Math Notebook for Students
|
|
- Jeffry Gardner
- 6 years ago
- Views:
Transcription
1 Mth Notebook for Stuets 50 Essetil Mthemticl Formuls Equtios b Peter I. Ktt Petr Books
2 Mth Notebook for Stuets Peter I. Ktt, PhD Correspoece bout this book m be set to the uthor t oe of the followig two emil resses: pktt@lumi.lsu.eu pktt@tet.et.jo The uthor ckowleges the work of Aubislivess which ppers i the imge of Pi o the frot bck covers of this book. Mth Notebook for Stuets: 50 Essetil Mthemticl Formuls Equtios. Writte b Peter I. Ktt. ISBN: 4407 ISBN-: All rights reserve. No prt of this book m be copie or reprouce without writte permissio of the uthor or publisher. 009 Peter I. Ktt
3 Mth Notebook for Stuets To m prets, brothers, sisters
4 Mth Notebook for Stuets 4 Mth Notebook for Stuets 50 Essetil Mthemticl Formuls Equtios
5 Mth Notebook for Stuets 5 Prefce This is little book for stuets who wish to hve the essetil formuls equtios of mthemtics i sigle esil ccessible source. I bout 50 pges, the 50 most essetil mthemticl formuls re liste. Ulike other lrge books o this topic, there is o ee to go through hures of pges thouss of formuls for the stuet to get the bsic equtios. The uthor hs serche severl books o mthemticl formuls tbles selecte ol those equtios which re essetil to the stuet. The mthemticl formuls equtios liste i this book re useful for stuets reserchers i vrious fiels icluig mthemtics, phsics, egieerig, etc. Ol the most elemetr bsic topics re covere icluig formuls for vrious geometric shpes, severl tpes of fuctios (trigoometric, hperbolic, epoetil, logrithmic, etc), the qurtic equtio, ltic geometr, erivtives itegrls, rithmetic series, geometric series, Tlor series. A comprehesive referece list is iclue t the e of the book i itio to umerous web liks to more formuls, equtios, tbles. The uthor hs ecie gist icluig umericl tbles for itegrtio, logrithms, etc becuse the use of such tbles hs bee supersee b the vilbilit of scietific clcultors computer lgebr progrms. Thus, ecisio hs bee reche to keep the book i compct formt tht iclues ol the most essetil bsic formuls. It is hope tht the uthor hs succeee i this regr. Etreme cre hs bee tke to esure the ccurc of the formuls tht re liste i this book. It is hope tht this little book will prove to be vluble source of
6 Mth Notebook for Stuets 6 iformtio for stuets i mthemtics, sciece, egieerig. Fill, the uthor wishes to ckowlege the help support of his fmil members without which he woul ot hve bee ble to prouce this book i its preset form. Peter I. Ktt Mrch 009
7 Mth Notebook for Stuets 7 Cotets Specil Costts 9 Specil Proucts Fctors 9 Biomil Formul Biomil Coefficiets 0 4 Geometric Formuls 5 Trigoometric Fuctios 6 Comple Numbers 8 7 Epoetil Logrithmic Fuctios 0 8 Hperbolic Fuctios 4 9 The Qurtic Equtio 5 0 Ple Altic Geometr 6 Soli Altic Geometr 8 Derivtives Iefiite Itegrls 4 4 Defiite Itegrls 47 5 Series 50 6 Tlor Series 5 7 Iequlities 5
8 Mth Notebook for Stuets 8 8 Coversio Fctors 54 9 List of Prime Numbers 55
9 Mth Notebook for Stuets 9. Specil Costts π.459 Rtio of perimeter of circle to its imeter e.788 Nturl bse of logrithms γ Euler s costt.44 Squre root of.705 Squre root of Squre root of 5. Specil Proucts Fctors ( ) ( ) ( ) ( ) ( ) ( ) 4 6 4
