 1] - , ] tc 1 ta n m a in R x R , an Į Į ge ' F ig 1 5 5 h a v e d is c u s s e d e a r lie r g r a p h interchangg th e c o o r d in a te a x e s As " w e , o f in v e rs e fu L is im n c tio n a ge o f f (x ) a b o u t y - Jč ' o r " by I L " (lv ) G r a p h o f y W e kn ow Thu s, - co l 1 x fu n c t io n 1 a in o f c o t ţ f : (0 th a t t h e do m x E R , 7c an ) d R R an , g iv ge E en by (q tc ) f (e ) = co t e is in v e r tib le r r T m > 1 - 卜寸三产 F ig 1 5 6 Scanned by CamScanner 1 y = l se c x , has do m a in 4 ( R ( . [1 1] U , 1, 1 ) a n d ra n y ◆ ' (n Ì , ) g iv e n b y f (e ) = - , ge [0 , Į 】 7c e is in v en ib le sh o w n a s b ya tt Þ y= ttlt » x 1 se c = E se c » a x is 1) . (v n A s w e kn o w f , R : ïĮ Į 2 ; y y = - co se c ( i ii {ū } ' 1 x do m i ) is in , a in E v e iī io ie g iv ( 1 R , en by j (e ) = co s e 1) co se c x go n o m e tr ic tr ig o a t fu n c t io n v e rs e fu n c tio n n o m e tr ic is fu m en nc tio tio n n ed, is th e n it m eans m e n tio n e d , it w aı i th e p rin c p ill m ea n the (j e ) , Scanned by CamScanner ! È EX A M P L E ; I ) S o h ıe t/w ı« ıU U in e q u a lity N U M ET R ı Tı ŁS C ı N EQ U A L ı s in 1 ľ > O N S O LU T ı A s th c fu n c tio n s in r : 2 , h n s lc ırs t p o s ltiv e ı »c ľ io d 2 r h ls gr a p h . Fy - s in H a n d v į - a re m ke n in e q u n v o c u lv e s o n x a {T h n t is n lity o n y p I. th e w h y it is In tc ıv a su llic i? m to s o lv e l .w 禹 y = 呼ln x a n d y 一 ij Ï Scanned by CamScanner a bo v e F ro m Th u s , o n ßg u r e ge n e ra , : s o lu tio " lis in g a b o v e 2 h ic h lm p lie L藍亟亟哩 w Į jIĪ Ĥ PLE d is c u s s e d so lu tio n S o to c h e c k th e 二三 兰 x ś 3 O n fig u re , S o lu tio n 0 7 7r , ' 6 0 r ig in a l 二 \ 些 : ś co s x n E 名 s , w ś o 2 rc s o lu he n 2 don 4 tc ś x ś 2n 几 十 3 3 f co s x ś F ig 1 6 0 n n + kn ow e so ta n x is fig u r e , 2 p e r io d ic o n th e lirtio n < x < or 2 , inequaliytan x s o lv e th e ' n E 名 2 x E ta n , 1 叻 < 2 w ith in te r v a l ta n x < 2 a r c ta n n Jc + ] ; n s oa ! w he n 2 2 2 ) 6 冗 十 竺 2n n 十 = 2n [ 2 ] in co s x 3 bo v e l g e n e r a is in g a < x < į p r e v io u in , It is d e a r fr o m x < 〈x 〈 n 冗 s o lv e th e in e q u a lity As ON S O LU T ı < h 竺 竺 竺竺 竺 e S C聞 Ž he n w s ln Jrc > G e n e r a l s o lu tio n n 2n n " 2n < x < l '" , 2 rr + ta n ' 2 n n + a r c t a ıı 2 x = ta n 12 . A rc u n 2 Scanned by CamScanner 1 in r < : s w ith pe ń o pro n ıfig u >u n · t < ) , d m re s in t n ow is p c r lo d lc th u s to c h e c k o n , w hen 4 V z ワ冗 〈 t 〈 4 日 幺 @ コ 凸 당 2n 冗 十 〈 十 4 4 = , ° i EXA M P L E n 4 + 2x s in 2x 0 Q 玊 x lity ." < co s 2n 〈 嘲 冗 十 4 12 13 ' " į S o lv e th e in e q u co s 2 ï 4 s in 2x " t " (Į " n n " の e » 四 . 国 겁 2× 2 Q " ° ; 了 pu ' " î " Scanned by CamScanner b ē " 墓 IN恼b l " te . " H e rë , .. s a B . p a r tic u la Ć " . " A · : cos A COS ť ' " 1 65 Wh e n f c a s e a r is e s cos A = + co s B cos = A B . + - A A C cos an C . d o r n = B - C (i E A A B + he n A , B , C w r + c a re n o n d is tin c t) tc = ?c /3 = 1 . 2 ' CO S : ĹE A so ı v e th e + COS B + Ţs C inequQı iy = s in ĝ x (o n ly H e re le ft h a n d s id e is d e fin e d fo r a ll x 2 ta n , he n A 1 co s x + 2 IO N w B - c) 1 ta n x e x ce pt x - = n 7c + 2 ' w h e re n z x i + ta n 2r 1+ ' y + ta n X 2 [L e t , ta n x · J] x Scanned by CamScanner ţ ţ' ' ; a re p lo t r in g ta n 2 x fo 2 In g iv e n c u r v e le t A , B , C s u c h th a t u s r x E ( n , n th r e e sa m e fu n c t io n c o n s id e r So ) c o n s id e r a n y th r e e p o in t s X w ta n 2 A + 2 ta n 2 he B + 2 ta n a C > 1 2 Scanned by CamScanner N G ï īQ U A T IO N S G R A P H ĻC A T L Y S O IV Ł ı e ħh a n d d ń 8 h t h a n d s id e an o l e q u a lity an d th e n um be n o l in t er a re EÏ aı so w e ı Fin d d ıe . num bď s in le t f (x »= o fs o lu (io n s o / an r d g (X 1ś T h u s to sket\ b o th c u rv e s w X ° f o = s in x 5 15 w ) s in x 0 1 10 5 ś 1 × 5 10 10 he n x E 10] [ 1 , F ig 1 6 8 Fm m ah a w ñ gu æ 只对 s in x a n = d g (x ) = i in te ts e c t a t 7 p o in ts S o , n m b e rs o f s o lu tio n 昌マ巴 フ EX A M P LE ' ı】 = '" " P ı? ' " ' " e ı eūstpmitve v a lu e o fĄ saüying ta n n x an d g (x ) = x + 1; x w H ch y s o lu t io n s co u x + - ld b e 1 lie s in th e in re r v a i sh o w n a s F ig 1 6 9 x = x + 1 h a s in fin itly m an bu t th e le a s t p o s itiv e v a lu e of Scanned by CamScanner łij(A M ÞLE F in d tlıc rıı ım l b e r o l s o lu t lo rłs s in X » x f th e e q u a tio n o 2 ◆x F ig h ic h Į y w a t a n EX A M P L E O N SO LU T ı F in d tiie L e t ; f (x ) = e " y p o in t , t h e r e fo n u m a n be r o f s o lu d g (x ) = 4 x , F ig Pro m J th e fig u re , it is P L E S O L U T IO N c le a r th e F in d th e L e t ; f (x ) = lo g y in te r s e c t a t n u m be r a n 1o x o s o lu d f h ic h w l + 1 70 re n o rio n s o e co u x o n X = ld b e sh o w n a s 1 71 tw o p o in ts , th e r e fo r e t w o f s o lu t io n s o f d g (x ) = , x w h ic h lo g 1o x co u = ld b e s o lu d o n s x sh o w n as ダ i l , 园国团困团团 园团 ■国■ 目 i 丁 イ ア 竺 "". (。 ,。x 团团国回国 = ° Į e e at o n e Scanned by CamScanner Àiilijptli 希㈹ W T ïO N 翩 ß kml>mggñ«phþ /T.yi#iij / (fi}x} ' ţ tp d rĦw ĺ h l§W v R ïņ # Į¢o fx , Ae w e 当加 和加 对 细声烛 殖紐 癩 h 声縣 翊 茗島 1 aĦ h Ï i{ ï e r vg ¢ §bIìu jd d r a w f?i# M }i 2 # ' , ブー 》p b w ł m d i fu !ā e tawi or w h 2 1r: P h lc b l¢d e fin e d fo r tbe In te r v a l o tJe n g t ı ä ti ダ 4 F l¥ 1 7 3 , X N ' , . XX ľ r! ĮF '" h r lh . . , " a. ' t ,a " iA t l'" . '" ln . '"\ , o ie a r n ıh e o te ł S « rd e n u a re a d víc e d 【 x + 2n rj , f s in Ą (s in F jas , 取 : 初 S x ś 2 1 S x ś 2 o 2 初 Ś × Ś 羞x ś an d 田 聞 2 Scanned by CamScanner 旧■ 席 ı (c o ı x } X ) f# P e r lo d tc w ith p e r io d 2 tr, d r a w th e r g a p h fo ín te r v a l o f le n g th 2 r o n e n a iıd F e p e a t fo r e n t ir e : : : - 摸 or 一 Ħ 角\ ぶ It h a s be d e fin e d fo en r o < 2 x < tc th a t 卜 h a s le n g th 2 7c so , its g r a p h co u td b e p lo tte d as ダ 一 い 灯 國 。 州 处 a b v n ic i 沸 相 差 の ー 穆 習 汇 冒 粼 F ig , ( i Th u s , th e c u r v e y - co s ı (c o s x 聯 1 74 ) 一 니 P L E S fæ tc ìı t h e ' g ra p h g r a p ìıjo (ta n w e sh o u x r ta n - ld d r a w ' < th e (ta n x < x (ta n x » ith p e r io d ?r g r a p h fo r o n e in te r v a l o f le n g th ) is p e r io d ic ta n fin e d fo y r ' ) w = th a t x ; Į 1c a n d re p e a t fo r e n tir e す ー . " h a s le n g th n So , its g r a p h co u ld b e p lo tt e d a s し ー 脳 Scanned by CamScanner L E X A iIIı P 国匦 LE 吟 昌 島 三 回 冢 酴 S O L U T iO N Jw w e tr w th t M jb r y - ł [c o i e c x ) = d an re p e a t fo r e n ti. . S x < U c o se c ' «c o s e c x ) = X ; ; dm d " Ś n x < 2 « X ś 70 Ŝ O o r く n n x ś X x ' 0 2 4 冗 " O O f 2 X " g th 2 x ' X ıı s, , n fı o os ec Æ (c o l e c x ) ' A ıy 也 m 猷 (a o ıe c x ) b p e r lo d lc w ith p e r io d 2 R th l* g r a p h w e l d a a w th e g ra p h fo r o n e in te r v a ı o f le n X or ı o o ie c ] x , {O , n 。 一 ( 흠] 。, } th a t h a s le n g th 2 n so ' , - " ph co u se c x ) d p e a t fo ld . : g r a p h fo : į ŁA Į r o n e in te r v a l o f le n g th 2 n an re r e n tire ť , h co u ed t ld b e p lo t Scanned by CamScanner F ig 1 7 7 Thw th e c u r v e fo r y - se c ı (s e c x ) ' ' V alu e s o fx As w (c o ,X ) is p e r io d ic ith p e r io d w 1r . e lm o w ' co t w (c o t x } hic h is d e fin e d fo W its g r a p h co u r ı e n g th i ld b e p lo tte d e , x (O , n (c o t x ) )an d x {x = e O {n rr , n x < < z n # } !i } as Jį ; m en do m d o n ed a h ıa n d : s ix c tır v e s a r e ra n ge an d th e n n o n p e r io d ic , b u t h a v e s k e tc h th e s e c u r v e s &į į Scanned by CamScanner H en ce sh w e , d s k e tc h y ı ou 1 (s in s in - ) x ly o n x Į 汇 飞 «lı » S k e tc h fo W ı m e ow r th e c u rv e y do , p lo tte d w e sh o u ld s k e tc h y a s sh o w n co s - [ y 1 x ) t h u s th e g ra p h fo r y , fn r th e c x = S o . Its g ra h p = co u so , its g ra p h も t ) o n ly , 1ś x 5 u rv e = y c o s (c D s x (ta n ta n - 1 w hen x E [ 1 1 ] S o its g r a p h , , co u ld b Ta n ld & (ta n ' x ) p lo t te d = JĻ = 1 . ) ' x V x ) E 1 1, 1] 1 80 an y x in F ig 1 8 0 F ig h - (i e ◆ (n j s d y , x (c o s an [ 1 1] x , - 1 , 1] ) x a in m ge y ra n e n ce 1 S , a iıd H (C O c o s - [ E 1 79 F ig ■ hen w d R a s sho w n Scanned by CamScanner W " 矚 ird ar y ge y sh o w n T bu W e (Ĺe h fo 】 cn o w < , or lś R y ' x ) g r a p h fo X r c o se c , y - do m ś 0 3 ) ( 1 1) ld s k e tc h 1 - X < , ) x = x on g r a p h c o u ld b e in F ig 1 8 2 y c x ( s , th e Or) s k e t - l Ś X w e sho u o o se c S o , its < 0 0 , ra n H e iıæ - (c o w ダ (Ĺ & l c o sec - a he n w p lo tte d as r ' (c o s e c sec (s e c in R 1 ly E o r x ' » x ) ( Ł1 ) 1 5 x < o o ) " x < = - ) R . Scanned by CamScanner ı ģ : Tw I fv ) s ln ' (3 x 3 4× ) fv o co s ' (4 × 3 3 ×l Scanned by CamScanner t ga p s a re n o S o In p . So . N o t d iffe e v io u s c u rv e t d iffe ţ m S ın tc h y .:· .ţ ' re n " re n - tia b le y - a s in t trm t p o in t ı , Ð w e kn ow it h a s s ha rp ed ge a t x 1 = an d x = l a tia b le c o S . n ıa · iıı 註 \ ) + . 