SCHOOL OF MATHEMATICS AND STATISTICS. Mathematics II (Materials)

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1 Dt proie: Form Sheet MAS5 SCHOOL OF MATHEMATICS AND STATISTICS Mthemtics II (Mteris) Atm Semester -3 hors Mrks wi e wre or swers to qestios i Sectio A or or est THREE swers to qestios i Sectio. Sectio A crries 4 mrks the mrks wre to ech qestio or sectio o qestio re show i itics. Sectio A A A Fi the sotio o the eqtio or > which stisies whe. (8 mrks) Fi the geer sotio o the eqtio e (8 mrks) A3 Aice throws o mss.6 kg t spee o ms. The kietic eerg o the is E m where m re its mss eocit respectie. I Aice ow throws o mss.56 kg t spee o.5 ms se the chi re or prti erities to estimte the % chge i the kietic eerg she mst gie the. (7 mrks) A4 A scr ie φ is gie φ 3 ector ie is eie φ. () Fi show tht. (6 mrks) () Fi the irectio eritie o φ i the irectio o the ector ( 3). (5 mrks) t the poit ( ) MAS5 Tr oer

2 MAS5 A5 Two qtities he mes respectie rices respectie corice.9. () () Ccte the corretio coeiciet etwee correct to 3 sigiict igres. ( mrks) It is ssme tht stis the ier retioship ( ) ( ) where is the me o. Ccte the est sqres estimtes o correct to 3 sigiict igres. Stte giig resos whether o epect ( ) to gie goo moe.(4 mrks) Sectio () Fi the geer sotio o the eqtio ( ) e. () Fi the sotio o the eqtio 6 9 8e. gie tht whe. / () I φ ( ) Hece ece tht show tht φ ( ) 5 / φ.. ( mrks) ( mrks) ( mrks) () A smpe o ights h the oowig ietimes (i weeks roe to the erest week): Ccte the me str eitio o the ietime correct to ecim pces. (8 mrks) MAS5 Cotie

3 MAS5 3 () itegrtig prts twice ete e where is positie iteger. si (9 mrks) () A ctio ( ) e is eie o the iter. (i) Show tht ( ) c e represete the Forier sie series ( ) e (ii) 4 Sppose tht ( t) { } si. (7 mrks) Sketch the ctio gie the oe Forier sie series o the iter. (4 mrks) stisies the het coctio eqtio t o or t > sject to the or coitios t t. () () Show tht the geer sotio is ( ) t t ep si. ( mrks) Sppose tht so stisies the iiti coitio 4 or < < t t where is costt. Sketch the iiti coitio or < < etermie the costts. (8 mrks) Yo m ssme tht si 4 ee o ( ) 3 E o Qestio Pper MAS5 3 Tr oer

4 MAS5 FORMULA SHEET Trigoometr t θ sec θ cot θ cosec θ cos ( A ) cos Acos si Asi ( A ) cos Acos si Asi cos ( A ) si Acos cos Asi si si t t ( A ) si Acos cos Asi ( A ) ( A ) t A t t At t A t t At si θ si θ cosθ cosθ cos θ si θ cosθ ( θ α ) where R cosα / R si / R si θ Rcos α Hperoic Fctios ( e ) sih e ( e ) cosh e cosh sih sech th sih sih cosh cosh cosh sih sih cosh ( ) ( ) th < coth > MAS5 4 Cotie

5 MAS5 Dieretitio Itegrtio Fctio e Deritie e si cos cos si t sec cot cosec sec sec t cosec cosec cot sih cosh cosh sih th sech coth cosech sech sech th cosech cosech coth si cos t cot sih cosh th < coth > MAS5 5 Tr oer

6 MAS5 MAS5 6 Cotie Fctio Itegr t th si sih cosh Dieretitio Itegrtio Forme ( ) ( ) [ ] ( ) o itegr o itegr or Prti ieretitio Chi Re. Sppose tht ( ) tht re ctios o t i.e. ( ) ( ) t t. The t t t. Sppose tht ( ) tht re ctios o the ries r s i.e. ( ) ( ) s r s r. The s s s r r r

7 MAS5 First-Orer Diereti Eqtios. Direct Itegrtio. Seprtio o Vries 3. Homogeeos Eqtios Mke the sstittio 4. Lier Eqtios ( ) ( ) C g ( ) ( ) g( ) ( ) to gie P () ( ) Q( ) Mtip oth sies the itegrtig ctor e P( ) e to gie P( ) P( ) Q ( ) e MAS5 7 Tr oer

8 MAS5 The Seco-Orer Diereti Eqtio where c re costts. The Geer Sotio is Compemetr Fctio (i) c ( ) Compemetr Fctio Prticr Itegr I the iir eqtio m m c hs roots m m the the Compemetr Fctio m m c Ae e c is gie i m m re re ieret m (ii) ( A ) e i m c p (iii) e ( Acosq si q) Prticr Itegr c m re re eq ( m m m) i m p m re compe ( m p q m p iq) i ( ) A C p c k ( ) Ae p e k where k is ot oe o the roots o the iir eqtio k ( ) Ae p e where k is oe o the roots o the iir eqtio ( ) Acos m si m cos m si m where si m or cosm is ot prt o the compemetr ctio ( ) Acos m si m cos m si m where si m or cosm is prt o the compemetr ctio p p k MAS5 8 Cotie

9 MAS5 MAS5 9 Tr oer Forier Series Sppose tht ( ) is eie o the iter. The Forier series or ( ) is gie ( ) si cos where ( ) K cos ( ) K si O the iter the Forier cosie series or ( ) is ( ) ( ) cos cos the Forier sie series is ( ) ( ) si si Vector Ccs The griet o the scr ie ( ) φ is gie φ φ φ φ The iergece o ector ie ( ) ( ) w is gie w The cr o ector ie ( ) ( ) w is gie w k j i The Lpci is gie

10 MAS5 Sttistics For t es ( ) ( ) ( ) K Mes i i etc. i i Vrices ( ) σ i etc. co Corice ( ) ( )( ) i i i i i ( ) co Corretio coeiciet r σ σ Lier Regressio Lest Sqres The est sqres it to the ier retioship ( ) is gie co σ The correspoig me sqre resi is ( r ) σ. ( ) i i MAS5