10 Mth Notebook for Stuets 0 ( )( ) ( )( ) i i Fctors re comple umbers (cot be fctore with rel fctors) ( )( ) ( )( ) ( )( ) ( )( )( ) ( )( )( )( ) i i 4 4 The lst lie bove iclues comple fctors.. Biomil Formul Biomil Coefficiets ( )... where the biomil coefficiets re give b!( )!! k k k ( ) 4! for iteger,! 0. Note tht k k
11 Mth Notebook for Stuets 4. Geometric Formuls Ares of Commo Two-Dimesiol Shpes Shpe Are Squre of sie Are Rectgle of legth Are b with b Trigle of ltitue h bse b Are bh Trigle of sies, b, Are s s s b s c c where s ( b c) Trpezoi of ltitue h Are h b prllel sies b ( ) Circle of rius r Are π r ( )( )( ) Circle of imeter Are π 4 Sector of circle with rius r gle θ Are r θ, Ellipse of semi-mjor is Are π b semi-mior-is b θ i ris Perimeters of Commo Two-Dimesiol Shpes Shpe Perimeter Squre of sie Perimeter 4 Rectgle of legth Perimeter ( b) with b Trigle of sies, Perimeter b c b, c Circle of rius r Perimeter π r
12 Mth Notebook for Stuets Circle of imeter Sector of circle with rius r gle θ Perimeter π Arc Legth s rθ, θ i ris Volumes of Commo Three-Dimesiol Solis Soli Volume Cube of sie Volume Rectgulr prllelepipe of Volume bc legth, with b, height c Sphere of rius r 4 Volume π r Right circulr clier of Volume π r h rius r height h Right circulr coe of rius r height h Volume π r h Prmi of bse re A Volume Ah height h Ellipsoi of semi-es, b, 4 c Volume π bc Surfce Ares of Commo Three-Dimesiol Solis Soli Surfce Are Cube of sie Surfce Are 6 Rectgulr Surfce Are b c bc prllelepipe of legth, with b, height c Sphere of rius r Surfce Are 4π r Right circulr Lterl Surfce Are π r h clier of rius r height h ( )
13 Mth Notebook for Stuets Right circulr coe of rius r height h Lterl Surfce Are π r r h 5. Trigoometric Fuctios Cosier right trigle with gle θ sies,, r. si θ r cos θ r t θ opposite hpoteuse jcet hpoteuse opposite jcet θ r For gles, we hve the followig reltios: π ris 60 o π ris 80 o 80 ris π o o π 80 o ris
14 Mth Notebook for Stuets 4 We hve lso the followig trigoometric ietities: siθ t θ cosθ cosθ cot θ siθ tθ sec θ cosθ csc θ siθ si θ cos θ sec θ t csc θ cot θ θ We hve the followig results for some commo gles: θ (egrees) θ (ris) si θ cos θ o o o 45 o 60 o 90 π 6 π 4 π π 0
15 Mth Notebook for Stuets 5 Formuls for the egtive of gle si ( θ ) siθ ( θ ) cosθ cos t ( θ ) tθ Co-fuctio formuls π si θ cosθ π cos θ siθ Formuls for sums iffereces of two gles ( α β ) siα cos β cosα si β si si cos ( α β ) siα cos β cosα si β ( α β ) cosα cos β siα si β ( α β ) cosα cos β siα si β cos t t ( α β ) ( α β ) tα t β tα t β tα t β tα t β
16 Mth Notebook for Stuets 6 Formuls for ouble gles ( θ ) siθ cosθ si cosθ cos θ si θ si θ cos θ t tθ t θ θ Formuls for hlf gles θ si ± cosθ θ cosθ cos ± θ cosθ t ± cosθ siθ cosθ cosθ siθ cscθ cotθ
17 Mth Notebook for Stuets 7 Squre of the sie cosie fuctios cos si θ cos cos θ ( θ ) ( θ ) Sums, iffereces, proucts of the sie cosie fuctios α β α siα si β si cos α β α siα si β cos si α β α cosα cos β cos cos α β α cosα cos β si si cos siα si β cos cosα cos β si siα cos β ( α β ) cos( α β ) ( α β ) cos( α β ) ( α β ) si( α β ) β β β β