1 {S e e E x a m p le 2 } J h ta n e = x 4 e = ta n 1 x » ◆ 。 隽 一 吸 Scanned by CamScanner 孤 t3he 評 ゆイ y ' O TS ' \Ļ ť â ns j 2 ıH n 1 41 , X < N . 歹 一 汽 卜 茫 一 龚 F to m a b o v e fıg u re it is c le a r y c o s: - 1 Ĺ ) is n ot d ifte re n tla b le M . - 0 r l v ııi u n ıil w ıı ıw ı ' 1 x 2 ) y y D e Ħn ln g d ta n = E ı f f į å · x i t ı» w Ŕl ." " an ' e y · (Į Į) ' ıe に個ママe l \ X Scanned by CamScanner S o , th e g r a p h o l . 1 x \ ł a ta n n Zx Z tn n ļ v x < 1< x l lß a h o w n n « x 2 w d iffe r e n tia b ı e n e ith e r c o n t in u o u s n o r h ic h is ' H e re, fo r d o m a in y at x - { 1 1} , ĺ ) Ĺ ) - ' ta n iyĘ x E R ex ce pt 1 3× 2 . 0 = X E R > x + 土 1 土 Scanned by CamScanner 30 箔 十 3切 九十 > n + s o , th e g ra p h o f y - 1 1a n () §/ Ï x - 1/43 ! 育く h ŁÏ I I )r: ta n ! X 6 3 ta n 冗 十 ¢ n Į 3 ta n . i ; t囟门 6 30 30 1 熠 ヰ 吕t莉n 4 1 . 3 ta n ļ x X く - < x < ' x J 么 」 3 X 〉 言 万 į/ I X a 1/l 3 Fı g 1 88 Scanned by CamScanner 叻 y - s in ı (s in 3 e ) = , 3e ş ' ¥ 3e 1c 2 do Fo r Fo r ra n m a in y ge S o , th e g r a p h y o - - f y s in ' s in ' - s in 4 ×3 ) (3 x 4 ×3 ) 1 (3 x · . 3e ś (3 x 4x 3 s in - ) " ı n x ś 2 ı n_1 s in l n x ś 6 [ 1, 1】 X . ž x 3 s in . 2 ' y ' , Į = 7c 3 s in 1 x 15 X Ś : : is s h o w : n as : ; : : : i : ; : ; : ;rv ' ' " t į{ t Iī . 3 4× ) ; : : : : ; .:.;: Scanned by CamScanner 3 co s 2n 3e 2 16 Ś 3e Ś 2 n 1 1C ś co s X = « 2n If ţ s in c e , o s e s co s x ' 3 co s ś o f y 1 co s - ś ı Jr ï rj × ś 1 co s 1 ' 15 X e ś co s X Ś 2 1 o co se ś o r n ś 1 H e re th e , in te r v a l c ha n ge d (4 × ]} 3 3x ) = , 名冗 凸C 0 5 3 1 co s 2 is s h o w 3 3 is d e c r e a s in g in [O , S o , th e g r a p h Seos 2 1【 Į ° X j 1 + co s b - 1 j CO S 3 cO S X n 2 n + 3 cos Z n l ' 3 1【 + 1 × co s 1 15 × x 5 2 ) n as F ig 1 9 0 ¢į , N u s , th e c u r v e į ţiĮ ;thich . ' ' " ţ Į į ĺ i æm § į5,ījE$ī.ŝĺìįİi) Ï . . fo r y - co s ' (4 × 3 3x e n t ia b le a t S k e r cli th e . I , ;c o s e c x . N h . x ) , į = J . (iD co s x c u rv e s e a c h x R (i se c x {ıtlc is e n ta n q u a l t o 1 , b u t fo z x r t しA d iffe r e n t do m a in a s } Scanned by CamScanner T lıw , (II) y = (iii) y - CO S X ta n x ld b c p lo th e y c o u se c X co Tx d tt e l; a = X f ı ı (2 m 1) å ; n f z 一 1 x ó R rln , Į + n n ;n 于 E z 邕 as V , į : 買 盪 į y 。 1 · . . 且 - ľ į y= 上 . 台 '"'" " ĝ X 入 围国 ¥ĝ eı n x co se c x y = " 2 i : . X 扣 & C0 8 X ee c X 自 ダ コ 弓 了 旧 , y . ta n x c o t x y = cos x e a r th a t y a s ln x c o s e c x = 1 , p le It b e c o m e s c ı ff t d i e n r e e i a r s d n m a i h o r t e ı s Ta n x c o t x - 1 b u t th e y a r e n o t e q u a , a n a n d r a n ße o n s w h ic h h a v e s a m e d o m a ı T h o s e fu n c tı o ns E q u a ıfu n c tı F ro m y - a bo ve se c x exa m a re e e l, qual o ns fu n c tı m 8x I s in x o l+ lo g ve p lo t a b o ) Ic o s e c x ı + 1) (v i) 1 + m u st c he c k c u rv e s w e 3 (lo g s ln x + p c r io d lc ity d o lo g co se c X m a in a n d ra n ) ge łX " , - , Scanned by CamScanner L 名 n n E Z ダ a bo ve S o , fr o m do m R a n ge it {n R a in , 2n \( E 2n co u + « « x < 1) x < x < p lo tte d be ld } n E £ n as (2 n + (2 n + 1)n sh o w n 2 )n in y - F ig 1 9 2 F ig ) S k e tc h fo (lı S o , it : lilo S k e tc h fo :i ' jį +jjĮ)Ţ : co u ; CO S X ld b e p lo tte d Ţī y " = 1 92 Ic o s Ą y r r = ii sı n l as 1 Is defined {a s ; s in 1 x is d e fin e d w I×lŚ 11 he n {a s ; 1 + {a s ; ( l× I 1) < {a s ; x u is n u 2 x 2 > 0] 2 · L I×l) I Scanned by CamScanner T t- H en ce tw o 1h p Th e . o in t s th e s, tı p o in t s A fo H w y Sho fo r v = , , in ll + x 1 2 j lr is o n ly n as fo r y ° " " ) - is o n ly G tw o ra p h fo d B r y ı o g = ' '" 1 v 4 * v 4 lo g " ], r e ln ' (\ ) 5 Fı 区 工9 4 l0 g - " - w s k e tc h an W ) S k e tc h y g ra p h ą lo g + į) İ) " * ever . ı o g 4 + " ' . 0 (4 x 4 6x + 1) 2 · '" Bx 1) " ' " 2 . : .°. . ° '' ". '" h, " , x > o an d a ¢ l) ソム 斗 " " R Ł a n Tı 】 u s , ť be g r a a h g e ı. p h is . - O » 0 ı 1 Þ 1 X 气 ı sh o w n a s 4 F ig 1 9 5 × D] Scanned by CamScanner Ă I w Ialn r l/ le n e v e l h Y 1 ll . , O - It f眉目C x l/ nd iı : O ıı 3 (lo g 1)i w . 姻旧国 i照旧国 D ı ti ı o Ħ1 目ı ı tlın g e y co sy - 佃网 照 c o s x ; r e p r e s e n ts t w o s tr a 朴e * e ( ı» - 2n 冗 土 x he n e v e r X ¢ n lt; II Ę Z 》 n 名 ig h t lin e s Scanned by CamScanner Ľ 2 F ig 1 9 8 in fin itte num b e r o f p o in ts o f in įiiiAı iı pLE àsuTıoN te rs e c t io n s ; (s o s k e tc h th e H e le s in y - cu r c jo r s in , y if a s k e d - s in x n u m ber + ( 1y s in . - n n y o f s o lu tio x n n s th e n th y a re in fin i e ) z : 酥 卿 二 一 = 5 燕 一 F ig 1 9 9 . . 燕 一 巍 Scanned by CamScanner ie tw o , Thu s , ţ $ËÀ M 飞 ģ+ in lin ite th e F p e r p e n d ic u la r s in y - e in x se t o g ra p h fo P LE r F in d th e n u m be r s tr a o ig h t lin e s a s sho w n f s o lu tio n s jo r s in in p ig 1 9 9 , 9 9x 1W = 2 . ı1 to . ." " fın d n um , ber o f (X ) = s" f s o lu tio n s ; w ' e sh a " ' " ) 500 9 9x = 5 ōi ll p lo t b o th th e ' c u rv e s a s / X L 2 F ig 1 1 0 0 e a r ly , tai fr o m th e a b o v e fig u r e , th e n u m b e r o f so lu t io n s a r e 7 ° P L E ıin d th e H n d n u m th e be r o f s o lu tio n s jo r ber n u m o f so ı u fıo n s co s x Jo r [x ] = = - x {x } w h e r e [* ] { · } , r e p r es e n ts [x ]: «x } = Scanned by CamScanner iŢĘl e į@ f m d th e fijįiţÑ an d x - å n um be r a r e th e o n o ly fs o lu tio n s o f tw o s o ı u d o n s th e v a lu e o f × g r a p h ic a w h e re tw o fu n c t io n s f (x ) a n d lly s a tis jy in g [x ] 1 + x 0 w he re g (x ) c o u ld b e X F ig 1 104 Scanned by CamScanner T h u s , fr o m th e 1 a b o v e gr a p h x . h e. r e A W ' " ' an d D is th e p o in t o f in x = Ia dd Cc i p an s o in t o f in te r s e c tio n te r s e c tio n 2 o z 5 = z 5 S Lx j Ś x Z in e w n X iĄ co s (a x + [c Ďju , . J if 1 2 x = , 47 f 05 b5 3 a n d th e equ a tio n x 2 + 2 4 + 2x 3 + 4 = 3 b ) = 2 x hos c o s (? + b) Scanned by CamScanner ł X',X- p o in t ly o n he n w F lg 1 1 0 6 [X O L U T IO N H g o n o m e tr ic Let y ; p o in ts A = Cen th e fıg u tr o id o f A Q (B . : s in B x ) R (C w B = A fo rm s in c s in , h ic h d an In A p re , hav w e o n , A s ħo m 书 ; s in s in x = x e re S e d B , Th r e e e s in c t h e r e a r e th r e e x QR = sh o w n C as , by P (A , s in A ) ) is X G i; s in = 3 ' A + s in B + s in c 3 慧 囊 F i8 1 1 0 7 Ų l; i g ure G " (A Ļ + B + 3 s in C ' A + s in H + s in c ı 3 H l> G I Scanned by CamScanner F ig \* P rm B,C n A s ho w th e n < c o s e lc A + B co se c B 1 108 + C c o se c C < 3 2 fig u r e ı\o 1 p fO P e o W . . 吕 e > h p e fO S o V O 冗 2 ¢in = p 2 B B B o r m 1r 2 B ++þ 1 109 n < B c o se c 1c < 2 : , B lo pe o fo q < s to pe o fO S d s lo fO R < pe o C co se c c A c o s e c A < Tc/ 2 a n s lo pe o ] 7o / 2 i a th o d 【ï n : d iffe r e n t q u e s tio n s l o no o f tr g e tr rrı Scanned by CamScanner : Ī \ ; ' 가丘 ニ ー ー 罾 ; C o n s tnu c t th e g ra p h fo T X (ï) f(x ) = « ; 4 X 〈 0 x O = r 4 y 찌 〉 = 2 t + t rt l 5 リ C f(x - h e re x ); w T he n ıE R , fo r f(x 2 闷 寧 If f ïs d e fin e d b y y th e g ra p h fo r f(x C o n s tru c t th e g ra p h 2 o f th e fu n c tio F in d co s n < a 1 ; th e th e ' n s ho w be n um (c o s x ) [x ] he w = i fx X [[×] = X l5 ; x < 4 . f(x Is f(x re g io n ) ) tX = x 2 + x + x ] g re a te s t in te g e r fu n c tio n F in d th e ) X F ; = th e n u m s in x = in te g e r fu n c tio 9 (w h e re [ ] d e n o te s F in d 8 l5 x ś 1 10 1 in v e r tib le ? If no t in w h ic h w be he n s o lu f he re fo r rr a > a o tio n s T th e [ ] d e n o te s re n o r re x ;w = , r g re a te s t in te g e r fu n c tio :) ( ) 3 < ] ) [t×] ta 7 th e g ra p h c o n s tr u c t [ ] d e n o te s g re a te s t in te g e r fu n c tio n 1 If O , ) o n s tr u c t 6 III 2 t = [] s o lu f tio n s fo r d e n o te s th e g re a te s t n ues o va ı ly f × g ra p h ic a ı w h ic h sa tis fy 2 i 5 1 F in d th e v a lu e o [ ] d e n o te s f × fo rw h ic h x 3 [x ] = 3, w he re ti th e g re a te s t in te g e r fu n c o n it is in v e r tib le N S W ERS 3 X Ż . 1 Ź 7 . 5 s o lu tio ns 8 In ïin ile 9 Scanned by CamScanner ı , 一 臨 回 密 室 可 X L m e e ts C ly a t P b u t is e g n t to c o n ta n F ig n o t 1 h æ L is ta n g e n t to C a s e v e ra V A t P b u t m e e ts C a t l p o in ts F ig 2 函 醒 醒 一 车 隊 W 一 趙 b n g e m to C a t p b u N e s o n lw o a id ' " C , . Ro e s l. g C a t F . F ig 3 . i · . 蛐蹋 籙蠶国園 目國 Scanned by CamScanner ı ıc ı lt d ı h a p te s e r w h ıs t u d y ı a d in s o f c u T he b e n fc u a tio n s o T ra n s fo r m rv e s a t d iffe re n t p o in ts rv e s C U RV A T U R E * ib e s tu d y * u cb fbe o n d in g rv e s th e c u r v e c u C o n sid e r a c u r v e a n L t A b e a p o in t near P e d a rc s, arc pq = & t d if r e r e n t p c u r v e s a p o in t p o n it th e c u r v e o n = f o in t s is k n o w n n ır e a s c tır v a i " o r " R a te a 二 " a AP o Q A a rc a n d le t Q b e 曇 四 回 ôs s + = a ゴ 燮 ロ目 x is p o in t 一 le t y an dV Qm altes w ith : ôy b e + is w h ic h th e ta n g e n ts at p a n d x a x is ôy is c a lle d th e t o y g le s an c a lle d th e ta a v e ra l c u r v a ge c n ır e i a m e o f th e f th o a rc e p q a r e p Þ X q 05 r ıs nm y 0 Ós . @ . ds cw a m e o c u rv a tu r e w e s f th e h a ll d e Ø c o N C A V IT V , ır v e c ı ne a t g = a p o r N F L E X IO N N TS O F ı C O N V E X IT Y A N D P O I ı m Ľ" 7 C o n c a y e ń\ u p w a rd s ' n e i hb g o u rh o o ry ?