Cve Functins Midterm 1 Review Plnmils Rtinl Functins Pwer Functins rignmetric Functins nverse rignmetric Functins Expnentil Functins Functins Dmin Lgrithmic Review Definitins nd bsic prperties Dmin f f x in 112 such tht text is defined hings t Lk ut Fr Denmintrs being Apprprite dmins f cre functins Fr exmple rcsin rcs hve dmin l D Lu hs dmin s tn hs dmin E t integer F Fu n even hs dmin Fr cmpsitins cnsider functins Fr exmple Finl rnge f internl rcs in x f E l Dmin 07 Fire C
rnsfrmtins f Functins Flt 7 x rnsltin up b FLN 7 x rnsltin dwn 1 Get Fc rnsltin left 7 x 7 ex rnsltin b b right b b f x btw Verticl stretch f Cx b 7 x Verticl b b cmpressin b b 9 Cx 7 be Hrizntl cmpressin b b 7 Gc 7 fe Hrizntl stretch b b f Cx tx Reflectin in X xis f cll te l Reflectin in xis Sfe rder Usull esiest t get crrect Hrizntl Hrizntl Reflectin in rnsltin stretch Cmpressin xis Verticl Reflectin rnsltin in x xis Verticl stretch Cmpressi
its f x pprches L s ntuitive 7 pprches but des nt im 7cl L equl frm bth sides x e.se S Given hrizntl strip Gentered t L t A verticl strip centered t such tht An E Lte O L 7 1cl C x text in x txi Verticl strip in hrizntl t strip verticl Given E Effetftists L E strip nrrw enugh 0C x cs t Frce S rts grph int tem L here exists 8 0 intersectin ss Using Ecs definitin t prve tix L Fix E Drw Grph s bve nd mke sensible chice f 8 0 i.e Mke verticl strip sufficientl nrrw Verit OC x les 17cl el CE
Exmple 7 Cc 2 1 x E z 3x 1 x 2 Prve 764 S K 72 Fix E SE S 32C s e L Chse 8 E s 9 2 1 Z h 32C 1 5 t Zc S E x 2 z E 3 A smller 2 31 21 cc 13 61 C E Z O C x z f e 134 11 Slee 21 7C C 221 c 12 41 c2 Kuti Slc CE Remrr ring t lgebricll reverse engineer n pprprite 870 strting frm tl LCE will be ver hrd in this cse
Using Ecs definitin t prve Lin Fi f L Y Drw picture nd chse E smll enugh t divide ner x fcx int tw disjint pieces Fr this E Prve tht Fr n S OC x l s FGC L1 c E Exmple fix l t x c l t x Prve Cim 7 x x l Chse E L t.tt 1 Z Never in intersectin 4 Let 8 0 nd chse x such tht 822C x ites nd 17cl it i 11 2 z N pssible chice f 830 Fr E z
mprtnt vrints A 76cg L x im 744 L im 744 L im 744 L CH x x Hrizntl n l Asmptte c pprches 7C pprches x grws x grws frmbve hrn belw mprtnt Definitin psitivel negtivel withut bund withut bund 1 cntinuus t it is 7cl defined 7 7 x exists x 3 7cl f x Vrints tel 7cl fix c t Fc k cntinuus Frm bve cntinuus frm belw 7 is c u it it is cutinus everwhere its dmin bve belw t endpints in ke Fcts Y All Cre Functins re cntinuus 4 Sums differences prducts qutients cmpsitins re cntinuus
it Lus C 1 l l 7C L nd Ling g.ci K x 3 tcxl g.cn Lt k x 9 x gcse LK F t k 3 7 Cx e Cmp w g c b Liz stel c fir 7cg Cx nfinitelimits im 7cl s 4 x insted 52 grws lt gives verticl psitivel withut bund smpttes im 7cl s S 4 x insted grws 52 negtivel withut bund wrning n bth cse limit Bsic Lws A nfinite its C _lt 1 1 1 7 Cx tx x 7 x 7
Z te s x 1 Fx FG4 L gg4 king Hx1tgc 7 HH i.f HH L Seim gcx7 king tst im gcx7 S fiz Hxlgcxi n ifeim gcx7 Lin thug e Lir ges im 7cg Cx Cim x z 7cl x 32 U mprtnt Cse Qutients s g c K FO L K V Clculte f Gc L nd 9 K V L t 4 0 1 K S O Mnipulte Qutient Ln nd ppl r LF DVE
7 Neither L S C j K t Ot L G O K d c im u s S v im L C O L S O t K O K O t 7cl 1 gds c gd gs g c termed in g x et Y O xis g x D g c g x f he Derivtive L f se h g Duble sided limit f ctin Fc Slpe f tngent f it exists we s wie t grph CK f x 7 differentible t t