18 Mth Notebook for Stuets 8 Lw of Sies b si A si B c sic b C Lw of Cosies A c B c b bcosc 6. Comple Numbers i, i bi is the geerl form of comple umber where b re rel umbers. bi c i if ol if c b. Aitio subtrctio of comple umbers ( bi) ( c i) ( c) ( b )i ( bi) ( c i) ( c) ( b )i Multiplictio ivisio of comple umbers ( bi)( c i) ( c b ) ( bc)i
19 Mth Notebook for Stuets 9 bi c i bi c i c i c i c b bc i c c ( c b ) ( bc ) c i Polr form of comple umber ( cosθ isiθ ) bi r where (,b) r b r θ t b θ Multiplictio ivisio of comple umbers i polr form ( cosθ i θ ) z r si ( cosθ i θ ) z z r si [ cos( θ θ ) ( θ θ )] z r r isi z z r [ cos( θ θ ) ( θ θ )] isi r Powers of comple umbers ( cosθ isiθ ) z r
20 Mth Notebook for Stuets 0 z r [ cos ( θ ) isi( θ )] Roots of comple umbers ( cosθ isiθ ) z r θ kπ θ kπ z z r cos isi where k is iteger. 7. Epoetil Logrithmic Fuctios Rules for epoets... ( times),, re rel umbers. m m, m,, re rel umbers. m m m m ( ) 0, 0, 0 ( )
21 Mth Notebook for Stuets m m Defiitio of logrithms log if ol if, Properties of logrithms log log 0 ( ) log log log log log ( ) log log log Chge of bse for logrithms log logb,, b log b
22 Mth Notebook for Stuets log,, log Specil logrithms log0 log loge log e 0.44 l 0 l l log log0 l Other formuls θ e i cosθ isiθ θ e i cosθ isiθ Reltio betwee epoetil trigoometric fuctios e siθ iθ e cosθ iθ e e iθ iθ Perioicit of the epoetil fuctio ( θ kπ ) iθ i e e, k is iteger More o the polr form of comple umbers iθ ( cos θ i θ ) re z i r si
23 Mth Notebook for Stuets Multiplictio ivisio of comple umbers i polr form z re iθ z r e iθ z z z z r r e i ( θ θ ) ( θ ) i θ r e r Powers roots of comple umbers i polr form iθ z re iθ z r e, is rel umber z r e θ kπ i,, k re itegers Logrithms of comple umbers i polr form iθ ( re ) l r iθ kπi l z l, k is iteger
24 Mth Notebook for Stuets 4 8. Hperbolic Fuctios e sih e cosh e e e e sih th e e cosh e e cosh coth e e sih th sec h e e csc h e e cosh sech coth sih th csch cosh sih Reltio betwee hperbolic trigoometric fuctios ( i) isih si ( i) cosh cos
25 Mth Notebook for Stuets 5 ( i) i th t ( i) isi sih ( i) cos cosh ( i) i t th 9. The Qurtic Equtio The solutio of the qurtic equtio b c 0 is give b the qurtic formul s follows (, b, c re rel) b ± b 4c D b 4c is clle the iscrimit. We hve the followig three cses:. D > 0 implies there re two rel but uequl solutios.. D 0 implies there re two rel equl solutios.. D < 0 implies there re two comple cojugte solutios.
26 Mth Notebook for Stuets 6 0. Ple Altic Geometr The istce betwee two poits with coorites (, ) ( ) is give b, ( ) ( ) The slope of the lie joiig two poits with coorites (, ) (, ) is give b m tθ where θ is the gle of iclitio of the lie. The equtio of the lie joiig two poits with coorites (, ) (, ) is give b The equtio of the lie with slope m pssig through poit with coorites (, ) is give b m ( ) The equtio of the lie with slope m -itercept b is give b m b
27 Mth Notebook for Stuets 7 The equtio of the lie with -itercept -itercept b is give b, 0, b 0 b The geerl equtio of lie is give b A B C 0 where A, B, C re costts., to the lie give b the equtio A B C 0 is give b The istce from poit with coorites ( ) A B C ± A B The gle θ betwee two lies with slopes m m is give b θ t m m mm The re A of trigle with vertices t the three poits with coorites (, ), (, ), ( ), is give b A ± where the plus or mius sig is chose to give positive re.