° d o f a p o in t p o n a cu rv e is a b o v e th e ta n g e n t a t P , it is sa id to b e co n cav e '" illc!șases a s jc in c r e a s e s = > > dx O 2 Scanned by CamScanner 1 1 。 G ca e o m e tr ı ly ı ダ , ge n t , 甍 屋 每 ぎ a 主 氐 G Þ X y + 0 F lg 2 2 lf t h e t c u rv e is b e lo w t at th e ta n g e n p jt is 2 as x d e c re a s e s = id to b e co n v e x u pw a rd o r c o n c a v e do w n w ar d , h e m a t l ca ıly d d e c re a s e s sa ) dx y 2 ? e o m e t r fc a lly yı v ge n t m . 目 ı 围 篷 F ig 2 3 If a t a p o ín t p , a c u r v e ïn t o f in fle x io n c a lle d v X chah v g e s its c o n c a ity fr o m u pw a r d s to do ic e v e r s a w n w a rd s o r v T he n 籬 Į ·y ļ ı Scanned by CamScanner į+ a į avi n ĺ co nc Fe p ¢e s e as; 门 声 自 ge n t 帅 自 ビ 固 唧 @ ı V Ë ha s ıe w r o o ts a m w e re E Ĵ tų6 M \ an d it b e c o M e d n ıin im h ic h fu n c tio n @ te Ĥ Ŗ o E m m m n lio r p e af w W an u ca v e c o iı p, - O) cre a se s o r (i e y i e a s es ld b e e s c le a r th a t t o n ı i ï ı» a x is a i e E a a se c E s n o dea w o u 出 2 6 o q a ıı dbo ¢e I { af B d an ŁŚ , u m in v a lu e do w co n ca v e n a iä p o in t o f inflexion 自如 一 y (x = 1 )(x 2) i (fo r p o in t o f in te r s e c t o n : y x = dy . dc ¥ 2 3x 2 Jc + o n x a x is ) 2 3 an d 一 L n inim im ıa t x - å 窮 . 靄丰 轉雖 Scanned by CamScanner 靄 um an d d e c re a se s w hen 二 醒 艷 = 3 12 一 Ĺl į i げ 、 3 :ł h 11 in Ł cï 7 " d dr 6 m H in im e re \ų h e n u m C 3× _ X 2 + V J 2 = óx ]2 as = O 口 12r + 11 ? 6 】 n a e a se s 睏 隱 攤 篱 下 一 - 口 w he n x < T 6 + X > v 3 ' deæ Co ea ses w u n ca v e co n ca v e do w ß 6 hen 6 < 3 3 pw a rd s n w hen w x < hen ß + x < > x 2 d F ig 2 8 2 ľ Skel. + an 1) 2 (x 2) 2 5x . dx 3×2 Bx + 5 an d = d JC 6x 8 ' 2 Scanned by CamScanner to l11i X ep ttth , ı \ı # Bx J 3 dx ı + ; b t ťì lł w he Ħ a ie ı x く te n w ı l< 1 or x ン 3 ¢ in đ 1 n p ¢c ı ı ļl tncave se s up w do w OONı Ve he n n w x > he n sk e tc h e 5 ニ Į , x < ° t x < 3 d ln F i呂 2 9 a s sho w n ― rLt S ke tc h th e y o N H e re ł etuT s x # = um w m ax ln ı Î O, 2 - 3× w he n he n y c u r ve x 2× y - 2 2× ― ― 二 妞 島 骂 2 9 3 3 O - 1 3× - ― (i) X y as - (iil d m in im u m w he n x = 0 y ina ease s w h e n 0 l dea ease s w h e n 0 Concav e up w x < as < 1 x < d an 6 = x > F ig 2 1 0 (iii) 1 1 he n ; x < 1 foncave d ow n w he n x > 2 is ske tc h e d a s sh o w n in F ig 2 1 0 yiļ jie s k e tc h th e g r a p h jiÐr th e x = 1, Tu n c tıo n 3 w hen f (x ) y - = Ix + 3 1(x + 1) O Scanned by CamScanner X ÄN iŕ l S k e tc h th e g r a p h jo r y e re X x = y = + 2 + 3 1 w he n y w he n 1 - 2 . 1 + x = 1 x + H ON S O LU T ı f (x ) 3 ? (x + 2 (x 3) i 2 + 3) 2 = 3 In c r e a s in g 2 x - D e c r e a s in g he n ; w w he n ; x 3 < < x < 1 3o r x » 1 , Scanned by CamScanner ł.i g tp . be r ı .M g fo r r u le Ji^ e . Dec" 3 , · ' 2 (ł 3 2 dx 0 à8 X w In lo » ¢ de d y a s; = 1 as ; 9x 3) 9 1 [ 36 x at ttlīı 3 = 2 3x (X2+3) 2 t u t】a rīrt i ı iJ itr + . < - 2 O e i w o r ks a s a s y m c u rv e x a x s a bo v e h a p te r 3 ta iır e fe r c p to te j e , , th e c u rv e w o u ld rie r e r m e e t jr a l s ŕcr · a C A L T R A N S F O R M A T IO N S G RA PH ı th e tr a lıd is c u s s w e sha w t ł üaforms t o (i) İ (x ) f m s to o r ?m (iD f(x ) to ? a n s fo r m s Oin) f(x ) o n s fo n n s t û¢) f(x )? a fo r m s to ţr) f(x )? a n s f rm s to ( f(x )tr a n s o s to (Ţii) f(x )tr a n s fo r m y コ f (x ) İ Q f (x į a ) (a f (x )) f (a x ) f( x ) f (X ) f( x ) 弓 オ 君 名 前 一 月 to W n e re I· Im eans m o d u lu s o r a - - - . Æŕ y . y . y . ° ra !u e fu n づ c tio n 芬 荘 - = f(x )tr a n s fo rm s to y f ({×D f(x )tr a n s fo r m s to y {f (X )} f(x )m fo r m s t«» y {f ({x })} f(x )tr a n s fo r ın s to \ f (x ) f (X )u a n s fo r m s to { } {f (x )} y 一 - 一 - w - n eæ {· } d e n o te s fr a c tio n a l p a r t = İ (x ) tr a n s fo r n ı s t«» y = f ı (x ıf o fx 一 三 一 = )y t = įż h b s o Į u te = (ń) y (ï ii) y ( y lliı ) s 「 巻 n s fo r m a t io n s a s jf (x ) r s to f ( l× I) s to ll ( I× l) I ţ x ) ( f t o l f s l m r o J f(x ) tr a n s t o l If (x ) I f s l m r o n s y f (x ) tr a f(x ) tr a n s fo r m s to iy l lf [l× D l lījjj) 八目tr a n s fo r m (in ) f(x )tr a n s fo r m (I ) f(x )tr a n s fo r m 4 一 ' (x ) r e p r e s e n ts in w rs e o f f (x ) Scanned by CamScanner its " , M pı o t p i しE w O N Lu T ı y u n e × e + it , s h o w y ı a ıo w 1, is n a s y sh - - e e ift e d " ţ + (e x i»o n " u y p w a r d s - e \ " e n t ia by 1 l ń 1n w ith t h e h e lp c t io n ) A ls o y 1 u n it , c o u = e o . fy h' e - ld b e p lo x sh o w 1, n ' " tt e is 0 " d sh a s ift e d d o w n w a rd s by a s 国 F ig 2 1 5 - Scanned by CamScanner ( 、 门 ロ コ 召 件 く ツ 罾 n r Z ı7 , , 一 一 一 勇 二 コ一 么一 一 一 际 一 一 一 障 一 ı ļ l D w W T ı W ı oN p lo t e kn o w , y y s in - - s in ' Jç ' x y (fn - fs in v e rs e ' x ? go ) + n o m l a n e t r ic d ) y co u - 和加 ' ld b e p . o x ) tte d 1 as " Ţ £x 8 1 a n d lòř Scanned by CamScanner 材 钭 认 臣 叻 囲 = s in ı Y = s in ı y d u p w w d s h y 1 u n it 1 Is s h ifte d d o \ m w w d s lry 1 u n it s h ifte 1 Is r + r . X F lg 2 2 1 : (II) f (x ) t r a n s fo r m ie fm s 蓊 G Ĺe ra , n s fo f (x t o f (x + f (x + r ıııs f (x ) p h ic a lly it ı{x co u ld b e ) o ) a + , f (x ) t r a to a a ) + s ta S h ift th e g r a p h is p o s it iv e a o f f (x ) th r o u g h ' ll u n it to w a rds ri g ' 离。 , is p o s itiv e S h ift th e g r a p h a te d o f f (x ) th r o u gh ' a ' u n its to w a rd s left as 主 丰 庄 一 期 仉 一 体 旗 Fı g 2 22 . ' : Ä ĴŁ S O L U T IO N 斗 Pı o t . A y s - d is c u ss e lx 2 y d is ı - l× l y , f (x ) sh lx - f (x ifte d 2 ' ' u n its t o 2 a w 1 a n ) s h ift a r d d y - lx + 2 1 t o w a r d s r ig h t s r ig h t 2 2 1= (x 2 1: x » 2 茬 藍 邕 靉 Scanned by CamScanner T a ls o y Ĥ lx - s h ifü e d 1is 2 + ' 2 ' u n h s to w a rd s . 江◆ 2 1 - @ a= p le p lo t . š s in - _ d . - s in . ; F ig 2 2 5 kxAiı lè so u rrro N ¢ p lo t W e Jr J id Bb þb y n a ıin ţ - - - kn y s in - ') y - s in l (x 1) a n d y - s in ı (x + 1} o w s in ı x c o tıld s in ' s in ı PĶ 2 2 7 (X (x + b e p lo 细 巧g 2 2 6 a e d a s s h o w ıı ' 1) is s M ft e d ľ u n it t o w a r d 'ľ u n it t o w a r 1 3 i$ s h ifte y 4 ds r ig h t d ß le ft : . : g 2 26 Ī Scanned by CamScanner 1 ) (llı ) fo ¢tpın ıfo r m ıte iı (ı 哄 与t r e t e h th e g ra p h 耐 八女) ぜ 1加 ë 争 비增产姓到§ 1 八X 」 ? - S h r in k th e g r a p h o f f (x G r a p h ic a lly 八 ×1 ła 〉 ' こ台 ) 'd łtlH ìe §a lü n g y be it c o u ld 〉 1 s ta te d as ax i§ì in e h tİw n ı flĺ m 藎 一 一 一 鱸 ı, Z , Z ņ p l" y SO LU T ı O N e : > S tr e y = W e kn o w t c h th e g r a p h 2 M n x 4 g jn x = y · a ln x and an d y s 20 f (x ) f (x ) # (ı f f (x ) 'a ' tim e s a lo n g y a x is a t r e tc h th e g r a h o f e jtıx 12 ' tl p m o @ n Jť , · . 腦 e § a lo n gy Id U iļ ith p e r io d 2 n · a x is , 一 一 一 一 一 驟 rı * a ıo . 丽 画 A bo ve cu rv e la p lo tte d fo r th e In te rv a l [ ー n , a n】 IF w , Scanned by CamScanner 月 月 ow ?åt ın fW Y x s h r in k tim e s a lo n 叫 ı r g (o r c o n tr a c t ); a > ) th e g r a p h o E 1 f f (x ) ' a 气 . 磨 コ ' 前 吵 口 x a x is 气 卫 国 薛 一 @ コ 齢 : n ro m ĪÏ abo v e fig u r e s in x is p e r io d ic w ith p %T . d s in 2 x w ith p e r io d g n Scanned by CamScanner 二 @ " : Ī ĒI " to s t r e t c is x a x is Sho w n as h (o r e x p a n d ) th e g r a p h o t , in . , . Ti . À , in F ig 2 3 4 F ig 2 3 4 IF 羽 一 园 m 齟 國 翻 " " fig\inx i s p e r io d ic ;Į w ith p e r io d 2 d 7t a n in ' s in () is p e r io d ic w ith p e r io d 4 n (2 × l 司 監耄,監,君ı 湖 团 Scanned by CamScanner 团 ıto d ï ıg y p to t ßIn ° l x (ln p lo t y a ln = ' (j Fı 8 2 36 . (v ) t r a ııe ",k": 二 JLX ) J ' ' a b " A Ľ ) w e , - a x le " f (x ) in y a x is a s p la n e m ir r o r 。" o u t y y " O R F lg 2 3 9 . . Scanned by CamScanner ' ı , L L ' · , ''" th , " istheimageinJax]s"p"n'""f'" ' , - N isa rı l a 40 , , . þiö t ttièlu m æl l » H s o L u T Io N y e re y e lo g . ( - ' " x a bo c lċ ' · x lo g . ( - ) is ke to ta } x ir r o m r Im a ge o fy - lo g A « u ty a )cis S h o w n aĮ ť y y , y l¢ - lo g e x 1, 0) 守 n ıa 4ı , , Ð pı o t th e h m H O N UT ı m im T o r Sho w n as a g e ū ıF y e re f y o i - - cu rve s in ı sh ı ' y s = th e is ( x) (x ) a b o u t y a x is 2 42 Ş ų , Víc . · Į 't N E ŁU ı 二 二 二 了 tr ın 豸fo n 彻 切 M )朋 T w o d ra w r tt m y , tn e 二 ra f旷 f (x ) ta k e - g = p 、 n lın a ge o fy - f (x ) in th e x L a x is a一s 寸 1 p/ \ xÆ » b y Jf ( 」 18 0 a bo u t x L a x la I* 智 . , ' 0 R 0 1 a一r pı 二 ' ヒ ヒ 钿 ·メ . i ēêīiĮţ ł ïĺĮ x , Scanned by CamScanner Pı o t th e As y o r fy - - c u r " lo g x is g iv e n lo g x in th e y \ - th e n x a x y is 劈 - a s lo g a n pı x e · 靈 + M m W 5 腎 Scanned by CamScanner Ħıı (I) ï m a «n a bo u t t c o u ld b p G ra p h ic a lly ı y · a x la . (ll) Im a ge a bo u tx a x la . s ta te d a ı Fı g 2 . 曾 矚 쮜 則 叫 姑 姑 Scanned by CamScanner 。。。 。 