28 Mth Notebook for Stuets 8 Polr coorites: The followig re the reltios betwee polr r,θ rectgulr (Crtesi) coorites coorites ( ) (, ) r cosθ r siθ or ltertivel r θ t The equtio of circle with rius R ceter t the poit is give b with coorites ( ) 0, 0 ( ) ( ) 0 0 R. Soli Altic Geometr The istce betwee two poits with coorites, z, z is give b ( ) ( ),, ( ) ( ) ( z ) z The irectio cosies l, m, of lie pssig through two poits with coorites (,, z ) (,, z ) re give b
29 Mth Notebook for Stuets 9 cosα l cos β m cosγ z z where is the istce betwee the two poits α, β, γ re the gles of iclitio of the lie with the positive,, z es, respectivel. We lso hve the followig reltio betwee the irectio cosies: cos α cos β cos γ or equivletl l m The equtios of lie pssig through two poits with coorites (,, z ) (,, z ) re give s follows i str form z z z z or ltertivel l m z z
30 Mth Notebook for Stuets 0 The equtios of lie pssig through two poits with coorites (,, z ) (,, z ) re give s follows i prmetric form lt mt z z t where t is the prmeter, l, m, re the irectio cosies. The geerl equtio of ple is give b A B Cz D 0 where A, B, C, D re costts. The equtio of ple pssig through three poits with coorites (,, z ), (,, z ), (,, z) is give b z z z z z z 0 The equtio of lie with -itercept, -itercept b, z-itercept c is give b z, 0, b 0, c 0 b c
31 Mth Notebook for Stuets The equtios of lie pssig through poit with coorites ( 0, 0, z0 ) perpeiculr to the ple A B Cz D 0 is give b i str form A B z z C The equtios of lie pssig through poit with coorites ( 0, 0, z0 ) perpeiculr to the ple A B Cz D 0 is give b i prmetric form z z At Bt Ct The istce from poit with coorites (, z ) the ple A B Cz D 0 is give b A ± 0 B A 0 Cz B 0 C Cliricl coorites ( r,θ, z) z D to 0 0, r cosθ r siθ θ z z r where (,, z) re the rectgulr (Crtesi) coorites. Altertivel, we hve 0 z (r, θ, z)
32 Mth Notebook for Stuets z z r t θ Sphericl coorites ( ),θ,ϕ r θ ϕ θ ϕ θ cos si si cos si r z r r where ( ) z,, re the rectgulr (Crtesi) coorites. Altertivel, we hve cos t z z z r θ ϕ The equtio of sphere with rius R ceter t poit with coorites ( ) 0 0 0,, z is give b ( ) ( ) ( ) R z z φ z r θ (r, θ, φ)
33 Mth Notebook for Stuets. Derivtives Defiitio of the erivtive of fuctio. Let f ( ) b be give fuctio. The its erivtive is give f ( ) lim h 0 f ( h) f ( ) h If we let h Δ, the the bove efiitio c be writte equivletl s follows f ( ) lim Δ 0 f ( Δ) f ( ) Δ Rules of ifferetitio I the followig rules of ifferetitio, we ssume c costt, f v g () c 0 ( ) ( c) c costt, u ( ), ( )
34 Mth Notebook for Stuets 4 ( ) ( c ) c ( u v) u v or equivletl ( f ( ) g( ) ) f ( ) g ( ) ( u v) u v or equivletl ( f ( ) g( ) ) f ( ) g ( ) ( cu) u c or equivletl ( cf ( ) ) cf ( ) ( uv) v u v u or equivletl ( f ( ) g( ) ) f ( ) g ( ) g( ) f ( ) u v u v v u, v 0 v