u to 8 9 ir q ) ぞ川 哺 liı X群 田 乞战羽 ı )n( ıøn lw a 8 ä a q o n t Ä o tjls · ― ' ' M ? T lt u o M 中? ?nN ? Jo J Á ゼ U T Scanned by CamScanner ― O i) T a k e im \ a a x is bo u t y i P lñ a ım on I u ııı\ ie H f (x ) , e re y ; is d r a w f (x ) ı ı - in t w n o s te 〉 If (x ) I ) @ ps (a ) In th e I s t e p le a v e th e p o s it iv e p a r t o f f (x ) {i e th e p a r t o f f (x ) a b o v e x a x is ) a s it is (b ) In th e n s t e p ta k e th e m ir r o r im a g e o f n e g a tiv e p a r t o f f (x ) { i e th e p a r t o f f (x ) b e lo w x a x is } in th e x a x is a s p la n e m irT o r , , , , , O R T a k e th e m ir r o r im a ge (in x a x is ) o f th e p o r tio n o f th e g r a p h o f f (x ) w h ic h lie s b e lo w x a x is O R T G ra th e p o u m ly it p h ic a ı G ra ph o r tio n o co u f th e g r a p h ı d be s ta o f f (x ) ly in g b e lo w f f (x ) G o a x is by 1 8 0° a bo u t x ax is te d a s r a p h f«» r lf (x ) I y + 1 x l y - fo o P lı, a , M Scanned by CamScanner )0 1 [a ń t a $ to U ij dı =cu ss fł ilfi¢M iį (* ) ls d lffe r e n tïa b lo fó r aı ly d R S Tt ı Miityf / {ij} I T Ť . 0 Fı s , Ş İk · p la n e m ir r o r 1 ł0 4 F ll 2 S S , , 2x t ' " Įļ 3 2x - (x 3, 3 ) (x ıe , th e , + 1 ) is a p a r a b o la ; ow pa rt be ı x a x is so to in th e Scanned by CamScanner L G ra p h fo r y lx = 2 2 _ 3 x 1i Fı g 2 57 , e a r ly a į bov 女真M ĖĽÈ E T r a s k e tc h th e g r a p h jo r y d ra w y o is d iffe r e n tia b le fo e c u rv e Is in - Iw x , e ta k e th e ll x x ir r o r im m { 1 3} R ls in - . a , l g e (in x a x is ) o f th e p o r tio n o f th e g r a p h o fsh x 세 니 區 荘 本 Fro 墓 m a bo v e y (lx ) 1 (x ) t r a n s fo ie rm s figure t ls in - to x is c le a r lis d iffe r e n d a b le fo ll x r a {m R c n E in te g e r } f ( lx l) f (x ) 〉 x l) f (ı , If w e (i) (ii) ef twa kn Le T o w y a v e th e rd of the , th e n to g r a p h ly in g k e th e im a f (x ) - a ge y- x i o p lo t y r ig h t s id e o f f (x ) in f ( I× l) - , f th e y ·a x is th e r ig h t o s ( if it ex i st s) is w om i as w t w o s te p s it is a x is a s th e fy ld fo llo e sho u p la n e m ir r o r an d th e g r a p h o yin g f f (x ) 】 tted . O R ' " ł : : d ' Ţ 9 8 o f c u r v e r x o a - Scanned by CamScanner F 18 2 5 9 . ĖXAMILE ı SO L U T IO N y (i) h - lo g ave s k e tc h th e A s w e ı × co u kn ow , cu rve ld b e d r a w n a ge o y in tw a x is a s p la n it is e m irT o r x . 1产 igigï gpplotthe ps a x is a s 1o g x v L 0 = fy o j× l lo g - s te o r ig h t s id e f f (x ) in th e y y lo g = th e c u r v e th e g r a p h ly in g T a k e th e im SO W T I m y cu rve 一 y - e l. l X Scanned by CamScanner X X Fı s 2 ea , , i pı o t th e O N S O LU T ı A s w e kn o w \J1 , cu r ve y th e c u r v e 2 - fo , y (x ) tr a m fo r m ı to (x ) ı y - 2 1× 1 x 2 & 3 a s sh o w n 3 is p lo tt e d 2x in F ig 2 6 4 a Fı 冒, 2 吕弓 2 64 , , 2 - r r ' 2 sh o w s l× ı e r ×l ı 2 ×ı ı o? § i iðb e 1× ı 3 o tt e d pı my t ia b l e fo r a a s sh o w n ll X E in F ig 2 6 5 {0 1 R L ° À '" I/(lx I)ı i : 1心国 团 Scanned by CamScanner a 卜 太 一 ' 厂 y « 毒 f(x )I ı 枷 Fı g 2 67 F ig 2 6 a , . S k e tc lıtlıe A s w e 如 ow cu rve th e g r a jo r p h fo r y = y 一 ll× l2 × 2 2 2 1× I 3 飞s h o × ı w ııa s ダ 毒 y 1 (i) y = x 2 2x 3 9 y 2 ×l ı - 2 0 1 1 3 = 2 2× × » 3 X 1× l X 。 ( 回国。 f 10 1, 4 j , 4) F lg 2 7 0 . Scanned by CamScanner 'P £ ◆ ĺiii'A S OL UL T ıON O L¢e M ,Mhttı eg?aphTw we kn o w the gr ap h frtr y y ĺ e . . ;e . ĺ" į . U . X Fı g 2 73 , , O tl) 。 一 Scanned by CamScanner O R r t io g le c t) th e p o m ove ( (i) R e i a in in g p o r t o t th e r e m (ii) p io d l it c o u ld b e s t a te aı G ra p h ic y o r n e G ra p h fo r y - n o as f th e g r a p h w h ic h lie s b e lo w g r a p h a n d a ls o its m iır o r im n in F ig 2 7 6 o n f th e sh o w , EX A M P L E a a x is g e in th e x a x is f (x ) F ig I x S k e tc h th e cu rv e IJ ı (x = 2 76 1 ) (x 21 SO LU T ı oN Scanned by CamScanner ら 、 r lH Z 7 9 . L Fı g 2 80 PLE E X A M S k e tc h tlıe S O LU T ı O N A s th e gra p h fo c u rv e r y - l× l+ ı y l = 1 x ı is ダ * ◆ y - 1 x 气 y = 11 F ig 2 8 1 × l ı 囝っy 乡 = 1 1지 키키 = 1 lx l Ĵ¢ F ig 2 82 F iı g 2 83 Clėty abovegue represts a s q u a re Scanned by CamScanner F ll 楚回廊 , W y į x l y = l · ı1 Fı g 2 85 , F lg 2 ߢ , , , , (*iı }y ı (X )tr a m fo r m . fe , , y = s f (x ) to ı vl II(X )ı Iy l- ll (x )h ı §p le lı e tl ín = lw o M e p # , O ta pł uc a lly It c o u ld b ë a ta te d a e $ ' ' " . - Scanned by CamScanner g X A M P L ¢ JO p lo t £fı¢ <u §O U JT ı ON c u rv e n fo F c jb r ly l le " = y " e - \ is s h o w n as y . 九 玉 盏 隆 ◆ 气 F ig 之石名 串 (玆) y 戦 - le ' 1 키ï l Je = ' I F ïg 2 A 9 , 匿 露 . 2 90 F ig ţi Į a m ¢l e @ S O L U T IO N (1) y « A e ' pı o t t he ¢w e kn o w cu rve e lj l ı t h e e ttr v e il = fo r y - e " is s h o w n as (ii) y - e X 1 争 y - le " 1 1 蘇 讎 雝 Scanned by CamScanner @ į 7 ' ' n ¢ P 加 3 " ıp L E Ex A ı lá soo u m o wu e H e re P【 ot w e c lır v e la ıo w lï l th e g r a = 2 94 :I s in " p h fo r y !jŜ - s in F ig 2 9 6 Scanned by CamScanner ?r N ı . X y = 1 12 y = 3 12 Fı g 2 97 . (Ū F ig 2 9 8 (x iii) y ĹŁ ı r m s to Iy ı ı n lx ı )I = . T he \ f (x ) tr a n s fo = s te ps fo llo w y = y = 间 f (X ) If ( l× ı )l co u 〉 e d a re ◆ G r a p h ic a ll it y f (X ) ld b e s ta te d a s - IJ I ı f (X ) I = 〉y + - ı f (X ) ı ı y I If ( l× ı ) = : ; : Scanned by CamScanner (ii) y (D y If (x ) I = 门 く X 0 1 » x ー 障 く F ig 2 1 0 0 . 气 F ig 2 1 0 1 前 Þ X F ig 2 1 0 2 P lo EX A SO LU T IO N H t 浈e w e e kn I爿 o w e 一 lx 一 th e g r a p h fo r y = e Þ X F ig 2 1 0 3 ( 3 O i) y - e X 1 今 ア ー e lx l 1 2 l 12 X ı9 lo g 2 1 1 /2 y - 1 0 2 Scanned by CamScanner 团团■画日 四国亡느 厂 x ı og 2 F ig F i8 2 1 0 6 E X A M P LE S O LU T ı O N e P lo t H ; e re y 2 10 7 1o g j× ll IJ I ı = - lo g x is p lo tte d as X Fı g 2 10 8 (i) y - l o g x » y - l× l lo 8 O i) y - lo g j× l y - Ilo g j× ll X X F ig 2 1 0 9 (iii) y - llo g J× ll― ― Fı g 2 1 10 IJ 1 llo g j× ll = F ig 2 1 1 1 Scanned by CamScanner ţ īĮiļĺ j@ ¢ t¢ a li ¢ N N w uuwT lo ñ L i o łı . ¥ s k e rc li th e e kn o w he t] c u rv e gmph lx l- lï × 12 lo r y - x 3 z 1× l 3x - 21 2 y ¥ 3 y Fı g 2 112 N y - x ł 3x 2 1 y - l× l2 3 1× l (ii) y 2 ı - l× l2 3 1× l 2 r y = ll× l2 3 1× l Ť, 2 I ll×l = 3 1× l 2 1 Fı g 2 115 , Scanned by CamScanner s te p 2 s te p 3 S te p 4 G ra in t e r v a ls M a r k th e . M a r k th e c o r r e s p o n . p lo t th e p h ic a lly ìt co u v a lu e o ld b e o d o in ts o n y ? is it le n g th w ith in te g e r s a s e n p f r a h o f f (x )} o n x l d in g in te r v a ls {w ith th e h e p o g p ls fo r e a c h o f th e m a r k e d in t e r v a fu n a x is f [f (x )] sho w n a s F ig 2 1 1 6 H er ch y = s s ° _ g 2 1 Scanned by CamScanner y Fı B 2 1 17 y - s in x > y - is in x 】 F ig 2 1 1 8 : Ip EX A ı ĹE 25 SO L U T ı ON c o u ld s k e tc h th e x H cu rve y - [x 2 11 fw h e r e [ j d e n o te s g r e a te s t in te g e r Tu n c tio n ) ś 2 e re b e p lo tte d y - x a s sh o w n 2 1 in F ig 2 1 1 9 L II J r W ' Fı g 2 119 Fı 8 2 120 Scanned by CamScanner rv e ー y iE li t w l\e r¢ re p r e se n ts a c irc le l l d e n o te s th e R r e a te s t iııt , Ķť fo r y Z a ぱ 一 、 : イ 门 一 T hu 月 s , th e g ra p h fo r y = [ イ ー 蓑 L 一 圈 龍 够 x ] ° Fı g 2 121 , (x v ) y 一 建 陸 nO m W H = 1 (x ) t r a n s fo rm s e re m a r k th e te , the s e in i nt er s e c t i on to y g e rs = o n f ([x ]) th e x a x is D r a w p o i nt s d r a w h o r i zo n t a l y v e r tic a line - s l lin e s till th e in t e r s e c t th e y g r a p h o f f(x } ( pa r a l lel to xx i s) to m e e t the n e a r s t ri 8 t e h f (X ) 庄 牛 工 中 F rg 2 1 2 2 . . , Scanned by CamScanner y f ( [× J) · ダ F lg 2 12 3 O R p lo t th e s te p 1 s tr a ig h t lin es pa ra lle l to y a x is fo r in te g r a l v a lu ' 3 X Z t w h ic h X N o w m a r k (h e p o in ts a S te p 2 k e d p o in t fo r x s a y if n < x < T a k e th e lo w e r m a r s te p 3 d by x h n e a r e s t v e r t ic a J lin e fo r m e draw a h o ń z o n ta l lin e to t e ' , - X - n + n es o 1 X fx - , ıth e + X - Ł o n th e c u r v e k e th e p o in t a t x c e e d in g in th is w a y n ra Łp r o q n an w e d ge t rv e requ ire d c lı . IXAMPLE so Lu T Io N p lo t th e H e r e th e cu rv e g r a p h fo r y - e y - e '× l f· J is s h o w n a s Þ y y = * in t׳ - s in b e p lo ×} 【 w lim t te d a s s b o w n 2 x ś × ś 2 X n in F ig 2 1 2 5 Scanned by CamScanner e ' " T he . c u rv e " ' fo y r - ° [ 】 co s × co u .. ' . ld b e p lo t te d " as X F ig 2 1 2 6 iİ ï İ E Pı o? ħ e c u rv e , fo a ır v g r y - y - ° [· 】 ° [X 】 25 2s . x S 2 ś 2 c o u ld b e p l. n e d a s X ー し F ig 2 1 2 7 (x v i) y H 1(x )t r a n s fo - ere , w e sh o u rm s ld fo llo w to y - f([x ])] 【 t w o s te p s Scanned by CamScanner s k e tc lt th e jiÈAM p L E H IO N so LU T e re fir s t c u r ve w e sha y ll p lo - ls in [×}ıw h e r e l j d e n o te $ th e w t th e c u r v e fo r y sln[ - w he n x E e a te s t In t ¢ gT u Tu n a ı fo n 9 0 Æı 【 , X F lg 2 1 2 8 Ff o m a b o v e fig u d e th a l re w e c o n c u O ś m ı w ł y iE TnjiiilÞ ' S O L U T IO N p H - y ś n = ? 【s í n [ ×J l y lo t t fıe c ıir v e h e r e t o s k e tc x t - - y r J [e ' j , fo r o [e'''j w s in - fie n w e sh o u a 0 1) [×j e 【 , ll 4 < to ld fo ı w oś × x < 2 ś tr t h e s te p s a s Scanned by CamScanner リ 歹 飞 尋 ı ı (ł ) y . I9 ' j 身 黜 凱 a i* Oś y x « O x < 1 Bho w n a 8 F lg 8 18 2 1 3 1 垂 , , G ( xv ii) y Fle r e , ra fo r w - ı lc ' l]; w 2 - r e p r e s e n ts o n - a ln x kn 1ś ow 1, 0 , 1 - 一 p h fo r s in x y - f Ca n dı e s p o in ts fo l v a lu r w fy H es o . h ic h e re , do f (x ) m a in o in te g e r s , f f (x ) a r e as [y ] in tegen se t o fv a lu e s ofŊ cu rve a re o n x - te g r a [J ] w e Ĺy ] ly In o , S k e tc h tile S( ra x < = E X A M P LE G hen x ) / G ra p h 1(x ) t r a n s fo r m s to [y ] f ( l th o s e f (x ) w e c h e c k o n y o l [y ] 1o p ı tl J f (x ) h ic h f (x ) In le g e r s T h u s, 2 13 1 , p h ro r ; y T hu s; 。 In ls s h o w n as s in x ś 1 b u t s in s in x ly = ce ; [y ] = s in x s o lu t io n s D - o m a in o f [y ] = s in x In F lg 2 1 3 2 . n x aı ダ 血 零 了 F lg 2 1 3 2 . . Scanned by CamScanner G p h fo ra r [y ] s = in x å D ' O ' B E ł ' (0 0) , , A F ig 2 1 3 3 F ro m ca n a bov d le s ), A e o r p h fo r , d O a rke , A B , C , , D E , , A ' , B ' , C D ' , ' , Is, t'h e" g r a p h fo r ' E , = S k e tc h th e A S O L U T IO N ra re M P L E : G (th e p o in ts m i g r a p h fo r Ĺy ] s n x fig u y s w e - s in kn l ow th e cu r ve U ] g r a p h fo y r s in = ' 1 s in - G x . sho w n x ra p h fo a s in F ig 2 1 3 4 [y ] r = s l ln an d 2 13 5 X l X F ig 2 1 3 5 F ig 2 1 3 4 F ro m (xv iı i) y As w a = e b o v e fig u re , U ] 1(x )t r a n s fo ha v e e a r lie r on l t y h o s e v a lu e s o s in x = rm s d is c u f × fo r w to sse hk h [y ] d y = - [f (x )] [f (x )] f (X ) E , ie , tr a n s fo m ıa d o n (x iv ) , an d w e kn ow W ] im p lie s in t e g e r Scanned by CamScanner (1) G ra p h fo y r ał n ı - Jc] Ħ H - F lg 2 1 3 6 D G (ı ra p h fo r [¥ 】 [s ln = x 】 X F ig 2 1 3 7 F ro (x ix ) y m a b o v e fig u r e it is c le a r f (x ) t r a n s fo r m = s to y - U ] + f (x ) N 4 G ra ew o b ta in e d fu n ı p h lc a ı y lt co u c t io n ld b e G r a p h fo g is g r a p h fo s ta t e ' y d = x f({×}) (w h e 军三汇 ļ. is in = r y - 】is p e r io d ic re l } de 〉 n o w ith p e r io d 2 te s f r a c t io n a 7t ıp a r t o fx , f ({ x D f ({ ×}} " as f (x ) G ra p h fo r y = f k ×») F lg 2 1 3 8 , F ig 2 1 3 9 Scanned by CamScanner G ra p h ßr y - ı国 。 F ig 2 1 4 0 . þ 1l ({ p lo t y i fo r th e n te . A l g , ph ' ' r le n g th o n e 1, pe a t fo No w , . - 1o . r e ta in th e [ų . 1) a n d . ' S k e tc h th e EX A M P L E A s d is c u s s e d in ({x } 1 l is i° Jï I ({ . } cu rve a . bove g. o 1j = . Am p le f (ť . } 1j y ' - . . x 国 ' 1 ({ · } a x is w hen ev er ({x } u ' 名 is p o s iti. ° 鸥 月 「 ヨ 印 巴 。 コ 目 庄 F ig 2 1 4 2 . \ ËIE lĐ s k lsġlut ıo n a s w e g r a p h fo r ŝê i ? Ī Gr ī : a p h fo : r e t c h th e kn 2 x is o w [Q 1 ) a n x y gra ph , - 2 × o fJ - l fu n c tio n a n d p d r e p e a t fo r r e s t p o in ts ex o n e n tia t r a n s fo r m w e w a n t to p h fo G ra T o r e ta in r y - x l x it to 2 【 , it × 2 « » g r a p h b e tw een x E [0 1 ) , 匪 庄 仃 一 - , , 达 a = byiw d Scanned CamScanner 国 (x x ) y H į y - 1(x ) t r a n s fo r m - e re f×j p io t d r a w EXA M p LE : &馅 + 卜 为 . . Tı ON G ph ra o - {l (x )} ln e s fo r a lıin t e g r a l v a lu e s o f y lı h h fo r b o e s a n d t r a n s la te t e g r a p v e r tic a l U n h r iz o p lo t th e o , to y s n ta s k e tc h th e c u r v e . A f s ln s w e kn th e o w g ra y p h fo {s in - r y d fo u n d a ry r th e y - p o in t o f in o an d } (w h e r e { } d e rıo te s th e jr Q c tio n a ı p a rt x sho s in x - a n f o x ) w n as x ダ 未 目 慧 コ バ 미 一 耋 團 围 レ ニ し娜 훌 ト F ig 2 1 4 5 参 A s to G ŕ 吕 一 iæį ' ra hen o ś y r e ta in th e c u r v e w p h fo y r - «s in x < 1 d s h ift an o th e r s e c t io n s o f g r a p h b e tw een y - O to J · » ¢ F ig 2 1 4 6 į oi E A s w e kn ow th e c u r v e y - x 2 , is s h o w n as Scanned by CamScanner v ių w > : tv la ln tł \e g ın h Þ fa ı o 8 v 1nnd . In ' ' O ll' " I lm p m n l, F lg 2 1 4 8 . ' S k , n Ì ï th e EX A H P L E ON s o LU T ı As w la ıo w e y m th e c u {, } , - Y . - " ' ť G sho w n a s ra p h fo r y - {e × » th e g r a p h fo r O ś y th e o th e r s t o O ś y < 1 To r e ta in < 1 a n d tr a n s fo r m F ig 2 1 4 9 . F lg 2 1 5 0 . (ï x i)y · f(x )tr a n s fo rm s to y . - . (f(fx })J S k e tc h tl w e kn o w = e , is p lo tte d a s sh o w n in F ig 2 1 5 1 Scanned by CamScanner 」 G r a p h fo No r y - e w ' y e - ch y to s k e t 0 × b e tw e e n re a l x G ra p h fo r y "re'"in th ' = ¢ 5 x < la n d re \j p e a ín JÐ (x } - e. X F ig 2 1 5 1 F ig 2 1 5 2 G ra p h fo y r «e « × - » » 一 王 1 2 国 回 F ig 2 1 5 3 国 回 国 ı 目 ı 国 H e re an d r E įK A , w e re t ° kn y th e g r a p h M ÞL E so u Io N G p b fo ra g r a p h fo r y th e o w r fo r oś y S k e tc h th e A y = s w e s ln kn ow th e - < e '× } , n ow to p lo t s tr a i h Æ n e s g 1 a n d tr a n s fo cu rv e c u rv e y y = = rm llto th e o th e r s b e tw e a x i s fo e n O ś y < r f in te g ra l v a lu e s o ? Ľ {s in {×}} s in Ķ is p lo t te d as x 一 一 玨 玉 生 X F ig 2 1 5 4 鳥 壅 Scanned by CamScanner ť G r a p b fo r y a iıı« × } - 2 15 5 « 8 { ×» » p h fo r y F ro m a b o v e fig u r e G ra y So , th e g r a h p (xx il)y = H e re o fy 1 (x ) tr a n s fo r e ta in th e - s in rm s gra p h graph fo r o th e r v a lu e s Gra phi c a lı d be y ït c o u ı s ta {× } a n d y to o fy } fy - = - - s in {× } O ś y F ig 2 156 {s ín { × }} a re s a m < L e f (x ) f (x ) o n ly w hen y - f (x ) lie s be tw e e n y [q 1 ) a n d n e g le c t th e te d a s Scanned by CamScanner S k e tc l] tlze A O N S O LU T ı G ra ı fo pı r y x - G ra p h fo r th c c u r v e s w e - fo r y 2 x = - x 2 is p lo , G S k e tc h th e A y o w {y } a E X A M P L E O N SO LU T ı kn s w e cu rv e s in kn o w cu rve th e c u r v e {y } y t te cl a s ra p h fo r «y } = x 2 s in x = - s in x , is p lo tt e d in F ig 2 1 6 0 a s sh o w n an d 2 16 1 x 图 F ig 2 1 6 0 N o w to s k e tc h {y } = s in x w e r e ta in th e g r a p h fo r o ś y < 1 an d n e fo r o thđ g le c t th e g r a p h v a lu e s G ra p h fo r {y » = s ln x ne 翻 g le c te d F ig 2 1 6 1 Scanned by CamScanner (x x lll) y A 1 (X ) t r a n s fo r m - s w e hav e e a r lie r d {y } - U (x )J is c u s s e d y «f (X )) (/ f s to = , tr ij n R fi» r m n t io n G r łıp h fu r (X X ) w , h ic h * ho w # íf (X )) y rx ) y 乔 园 四 捌 佷 碧 F ig G r a p h fo Thu s, «y » h r a g p re m r {x } = a in s s a m 2 162 Fro m a Fı g 2 16 3 b o v e fig u re w e c a n se e y - {x } a tta in s a ll v a lu e s b e tw een [0 1 ) , e X F ig 2 1 6 4 iliE ijÞ S k e tc ìıth e . Ņ O LU T IO N As w e kn o w cu R vE th e c u r v e {y } y - = {c o s x } {c o s x } is p lo tte d a s sh o w n in F ig 2 1 6 5 X L / : : Ŝ Ś Scanned by CamScanner : : T hu s, V0 Hl Į t\r e ; Y g r a p h Ñ } r tv } = (m - (e o ı e # b e lw }ı s ,ť R ť co n A lo w lt h H ı ı ı Ñ V = {c o * x } a n d ( y) (.... , ' } s f(x ) t r a n s fo - cha d is c u s s e d in rm s to y p te r 1 ' - y f (x ) ' f ( ť ) is th e - m ir r o r im ıg e o ł ly - f (x ) a b o u t y - x , O R ˜ " ra w " " 2 16 6 (x x iv ) y G , } igj į 蓁 0 ı ), ı In te r c h a n g e ı y ït c o p h lc a ı x an dy ld b e u a x is w s t a te d h e n fu n c t io n ls b ue c tlv e " a s 一 走 ふ 园 一 中 二 十 一 f (x ); w h e n f (x ) > g (x ); w h e n g (x g (x ) ) > f (x ) 如月甲 中 th e 4 gra p h □h( X ) o f f (x ) rı J in : irn f( : , u rn w (f (x ) en its g r a h is lo w er an d o th e r w is e ske t g (x ) , : Scanned by CamScanner . gra ph th e draw To lI nf Jrim « * ł 4z n * o f l y - {f (x )r ¢(x )» im a x o r h th e fu n c u im f r x ) a n d z !Å Ĵ M d ]e k p o in tã o í in tle r a e a io n s ílrs t d r a w th e g r a p s í rf h Tn b e t w e e n th e s e p o im s e it h e r fin d a n y tw o s e c t łt l v e Ħ»ln tš { í l n t # r te rï w e Tł h I o r d tii ın * x {f (x ), g (x )» w e t a lte t h a G e a e g n ıe a a s o f f (x ) " r l tx ) < g (x ), t e n , n T ) x j( W e ı . ą a fi y s lm lı f(# )< * b e tw g {×】 IXAMPLE tiu h « e jn (í (x )i g (x )}, w e t a 】 ĺe a n y tw o c m t J v e p o ln æ o f l ó e < X i J O n o r d e r to In een O m P lr » t p lo t b o th y b y a d o tte d u r v e a s s o LU T IO N y co s x - lio m th c a n d th e n w a p li r S k e tc li tlie w s ln x a n - ean n th e In te r v a l g ra p h ı th o s e d a rk e n ďy be 7【 , - m a K (s jn J¢W } , f (x ) d ( } n ŕ x a x ( ) ĵ ı f g «e y g m x e n ts rJ n f fo r w h ic h j d n ge ) Þ d o tte d lin e s fo r X hic h 八幻 》 g (x ) o r g (x ) 》 f (x ) pro m ad a ce n j in ters e c tio n s G ra p h o a re fm U g u re t A a x , C J¢ , th e p c d ııt o f . (a lł】x F lg 2 1 6 8 eo # x } " \ T F h; x Scanned by CamScanner " , 。 . . 一 コ 園 園 罔 F ro G Ħ m ra abo ve ph o f fıg u m in re w «t a e o ıı x » b t a in c o t x th e g ra p h o fmi n«ta n tx co } » ťH W - 四■■ t $ 2 2 F ig 2 1 7 1 (E X A M P L E S ke t c h tiw p ir s t p lo t e a ır v e y - m in { l× l lx , 1 1 lx , + 1 1} g r êıp h F ig 2 1 7 2 : J Ī Scanned by CamScanner " i A s fr o Fro m a m th e a bo v e bo fig u T o s ke t c h th e v e c u rv e g r a p h fo r y - m in ( lx re cu rv e b e tw e e n tw o p a ra lle l lin es y - x + tan dy - x 1 1i ' (c a lle d b o u n d e d m Scanned by CamScanner 習 U \ 田 陬 ㈹ 铺 ㈹ 缽 日 。 