35 Mth Notebook for Stuets 5 or equivletl g ( ) 0 f g ( ) ( ) g ( ) f ( ) f ( ) g ( ) ( g( ) ), ( u ) u u or equivletl ( f ( ) ) Chi rule u u ( ) f ( ) ( ) f ( ) or equivletl ( f ( g( ) )) f ( g( ) ) g ( ) u u u u Derivtives of trigoometric fuctios ( si ) cos ( cos ) si
36 Mth Notebook for Stuets 6 ( t ) sec ( cot ) csc ( sec ) sec t ( csc ) csc cot ( u) si cosu u or equivletl ( si ( f ( ) )) cos( f ( ) ) f ( ) ( u) cos siu u or equivletl ( cos( f ( ) )) si( f ( ) ) f ( ) ( u) t sec u u or equivletl ( t ( f ( ) )) sec ( f ( ) ) f ( ) ( u) cot csc u u or equivletl ( cot( f ( ) )) csc ( f ( ) ) f ( )
37 Mth Notebook for Stuets 7 ( u) sec secu tu u or equivletl sec f sec ( u) ( ( ( ))) ( f ( ) ) t( f ( ) ) f ( ) csc cscu cot u u or equivletl csc f csc ( ( ( ))) ( f ( ) ) cot( f ( ) ) f ( ) Derivtives of epoetil fuctios ( e ) e ( ) l u ( e ), costt e u u or equivletl f ( ) f ( ) ( e ) e f ( ) u ( ) u u l, costt or equivletl f ( ) f ( ) ( ) l f ( )
38 Mth Notebook for Stuets 8 Derivtives of logrithmic fuctios ( l ), 0 log e ( log ), 0, u ( l u), u 0 u or equivletl ( ( f ( ) )) f ( ) 0 log u e u ( log u), u 0 or equivletl ( ( f ( ) )) f ( ) 0, 0, ( ) ( ) l f, f log e log f ( ), f ( ) Derivtives of hperbolic fuctios ( sih ) cosh ( cosh ) sih ( th ) sech
39 Mth Notebook for Stuets 9 ( coth ) csch ( sech ) sech th ( csch ) csch coth ( u) sih cosh u u or equivletl ( sih ( f ( ) )) cosh( f ( ) ) f ( ) ( u) cosh sihu u or equivletl ( cosh ( f ( ) )) sih( f ( ) ) f ( ) ( u) th sech u u or equivletl ( th ( f ( ) )) sech ( f ( ) ) f ( ) u ( coth u) csch u coth f csch or equivletl ( ( ( ))) ( f ( ) ) f ( ) ( hu) sec sechu thu u
40 Mth Notebook for Stuets 40 or equivletl sech f sech ( hu) ( ( ( ))) ( f ( ) ) th( f ( ) ) f ( ) csc cschu cothu u or equivletl csch f csch ( ( ( ))) ( f ( ) ) coth( f ( ) ) f ( ) Higher erivtives f ( ) Seco erivtive f ( ) Thir erivtive iv 4 4 f iv ( ) Fourth erivtive Differetils f ( ) Rules for ifferetils ( u v) u v ( u v) u v
41 Mth Notebook for Stuets 4 ( uv) u v v u u v u u v, v 0 v v ( u ) u u ( si u) cosu u ( cosu) siu u ( t u) sec u u Prtil erivtives Defiitio of prtil erivtives f f lim 0 Δ lim 0 Δ f f ( Δ, ) f (, ) Δ (, Δ) f (, ) Δ Seco prtil erivtives f f f f
42 Mth Notebook for Stuets 4 f f f f The ifferetil of fuctio f f f. Iefiite Itegrls I the followig formuls, ote tht the costt of itegrtio is ot show. Furthermore, ote tht costt costt. l, ( ) f ( ) f
43 Mth Notebook for Stuets 4 ( f ( ) g( ) ) f ( ) g( ) ( f ( ) g( ) ) f ( ) g( ) Itegrtio b Prts usig this ottio u f, v g, u f, v g ( ( ) ( ) ( ) ( ) u v uv v u ) Other itegrtio formuls f w ( ) f ( w) ( f ( ) ) ( w) ( ) g f f g w, w ( ) w w w, w l w w Iefiite itegrls of epoetil fuctios e w w e w w w w l, > 0,