、 胪 》ct f ig lW . 2 6 H E 阔 歹(幻, g (幻 T e i s E « ıd t a p w o a c h b u t w e c E m s e L q ]m in in m ïn a r E n a m S 【æļ p L X A ııP L E e に s ou m H e 阵 w 一 d g i «ĵ c ) Ś h S lie d ıd ı e a m (x ) ł = y 号 fan y on e o f th e m sa y a ś g an d m a x im u m o fa n y (x ) ś b (X ) Ś b f (X ) O - f (x ) 君(x ) = h (x ) th e ft ı llo w in g s t e p s if m in im u m m a r in n m ıo a C h tpr Tt S le p & 西 r n ï ı\ı se = \ 2 17 4 h (x ) = 0 x s in = 岫 ド n i oı « 1ś a eĻ x ◆ ー 一 a tū c ī J s n i x = ' s in x xsin ' ś l G) ś y ś x i o 。 一 。 一 辩 座 三歹= O W 南e n X 2冯 = 凤 鸣风 艮 二 盐 ; ii Scanned by CamScanner y D ra w x - w th e g r a p h o hen , f th e fu n c tio n y W - X o N uTı A s w e kn 1 ś o w at x - ś 1 s in x 4 y - tie s b e 1 tw e e n 1 to X t o s k e tc h t h e c u r v e w H e re s in x ą y is n o t d e fin e d bu X 1 h e re 1 (i) ś y ś t a s 今 O s in x 콰 ダ ° 今 1 (ū ) X Scanned by CamScanner Et X A M 卜しE S k e tc h t h e (iio y S C) L U T ı O N (I) A s w e - kn c u r v es o l[× ] + J m , fo r a n y x E i þw he re [ ·] [0 , 1 ), w [x 】< 1 x hav e , s. Jn Fr o m a bo v e w e hav - 一 y 一 , 分 w h w hen - , y ' th e c u r v e fo r y - Jx x < o 0 5 x < 1 18 en 2 ś 2 x < x JF i , hen O ś x < 1 w hen 1 ś x < 2 - w , 3 < [x ] is p e r io d ic w e a s s h iftin sa m , hen (ıe は 四目 ˜ 1 ś w (ıe 台 长 R x hen w , 一 一 ll r a A n d so o n p lo t e to y 一 fo , U T hu g r e a te s t in te g e r jiın c h on ś x ś 2 J Ħ o w r e p r es e n t e x N in g th a t o w O ś A ls o lo w f th e To ı 2 5 x < t o w a r d s r ig h t b y 1 u n it g t o w a r d s r ig h t 3 s a m e a s s h ift in , g ith p e r io d ' 1 ' S h o w n a s ー by 2 in F ig 2 1 7 8 レ 一 匡 理 露 X F ig 2 1 7 8 . l + y į i \ n: . , g + 2 + lx i , . hen w , w , r l Î J i w w . 1ś hen O ś hen l 5 he n 2 ś 1 x < x o x < < x < 2 3 An d so o n : Scanned by CamScanner u n ir Thu th e s, » 0 g ra p h fo r Y - [× J + . m / is o b t a in e d b y [x } < a o r th e g ra p h o f y = Jx a cco [x ] b y rd in g as r 一 薯 兹 ° 1 ー 卫 「 目 ゴ 一 曲 ー 日 日 ˜ 鼎 , 학 隅 了 F ig 2 1 7 9 Th w th e c tır v e Fr o m a bo w y - fx l + ì m is o Th u s Th e . fy - - e co u ld d is c u · × J r y [× J + - 1[ 】 x + i is d d iffe r e n d a b le fo o b ta in e d b y a b o u t - Jm s s th a t c o n t in u o u s a n l[ J J i l [x ] + g r a ph fo y r rv e w c tı T h e g ra p h fo r y gw h fo an re r a ll x fle c tin g th e p o r tio n ly in g b e lo w a x i s o f th e d k e e p in g th e p o r tio n ly in g a b o v e a x i s (a s it is ) di l i I F ig 2 1 8 0 Scanned by CamScanner i:H 丁 一 瓤 サ S k e tc h A M p L E H S O L U T IO N (b ) W m th e c u r v e y - (1 x 2/3 3/ 2 ) e re n en a bo v e ー X F ig 2 1 8 1 园 開 画 國 脳 X } 。 园 關 斟 蟹 瑙 画 Scanned by CamScanner (e ) - rr u ON SO LU T Ł A s w e kn w o w th e g r a th e ph Jo r g r a p h fo r y - x 1 F ig 2 1 8 5 . . Scanned by CamScanner F lg 2 1 8 7 , Fı g 2 18 6 (e ) X 一 一 一 ー ド が F ig 2 1 8 8 ー 羽 . į f · 飞 ' ? . -+ F lg 2 1 8 9 . . Scanned by CamScanner (b ) 夕 上 一 X 。 。 0 1 ĺr F ig 1 今 = ii Þ x 1 X y 。 1 = 1 2 19 1 ブ ー 2 1 X F ig 2 19 3 Scanned by CamScanner łA & ĻŤ Ť T : s k e t c h th e EX A M P L E Į!jı . (ı) y e = Tp " a e ' t 0 N LU T ı (ı ) H e re y li t e - y d is c u (i) (ii) ł - ss; w hen w he n (iii) f ( O = y ) x y x y is In 0 - y 4 e - O 2x 1 0 d a ls o : 2 dx e x . J 2 O < e X e - 0 = in g fo r a X ; ' X k o th r o u ; x e x a s s e s 4 x = gh o r lg ln ) O ll x = : = » f( ) dx y lx ï 丁 x = c re a s > . (It p (v ) : lı ' e T o J ı' s T t tl h w w e n h x e n is p lo tte d 0 > x < O 2 e · c o n c a v e c o n ca v e a s sh o w n u p d a n o w d in n c r e a s in an d in g c r e a s in g fr o m a bov e d is c u s sio n in F ig 2 1 9 6 : Scanned by CamScanner o Tb us fro F İg 2 1 9 7 m y O a y く e 0 / 2 » * 间 POtdng o fy = Ff 여 イ1 ¢ 2 197 a e y e l+ - ;x ž O e X X > F ig F ro m a F ig 2 1 9 9 2 198 b o v e Į íg 2 1 9 8 ア ー e 1 Jv 4 tx l + 11 is s a m y = 1 + Pł m ng o f 11 + e ly l 11 + e y - = '. ı þ4 x e e X 1is e 》 y - + e e sho w n '× l e l× l e l + e 'x ï x 0 X x < - r X - 1 y G m p h fo e · - Th u s , th e g T a p h fo r , e as e " 0 1 x in P fg 2 1 9 9 l Scanned by CamScanner " #\]ı¢) r h p ť l fr íł ! łv ra ł n \ n lı (0 , 0 ) a * (ł I l l cm m łe , t¢rw tt d p (0 , 0 ) ¢ h u li ı ı n t «¢ y l»a ı » ï ty lıı »ı ttrtle rm rT ° , ñg u t a h # tle P R r l lđpIßeents }łła ın a lv # n lh ¢ a r e tłb s u n d e d b tt tw rT ĺ n rT r IU UUU!!HŅ !!!UlljļU # re R , İe , û ñ n lw n c u r v ¢ ' " y " , Fı g 2 20 ] , £X A M P L t ĝ Ĵ S k e [c ì\ tire u rv e o \e lb l[o w f tł ln g + ıO L U T ıO N (n H e re to e to ¢ ıı kn o w w e y iz 1 x + y g 1 ły 2 l> x + 1 j = x + y l n r an ły x l lx h, d 1 1 y < x , ll d r a w th e g ra p h fo F J now w e x ły - 1 n n d x ły " l « o n # ld e r e n y fix e d p o in t e a y tı sh¢ (0 0 ) n m l c h e e k x ły 2 1 a n d j ho ı dl e F n o E x + y f P lr e t w e ¢lm , l , . A#: x «h a d e d I , y ? p a r t l# a g a in a $ l 0 2 Į (fa lße ) 1 x + , a w a y fr o m y = O£ In o rl吕 1 (fe l» e ) 1 ä )m d e d p a r t l* a w a y fro m # h cıw n a » 1n P lg 2 2 0 2 , ı t* ı (ı ) T o ıı oh ltr e kno w - , , F 18 2 2 0 2 口「1吕ln , , , y l8 1 Ix 5 1 yı - ] - x s - y $ I - Scanned by CamScanner ' r e s h a d e d p a r ts is t o w a r d s o r ig in a s ï n fig u i d 0 n t f 0 i o s i a e x ( n ) x y p ; an y ł y lś 1 , u tth ıg p 10 叻 0 1ś X 1 lx ) To (llı s lx l+ ly lś k e tc h ly I H e re = I× Iis 2 F ig 2 p lo t te d y lś 1 2 203 a s l O i) (I) F ig Fr o m a b o v e gr a p h ly lś as a t (O O ) (iv ) S ke t c hı n T o s ke t ch g g to w o I f Iï 2 f ïx 2 = I× l , e ca n c h e c k sh a l× l+ lJ lś o r d in g o f 2 X sh o w n a r d s o r ig in 1 1y 1 I× l ly J l× I w (tr u e ) 0 5 2 , s h a d in o 2 204 1 as in F ig 2 2 0 5 1 w e p ro cee d as F ig 2 2 0 5 . Scanned by CamScanner (lv ) F ig 2 2 0 6 R P SO S k e tc h th e LE U T IO N (i) A s w e kn c u rv e ow H+ 2 l2 × 5 4 \× p ly l 上 - 一 Þ E ı ı ı ı F L1ĺ X y ï5 4 x x + Y l+ l ı ■ 收 け F ig 2 2 0 7 T hm , to s h a d e th e p o r tio n ū) T hu s , to sh a de th e w p o r tio n hen w hen 2 Ś x x ś Ś 2 2 an or d x Z S ż 2 y an į á - d y ś ııU F ig 2 2 0 7 V ' 2 o r y 2 show ia n as F ig 2 2 0 8 琊 园 小 囲 矓馥 釀鑾藏 F \g 2 2 0 8 Scanned by CamScanner tc fı t h e curvejorı (io So L U T IO N (1) H e re 2 l' 'lJ 5 1; l+ 2ll × ' × Ľ Iś 2 ś 1 a n d ı j ı ś 爿 - 鳥 T h u s , to p lo t lJ l+ ' 2 月 2 ś 2 lx l , w e p ro ce e d コ 勇 a s 飾 e 汽 凋 Sho w (iv ) n a s; 售 3 怎 뇌 准 0 F ig > 洹 x 涛 2 209 一 倡 n e ect gı X 诩 卅 1 /2 Scanned by CamScanner 瀛 自白、 IĄc rc fig u , 1 0 Ś 1 (11) P lo trin g o 0 Ś o r ×ı 2ı Iy f 2 l× l 1+ 1 く二 lx ı an b o v e fig u a 2 l×'IJ · 2l 1 + ı ' 2 tr u e , t h e r e fo r e , sha O, y - d e d to w sn o w n in as F ig 2 2 1 1 lo g f th e Tu n c tio n f (x ) lo g 2 (1 o (1 2 2 x do m a s x x w 2 hen ) , l < x < 1 ( 1 1) an , 1 = ll X - 〉土 - ) e x is ts 〉 d ifr e r e n t ia d n g y 分 o o ge t w e U s in fr o m a r d s o r ig in 2 = I < Ś 1 he gra ph y IJ I 2 ò x e r c rı t : O In 1 F ig 2 2 1 1 a ls o - ś 1 1 5 ly ı · ' h ic h is x re 1 × I5 ı a n o w lJ Iś d 2 fr o m 1 p u tt in g to w a r d s o r ig in a s (iv ) is s h a d e d ı E g b e r lin n u m , dx - 2 a bo v e y = 2 e (1 lo g (1 - e r u le X 2 ) is (1 + x 2 2 x 2 ) < j 0 fo r a , co n ca v e do w n fo rx ( 1 1 )} , r e s u lts w e c an ]o g (1 2 X 2 ) ( 1 1) (iv ) d ra w a s sho w n in F ig 2 2 1 2 II I ' l! « ı 0 g 2 \ı A Fı g 2 2 12 , 叫 罗 公 园 Scanned by CamScanner ' :ëji-P.E ? so H N L U T IO . = lo g (s in R ) , s in x ) is d e fin e d x is p e r io d ic , w hen e n t ir e n u m be a s (币) r lin X a s s in x > ith p e r io d 2 w ) x fo r y e re Io g (s in - S in c e , (ii) s k e tc /ıt /ıc g r u p /ıo í O (2 n X rrł (2 ıı + 1 )rr ) n hen w ) = (n x n )as e x is ts w he (o x n e - 〉 O 〉 = y O y lo g (0 ) = - s i n x , n ) a n d th e n p lo tte d _ n 1 X = 2 (iv ) d x CO S X . = co tx s in x c gy > dx 2 回 (v ) d y co se c . d x < 0 fo r a . ll x Q, hen w w hen (Q rr ) in g a n d c o n c a v e d o w ıı d e c r e a s in g a n d c o n c a v e d o w n hen l x (2 n n p lo t th e c u r v e o n y w in T hu s , an 2 g o to x , (O , rr / 2 ) (n / 2 , n x x , Ŝ å X cre a s O, e (2 n + 1 ) 7c ) S h o w n as in F ig 2 2 1 3 Þ X 2n '" w y S k e tc h t h e g r a p ìıTo lo g = 50t llTı o °u N (i) H e re , y - lo g s ilıx e i - s , s in x > O s in x a n e is d e fin e d w hen , s in x d s in x # 1 i Scanned by CamScanner 1 y is p c r io d lc w ith p e r io d 2 n - lo g to , s in x d is c u g r a p h fo s s th e dy (q x E r 十 (lo g a x s in x {n / 2 } シ ー 打 心 dx . 2 an ) > O w he n x < o w he n x e 2 ( 2 co se c ) x { (lo g s in > fo 0 r a ll x 2 ) co se c (0 X in 2 (lo g s in x tc ) Į , c r e a s in 1 s, lo g p s in x g r B k e tc h fť »r tł \e CO T X 냐 ヒ 2 (ı og s ln x e n tı re ı \u rn lw ,ı l nc ) c īlt A )4 s ln x )c o t z ť } 2 ) d an d e c r e a s in g 1' l 2 (lo g + x co tJ s in d 凇 嘟 (lo g . y s in x . CO S 黑 s in x r o r T hu ) 1 = dx ı c , an d c o n c a v e: u co n cav e u p p AJ G x e ĹĮ , o, " Į) ľ F lg 2 2 14 , , liı.P i e , S k e tc h th e c u r v e (ï U ] = co s x x E (iı) b r ] = [c o s x ]; x [ 2 [ 1【, 2 2 n , n ] 2 ] 7c w h e re [ j d e n o te s greatesmtegerjhn(ti''" X Scanned by CamScanner [y ] he n w Thu is p o co s x = 2 . İ x s s ib le o n İ İ , ly Į 7c, į 〈 O , s, CO S X = CO S X = x co s T h u s, 1 = 1 ś y ウ 0 O ś y < 1 1 ś y < [y 】 O 2 p lo t t e d is c o s x = a s U 2 2 16 F lg s b o w ıı In ニ F ig 2 2 16 令 n g S k e tc h ı o f [y F ir s t to p lo t y ] = [c o ] s x ic o s x ] S h o w - n as r 二 ダ 2 0 F ig 2 2 1 7 F ro m a bo v e fig u y re - [c o s x ] 3 y 1h u s w , hen = < l× o ; o 1 l× I < = y - O y - 1 y = 7 Iś Å Q, 2 1c 2 an d 3 T 1【 ś I× l< 2 1c 1 İ Scanned by CamScanner g T h u s, 卧l ; ËX A M P L E S k e tc h th e ra ı fo pı r [y ] [c o = o x ] c u r v es 嗍 四 叫 ON SO LU T ı (1) y - 4 kn o w [x ] c o u ld b e p lo tte d As w e w ith in tw o s t e p s 。 ブー 4 ˜ x 〉y - 4 [x ] 段 逐 (ln æy X H te Scanned by CamScanner G ra p h r [ı] !n = 4 [x 】 līr n de d 」 国 ユ ユ 김 一一- 3 2 220 F ig In (U ) T o e a bo v e c u r v s k e tc h lo w e r [ ly IJ 図 = 4 = bo u n d a ry in c lu d e d a re a n d u a r e e x c lu ppe r de d [ i× Il 4 [x ] 今 け] 口x l】 4 = 今 ] 디ア 」 = 4 디x l] X F ig 2 2 2 1 \ . , d in a te s x (ın s h a d e t lie r e g io n w h o s e c o o r i CO S X uTı o N H e re . co s x co sy . o co sy > dy e u a tic s a tis fy th e q O it te n a s , c a n be w r 2 s in Cy ) Jc s in Scanned by CamScanner 昀 o r 囀 H e re th e e q u a tio n , d iffe r e n t v a lu e s o fk k Say ie , th e s e t o f p o in o w ho 0, 1 = y > , th e s e t o e ra f p o in ıp o in ts ts w an x - f th e s e c o o r d in a t e s s a t is y O S im ila r ly , fo r g e n N ts w = h ic h 2 k lc s a tis fy < s in d in e < x + y « 2 rc x + y < (2 k + ( ) . y - qu a X + 2Æ lity Įiļ 1 )n o X Scanned by CamScanner (? h ic h 2 :彦 t tıo s e ķ ı o ı p o in ts o f T \ť ľ Fıo IXAı *P i s o L mc + w E q (ii) (Zp - a ıa b o v ı ı )X n x sa e nestłl g r a p h fo L E S k e tc h d le A ON s to s k e tc x h ) R< r c o s x > ' c o s 3 ľ cu m e . , - y is + Y (2p ) r < 2 s h o \m (ii) R (ū i) in F ig 2 2 2 2 3 z 3 A . " . . 1 1 + « 2 { 2 n + ı) 8 z ' v (2 n < 2 ?ınmetric a b o u t y (a s w h e n x is r e l d b un ro p ace y ( x ) c u rv e re m i T m a e tr ic a b o u t x a x is s s e j un c a s h ( w Đ e i n 6 y s r e p la c e d b y ( ľ ) c tır v e r e J n c u r v e fs sĮ etricaboutorigin (a s w h e n x is D r e p la c e d b y Y a n d ľ b y x c u r v e @ x O > į a ľ (iï ) W n e n is pe = , (0 Y 2 73 d iã e ï e i ıt ia t in u s in gr a p h (W F n o n ıa bo g b o th g E qs ĺ s id e y _ 2V s. - (a z 3 (a 2 /3 3 (a z e ) a in s s a m e ) re m a in s s a n ıe ) = x X ' x Z 3 ı Z 3 ) z 3 3 Ŝ 2 ) . ' " (i) (u ) a n d (iii)] . yd ve a - 2 v or _ = a in s s a m g a in . w e _ x Ļ 3 (a 2 73 x z f3 ) 2 ı 9et 2 73 2 ) Scanned by CamScanner s in r h e r e t h e q u a n t it ie s x a n d y a it is p e r io d ic w it h p e r io d 2 n re E lim in a t in g co s t n n d w , Thu su fr o m d e fin bo e d fo r ł ıll v a t to d is c u ffic ie n s D o m R a n v e e a in ge o qu f fu o o w n th e b a s is S t u d e n ts p a ra m , e s o w e ge t fr B fo 3a s in y th e 2 73 fu [0 2 t , _ Q 2 13 3 n c t io n s c o s r a n n ] d s it ra e tr ic o a re if p o fabov e in fo x = a co s y - b s in a 3 3 d v ic e d a t io n rm w + [ ž ( aï e . n c t io n 2 r , + 10 2 ï c) n c t io n c o s 2 /3 u t s in c e s s th e c u r v e 3a x t s in a ] tij t (İf】 , = O lu f fu = N a tio n s a t co st s k e tc h e can t t to c o n v e rt th e c a r te s ia n in to s s ib le F ig 2 2 2 4 į E Á Ńip ĹE s k e tc h th e . T aU ) . 0) S k e tc h ı n g o cı ır v e s f f (x e ) = y = . (- ) = f (x) , e ve n or sy m m e t r i c ab ou t y- xi s Scanned by CamScanner ( y 2 ; 0 C < C 1 0 Ð 3 th æ x ı âR ? e + {U \ an @ ) (iii) . . an d m l] g r a p ł ıo f F ig 2 2 2 5 IS - 1 . S B e ıc l* ıg y e x » = e - 2 n ガ ĝ d2y 矿' dkz e x '" 2 F ig 2 2 2 6 fıo ın E s q m . (í i) (iii) a n d (iv ) g r a ph , o fy - Scanned by CamScanner ļ id ı lc fu a nc tio n s " p r e s e n te d b y re . co c a ll w th x eF + = ť i ? i lc c o ta n g e n t w h e lr e if d l p e rb o ı ic ta n g e n t a n d h y p e r b o ı ' h ic h ı ı e q ila tio n o f h w rbo ı a dr e ı n hx ' a n d c o s hx ' * l d - 下了 - · ◆ ) (iı ť x - co s h t, y . s in ht ı T a re ic fu h y p e rb o ı n c tio n s , Scanned by CamScanner S k e tc h t lıe c ım P L E iĖi[A M dLuTı oN h : 甲t i in . ^ s in ce a 2 co SX 2 co s x 兀 5 17 3n 6 2 . dx H e ie 冗 O X 功 2 s in 2 x (1 ż d 2 s in 2 = (b) 2 dK " = " at x is = m 51 an d w e = y - X 2 (c o s x n it is , 2 s in su tftc ie n t to in v e s tig a te th e co s x x ) ) hav e co s 2x h e re = 2 d ]c > - 0 at x h e re y w ) y 2 = 3 一 , 4 w 6 a t x u m s in x - 冗 in im m . 冗 a t x u m 2 X at im a x it h p e r io d 2 3 = x 2x co s .. J - at w + = (h ' x , dx dy · 2 s in - fu n c tio n is p e r io d ic l [0 2 n ] . (a ) d 2y y th e , fn te r v t c c ' 4 = 1 ダ 0 2 X : (m a x im ) u m o IË 5 2 1子 3 at x an d = Jf Th u s , w e ,! c u rv e fo r ļ ĹĹE SO LU T ı ON e - y - 2 s in s k e tc h A (a) Syn u n e tr ic bo u ty 0 > 2 ) u m X CO S + a x is y - ' W 2 y f (x ) = {a s f ( as x as y L/ y - 2 s in x + c o s 21 I F ig 2 2 2 9 )c th e c u r v e s th e c u r v e a 6 , (m in im 3 - hav x x e ) = ss ia n c u r v e ) (G a u 2 f (x )} O = 〉 》O = 〉x 今 - 2 x e - y - - 〉 1 〉士 図 (iii) Scanned by CamScanner 1) dr ' (iv j (e ) y ì e = 2 a s s u m e s m a x im X Ş į an D d īn a bo = a in o ı ıı Ra F r, o O = n y - u m a t 1 R (v ) (0 1 】 ge , 团团国ı ve ulul F ig 2 2 3 0 E X A M P L E S O L U T IO N (b ) f ( M om x @ H )= a bo v e co n S k e tc h th e f (x ) h e n c e f (x ) is , y 1 ı. (a ) e re X cu rve , d itio n s g r a p h fo r o y + x 2 m e t r ic a b o u t o r ig in d d , sym (i) O " . (ü) x - 1 + x z Scanned by CamScanner 1+ 困 c o r ïc a v e d o w n r c o n c a ve u p cave F ig S k e tc ï t th e AM p L E S O L U T IO N (i) s y m :m T o p lo m e tr ic a a ge a bo u cu rv e y 2 t th e c u r v e w e d is c u (a s y + bo x ? u t» a x ís is fo r y - x . do lııc w n o n ca ve u p 2 233 3 (S e m ic u b ic a l p a r a b o la ) ss c u rv e re m y (Ĺ e , to d is c u s s a in s s a m c u rv e e ) w h e n y O) 旧 一 殛 じ じ 国 ー 阳 囲 一 一 邕 田 打 Scanned by CamScanner蓄 . . Ţ ĺ tl) S N \n tm N w m ș ı iĄi" tï ie a h i\\\ıN m td y a x lt , T' ł īl (b ) W h e n v 9 , i o y 醒 目 、 圈 圈 F r o tn ab o v e d ls c u s s lo n , : x 国 國 甲 Fg . 2 2 35 . 座 臨 曝 Scanned by CamScanner 1 S k e tc h th e c u rvre 吕 (w hte t in te g e r fu n c llo n ) g re a te s re [ ] de n o te a th e (v i) y t 2 = ı " (v iï) y lo 9 2 (la in x l) (v iii) Iy l lo g z ls in x ı 1 11 = 4 = 4 (iin ly l 2 = ĺr r j ly 1 lo g . = ) " 3 S k e tc h th e c u rv e y - 4 S k e tc h th e c u rv e y - (v i) y - 1 2 ' I (v ii) ly j le x ] (v iii) y x 【 × 2 = S k e tc h th e 5 1 (i) y (iii) - y = - (v ) y (x ) y 2 fx j) (x - 6 - C o n s tr u c = f×D (x ii) ly l (x = w he 2 t th e g ra p h S k e tc h th e 7 x + - (x ix ) Iy ı x = + [x ] c u rv e s (i) y J s ín x (ii) ly l J s in x (iii) y - ls in x l+ lc o s x l (iv ) Iy 1 c o s x + Ic o s x I (v ) y s in z x 2 s in x lx l , O, w he n w he n 1 x f th e fu n c tio n l× 15 l× l> 1 1 x ı 2 ı 2 8 F in d 9 num n x F in d th e r o [0 4 rr ] , num u tio n s so ı f be be r o 10 S k e tc h th e o f s o lu tio in s s in rr x - = 11 = o = f×] W he Ł S ke tc h th e x c u rv e s X (x v ii) y ) 2 (v i) y x f(x re Ą ×l = m (x i) ly l e - (iv ) ı y l 3 x 1 + (ii) y × l ı 2 '× l e 2 c u rv e s 2 x ĺir ) 2x x × le ' l 1 s in - f 2 o co sx = ls in x ļ f × ll Ito g ı c u rv e s in y = 2x 十 COS X 2 = = A N SW ERS Scanned by CamScanner N N o A b «o ı u ı e m a x im u m lu e o ï la n y w h e re ı e r vł g re « ı A lD o a lo c a l m a x im u m o H X . No i-iix .' " L o c a ıh ı n lm a lı e rvt ı u e o t fn e a r b y em a II N o 6, . ' a C e L o c a lv l a d A bso ı u te 一 ー b b a o lu te (G lo b a ı ) E x tr e m x a id e a m a x im u m い f ' u n d e fin e d ー ー ト ト ー; 国 国 平 ~ 一 Lo 茶 麺 强徽 W oa ıv ı a b ıo ı u te (G lo b a l) E x tre m a Fı 8 2 . m ı rf' c h ł ı n * e a fro m p o ņ lw e to n e i w tve at C ? ' > O fo rx < C an d i ' < O fo r x > C ), th e n j Scanned by CamScanner ha s Scanned by CamScanner į iı S Y M P Ţ T ĘS I S Į B [ G Y Ļ R P 0 Į B !Ţ S B ÉĐ . 圈里參二 0 玎景甘皿 で聚焦口壩 日 ı 君 ı n th ı * c h a p te r w e s h a ı ıs t u d y Pı o ttin g o f R a tio n a ı a n d ı r r a tio n a ı fu n đ io n s ı n te r s e c tio n o f C u r v d e an S t r a ig h t lin e a t ı n fin ity b I ıj th e te n p e r p e n d ic u la r d is t a n t o in fın it y ds A th e t o gh h i . s tr a i Æ n e t s s tr a A is ce o f th e p o in c a lle d ig h t lin e a n a s ym ıI A SY M P T O T E S t p o n th e c u r v e p to t e t o a cu rv e a pp ro a ch e s z e ro a s th e fr o , th e m if th e p o in t M lin e te n d is t a n te n ds ce to ds ô fr o to z e r o w m hen x or y bo th th e v a r ia b le p o in t M o f infinity sho w n as °t Le t y T an = f (x ) b e a c u rv e an d le t (c y ) b e , g e t a t (x , y ) is g iv e n b y a p o in t o n it . 