44 Mth Notebook for Stuets 44 Iefiite itegrls of trigoometric fuctios si ww cos w cos w w si w ( cos ) t ww l w ( si ) cot ww l w ( sec w t ) sec ww l w ( csc w cot ) csc ww l w sec w w t w csc ww cot w t cot si ww ww ww t w w cot w w w ( w si wcos ) w cos w w sec wt ww secw ( w si wcos )
45 Mth Notebook for Stuets 45 csc wcot ww cscw si ( ) cos( ) cos ( ) si( ) Iefiite itegrls of hperbolic fuctios sih w w cosh w cosh w w sih w ( cosh ) th ww l w ( sih ) coth ww l w sec h ww si w ( th w) t ( e ) w csc h ww l th coth sech ww th w csch ww coth w th ww w th w w ( e )
46 Mth Notebook for Stuets 46 coth sih ww ww w coth w w ( sih wcosh w ) cosh ww w ( sih wcosh w ) sec h wth ww sech w csc h wcoth ww csch w Itegrtio b substitutio F F w ( b) F( w) ( b ) wf( w) w, w b, w b F ( ) F( cos w) cos ww, w si ( ) F( w) ( ) F( t w) F sec sec w w, w t F F ( ) ( w) e F w w, sec wt ww, w sec w e
47 Mth Notebook for Stuets 47 ( ) F( w) w F l e w, w l 4. Defiite Itegrls Defiitio of the efiite itegrl b f ( ) lim[ f ( ) Δ f ( Δ) Δ f ( Δ)... f ( ( ) Δ) Δ] Δ where the itervl [ b], is subivie ito equl prts of b legth Δ. Properties of the efiite itegrl b f g b ( ) ( )] g( b) g( ) where f ( ) g( ) g ( ). b [ f ( ) g( ) ] f ( ) g( ) b [ f ( ) g( ) ] f ( ) g( ) b b b b
48 Mth Notebook for Stuets 48 b ( ) c f ( ) cf b f ( ) 0 b ( ) f ( ) f b c b ( ) f ( ) f ( ) f Me Vlue Theorem b ( ) ( b ) f ( c) b c f, where c is betwee b f ( ) is cotiuous i [ b],. Approimtio of efiite itegrls Subivie the itervl [, b] ito equl prts b the poits 0,,,,...,, b where ( ), i 0,,,,...,( i f i ), Rectgulr rule of pproimtio b f ( ) h( ) 0... b h.
49 Mth Notebook for Stuets 49 Trpezoil rule of pproimtio b h 0 ( ) (... ) f Simpsos rule of pproimtio b h 0 4 ( ) ( ) f Selecte efiite itegrls π 0 si ( m) si( ) 0, π, m m π 0 cos ( m) cos( ) 0, m π, m π 0 si ( m) cos( ) 0 m m, m eve, m o π 0 0 si π π cos 0 si π 4
50 Mth Notebook for Stuets si cos π 4 t π 5. Series Arithmetic series ( ) ( )... ( ( ) ) where l ( ). Emples of rithmetic series... ( ) ( ) 5... Geometric series ( l) r r r... r Emple of geometric series r r, r
51 Mth Notebook for Stuets 5,... < < r r r r r r Emples of other tpes of series ( )( ) 6... ( ) ( ) l π π... 4 π π
52 Mth Notebook for Stuets 5 6. Tlor Series f ( ) f ( ) f ( )( ) f ( )( ) f! ( ) where the remier! ( ) ( )( ) R... R is give b the followig equtio R f ξ! ( ) ( )( ) ξ is betwee. Also, the fuctio f must be cotiuous with cotiuous erivtives. The ifiite series give bove is clle the Tlor series for f bout if lim R 0. ( ) If 0, the the series is clle Mcluri series. Emples of Tlor Mcluri series 4..., < <..., < e..., < <!!