艷 Scanned by CamScanner 月 - i 4ľ p to t if a w iöw , ł h e e x is t s , t e n x ' 夕 dy 9y o re d u c e s t We h ll d i 0 sc u ss s a 1 + dx , Y x m = fin ite lim it ĥ j L ? (i) - nī + th e fo llo w in B c an d y is a sy m p to te i w i Let the e q u a t o n (a ox it ' be a iT a n N o w , ifQ o k ıln ia e Ē tw o r o o ts o H en ce , a iY + N a in if b o th th e t e r m , n d x øo w ñim a k e th r e e r o o ts o k alle t to x a x a sy m p to te s a sy m p to te + + a . c o n s is t in y " ) + (b l x e rs o g x " fx is a 1 " + b2 x n 1 a re a bse n b se n t , th e n a 2 y f E q (ii) infinite h e n ce 2 " + y b .y + ' " ) (i) fo llo w a s s: t , th e n a ï r f E q (i) in fin ite a s c o e ffic ie n ts o f b o th O is a n a s y m p t o t e p a r a lle l t o x an c f e q u a tio n o = x d , y Q 2x an x a x is be rv e m c a se s g e d in d e s c e n d in g p o w Q, i e - le l to ı fcu o y + + a lx ca n ra y c dx te p a r a lle l t o x a x is A sym p to a r a lle l to y a x is m p to te p (ü) A sy te o f a lg e b r a ic c u r v e s o r o b li u e s m p to q (iii) A y e b y in s p e c t io n t o t ı n p (iv) A sy s e c tio n o f c u r v e a n d it s A s y m p t o te s (li) In te r m to te b y E x p a n s io n (b A sy p o s it io n o f th e c u r v e w ith r e s p e c t t o (Ýū) T he p lsl ţn A s ym p to te p a sa , 2 a z b2y + y 2 + " x an a x d o, i e - " x 1 , co e ffic ie n to l f xı . o a re z e ro is + c2 bz y b1 + + - o, i e c2 - o , w co e ffic ie n t o f x ill g iv e n 2 b e in g t w o a s y ın p t o t e is ı letho d to fin d a s y m p t o t e p a r a lle l t o x a x is To fin d th e a s m t o t e a r a lle l t o x a x is e u a te th e c o e ffic ie n t o f h ig h e s t p o w e r o f x to q y p p Ifthe c o e ffic i t o te p a r a lle l to x a x is (h o r iz o n t a l) e n t is c o n s t a n t , th e n th e r e is n o a s y m p IÐAM E' (n J A s y m From p to te p a lı e l to y ra a b o v e a r tic le ?ow er o f y to z e r o Ifthis c o e ffi , if w a x is e n ee d a sy m an llsoluT ıo N M H e re p to t e p ; a ra cu rve is n o a sy m a x is , e q u a te th e ll l to y p t o te p a r a e a x is co e ffic ie n t o f h ig h e s t (v e r tic a l) y 5) = 1 y (x lle l t o lle l to y p to te p a r a th e r e c ie n t is c o n s t a n t , th e n S k e tc h th e z e ro x a x t o te p a r a lle l t o y a x is is y = o x - 5 Scanned by CamScanner ' Th u s x , = S a n dy a x le a r e a e y m ' p to te s sho w n a8 in fig u re 4 A 团团阳 O = a y m p to to 虿 ソニ正厂 5 Fı g 3 2 £E he ' " ' " ' " y y re 〉土 y ^. ta n " - = ' A s ı' e , (a s qu y Sho w n a t in - g h ig h e s t p o w ta n x = » y er o co t x = fy 1, In jï n ite n u m ber o f v e r t ic Q l (w )m p to te ta n x as 一 . . " ' " 冗 들 , å , 土 土 咢 。 . . - h e re w O) co t x a s Þ X 一 。 。 己 圧 人 、 く i E T Ť ł - ı oN S h o w th e H cu rve y e re e - y y = '' ' e has . . .. . .. . . . . . Ta te ıa s ym p to 1/ x 二 O e 。, , 手 O as Ťģ¥ Scanned by CamScanner a in g Ħg u ąjo in re y v x = e = lo g y 1 X ' . . 1 X = lo g y à I■ c h x S n w (lo g y ) = 1 )letho d to fin d o b H q u e a s y m p t o t e Su p p o s e y m an ge m x + - it in d e s c e Sohne th e s e tw N o ta o e qu 1 H e re , 2 n is a n a sy m d in g p o a t io n s a e , w e rs fın d m ill fin d , F in d th e n o n a sym or (I) ı p to te o f th e c u r v e P u t y - m x + c in th e e q u a tio n o f th e c tırv e ł\n d o f x E q u a te to z e r o th e c o e ffic ie n ts o f t w o h ig h e s t d e 8 re e to n n s ıs y m p to te s a n d c P u t th e m In y - n t x + c to g e t ł le ıo r p a ra ı ues ofm n a ry v a ı lıim a g ı w e w N e g le c t W+ c x n p t o te p Eq u a ti n a r 封山区 t o g h ig h e s t p o x p to te s 2 " a x w er co e m n o n re th e c u r v e y e st pow er co e . . - p to te : a ' is ffic ie n t o fx ffic ie n t o on ı y " e tc h "" ţį I to z e r o s fy in o to z e f x Jg r + 1 = 0 (w h ic h ls 0 parx Uatingh ig h a sym o " l () p e a te d n o t tr u e ' +ţĺ in Scanned by CamScanner J「 N X 手上 t o tr a D e t h e c lır v e o w m 二 S ]m m e tr ic a {势 D o m a in e fv l) R a n g e dr bo R (a s o d d fu n c t io n ) {아 = x u t o r ig in 2 1u [2 , , ' - - ) {tls in g m ı n ıb e r lin 中 e r u le 手 ° - dx > q w d Jc whidr sh o w hen w s Ym ym he n ax ln 1 at X at X 2 dx 2 O, > 1 x < or X < X > 1 1 {0 } 1 1 = x w < . , = 1 3 he n (C O LIle a O x > c a \ų e c o iı y ļ ; . dx 2 < 0 , w he n X < 0 ( ) ve up do\m ) Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner r \ 」 、 一 ' 2 兹吧 , + )¢ f (x ) an ㈹ = d ť ' f (x ) 1 = f (x ) ' Ć į ! d A rı . ' lim X 一 吕h o w 一 a s ' f (x ) lirıı 2 X i yae 产(x ) = 切 an d C 厂汇幻 hm X = . 会ビ 一 y w ķ 亲 仁 毒 y - fb ¢ 、 一 M X F ig N o t· A c u l p c a n e ith e r 3 9 F ig b e a lo c a l m a x im (1 ) o r u m lo c a l m in im a a as in 3 10 (2 ) (c)k »la te d p o in t lfthe re j g a te ¢ m u a re n o re a po l p o in t o n th e c u r v e in th e n e ı n t ig h b o u rh o o d o f a p o ín t p ís c a lle d n is o la t e d o r a 一 一 ˜ 一 一 一 ls o ı a X - 0 一 Ta U aq dL lL n r a lc c u r v e m q rwła tin Tı oN 一 t n l th e o g to ? LŤ O = ToL u ge a lg e b " 1 n The p a s s e s th r o z e r o th e S ho w e qu rh o n a t io n o u lo w he t io n o f ta n g e n t o r ta n g e n g h th e o ń g in , th e e q u a dtio n o fımth e c uıır v e h t d ,e w e sc m °,e rte nr mn s in t ee e q u a c" f th e ' " c tır v e y 闷 、 3 11 F ig . 4x ' + 9x ' h" " de a r. ń gm . t s a t t h e o r ig h ıis '" " " " is , 2 . 4× 2 + 9× 4 (i) 一 厂籀國 Scanned by CamScanner Scanned by CamScanner [J Lj ĺ f ř. 06 Ħ The do u b le p o in T be d o u b le p o in t w iĽ b e an t w ill b e a cus j is o la te d o in p if j Ex A M p L E x Ĵ j - . Į ħa h e r th e p o in t is - . or th e n o de f. . c tm e o in ts a r e a t x is o la o r み Pb s s ib le d o u b le p O - ー te d . f , o > Ħ if 】 f. = . . o (J q y } th e n it w iu b e 3 ロ t, y p ĺf f Hw F . . + x po = 〉 2 + 2 4y x u 3 + lt ip le p o in o - . Tin d to fo rd e r gre a te r th a n 2 thedoubıepointand in t 2y 4 ( 1 . + y a m O , a n = 0 二 〉 y ヱ ー 2) d f( 天 。 1 2) = , a t ( O 1, 2 ) 一 4 Scanned by CamScanner 댜 一 七 喜 す 不 坊 一 车 ち 2 = ー üi) 间 ah o ( \ : W x 2 ) ia At ( Ł 21 Po r inr h a do u bı e p o j ( Ļ 6 ( 1) = 3 x 2 x Y fo r a ll \ w o Tw o b ra n c h e s h an che s ijĝ e S O LU T ı O N L et itiv e + b s titu t e s tı 2) or :" T y s a tis fie s th e 2 - g iv e ııe qu a ti» in Ł , t e o r ig in t o \ = 4 x 2X Y + X 土X = 2 = = \ y x y 2 . Y - 2 in th e g iv e n n e qu a tio 0 ?ii fX o f (iv ) p a s s th r o u g h o r ig iŊı 2) o f Đ p a s s th r o u g h t 1 , F in d v a lu e o jo r th e y cu rve f (x , 2 . = = = x s in x 2 y )= y j f, İ . s iııx o r ig in is no de , ° p a te d ı o r is o n° po i x s in x x co sx 2y i X COS X + x s n - CO S X Scanned by CamScanner Scanned by CamScanner ら、 (c ) p o in t o f o A do u b le e x io ï n fı 5c u c u sp o n f b o th t h e SP e c i Ţ X F ig 3 1 6 m S k e tc h th e E X A M p L E 1 T he 2 C is c u rv e s y ın m e t r ic a l a 2 y cu rve bo (a u t x ) + x a x is x = 2 (a x } i a t a p o in t (a O ) h o r ig in a n d c u ts th e x a x s p a s s e s th r o u g e t th e ta n g e n ts s o f E q (i), w e g l w e s t d e g r e e te r m o h e t o r e z t o E q u a t in g 2 2 o r . y = į x x y , 3 u rv e 4 E qu 5 F ro 6 A a t in m s x g t o z e r o th e h ig h e s t d e g r e e he n y e x is ts w in c r e a s e s fr o m o to a Ţfijţ = L ţïtŢ . .: , w e g e t th e X y (i) Eq te r m a < = > y - x - ś į = ń gin p to te a X a > = in c r e a s e s u p a sym at o D p to o m a in a p o in ( ¢ t th e n a ] d e c re a s e s to z e ro = U Scanned by CamScanner M U I R t { »R 1 }ł A( ı N [i ( A ı t l i l iA N l {j tlA fï o h i M sym (a j A (b) A sym is s y ın m e tr ic a t i c u r v e is s y m m e r c a - c tu v e (c ) A aiı ı s s a n ıe (e ) T u t x ų a x la . 1e la bo u t y a x is , lf la bo u t l a bo e tr ic a ı sy ı ıu ı is be c tır v e m s y ın m e tr ic a is c u rv e (d ) A la bo e tr ic a is c u rv y u t l in o x = y - , if x Y iıi Fe p ln c e d b y . f( , , o n ic x ) a r id c y 十 : :二 u r v e r e m a in s b a rm e . F (x ) In te rc h a n g ln g x a n d y c u r v e r e m a i n ıs i on n te r c h a n g in g × b y , x « y an d y by p p o s ite q u a d r a n ts . i e f( , x ) w u e 只以 = Ŝ Ĺ- k fo r o r ig in L Ch e c m d iıe t h e w yes ec ı cł , k fo r ł p o in t o f in te 株式 x = lie s o n t h e c u r v e o r n o t lń p le p o h ıt s (S e e A r t 3 2 ) r o r ig in m rs e c O an u t io n ith w d fin d y a x is a n x pu t y d y a x is a n d fin d O - x A ls o o b ta in th e ta n g e n ts a t su d ı Ł A s y m p to te s Fin d th e p o in t ain a n at w h ic h a sy m p to te m e e ts th e c u r v e a n d e q u a tio n o f a s y m p to t e d r T o c h e c k in h ic h p a r t th e w c u rv e lie s to l? ic ily a n d m a x im a r m Fin d 41 d an c he c k th e in te r v a l In w h lc h y In c r e a s e B o1 ' d e c re a se s a n t ıth e p e in t a t w N ie h iĤ ? smaximum or m V C o ï c a v ity a n d t T he in te r v a l in U tln g a ll th e a in im u m c o n ve w bo h ic h , i v e r ė s tlts inte ċ a ín sk et ttië ë u iv ë Scanned by CamScanner Scanned by CamScanner gļ à gy O > , O < dr w 。 O R , 一 厂 w he w he n n ( X a (0 6 ) Q ) , : o r he re T hu s , th e c tı rv e fo r y 2 (a 2 + x 2 ) = . 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