53 Mth Notebook for Stuets 5 ( l ) ( l ) l..., < <!! 4 4 ( )..., < l 5 7 si...,! 5! 7! < < 4 6 cos...,! 4! 6! < < 5 7 sih...,! 5! 7! < < 4 6 cosh...,! 4! 6! < < 7. Iequlities Trigle iequlit......
54 Mth Notebook for Stuets Coversio Fctors Legth kilometer 000 meters meter 00 cetimeters meter 000 millimeters cetimeter 0.0 meter millimeter 0.00 meter ich.540 cetimeters foot 0.48 cetimeters mile.609 kilometers cetimeter 0.97 ich meter 9.7 iches kilometer 0.64 mile Volume liter 000 cubic cetimeters cubic meter 000 liters cubic foot 8. liters
55 Mth Notebook for Stuets 55 Mss kilogrm 000 grms kilogrm.046 pous pou 45.6 grms For etile coversios of other uits, visit the website for immeite olie coversios. 9. List of Prime Numbers The followig tble lists the first 0 prime umbers:
56 Mth Notebook for Stuets 56 Refereces. Spiegel, M. R., Schum s Mthemticl Hbook of Formuls Tbles, McGrw-Hill, Seco Eitio, Spiegel, M. R., Lipschutz, S. Liu, J., Schum s Outlie of Mthemticl Hbook of Formuls Tbles, McGrw-Hill, Thir Eitio, Jeffre, A. Di, H. H., Hbook of Mthemticl Formuls Itegrls, Acemic Press, Fourth Eitio, Abrmowitz, M. Stegu, I. A., Hbook of Mthemticl Fuctios: With Formuls, Grphs, Mthemticl Tbles, Dover Publictios, Zwilliger, D., CRC Str Mthemticl Tbles Formuls, Chpm & HllCRC, st Eitio, Tllri, R. J., Pocket Book of Itegrls Mthemticl Formuls, Chpm & HllCRC, Bllst, D., Architect s Hbook of Formuls, Tbles, Mthemticl Clcultios, Pretice Hll, Burigto, R. S., Hbook of Mthemticl Tbles Formuls, McGrw-Hill, Fifth Eitio, Luerer, B., Nollu, V. Vetters, K., Mthemticl Formuls for Ecoomists, Spriger, Thir Eitio, Spiegel M. R. Liu, J., Schum s Es Outlie of Mthemticl Hbook of Formuls Tbles,, McGrw-Hill, 00.
57 Mth Notebook for Stuets 57. Boljovic, V., Applie Mthemticl Phsicl Formuls Pocket Referece, Iustril Press, Brtsch, H. J., Hbook of Mthemticl Formuls, Acemic Press, 974. Iteret Liks. S. O. S. Mthemtics: Tbles Formuls The Worl of Mth Olie Mth.com Mth Referece Tbles Mesuremet Formuls: Geometr emesure.htm 4. Mth Formuls, Mth Tbles s_mth_tbles.htm 5. The Euctiol Ecclopei: Mthemticl Formuls rmuls.htm
58 Mth Notebook for Stuets Mthemticl Formuls for Ares Volumes 7. Commol Use Mthemticl Formuls muls.html
59 Mth Notebook for Stuets 59 Notes
60 Mth Notebook for Stuets 60 Notes
Sharjah Institute of Technology
For commets, correctios, etc Plese cotct Ahf Abbs: hf@mthrds.com Shrh Istitute of Techolog echicl Egieerig Yer Thermofluids sheet ALGERA Lws of Idices:. m m + m m. ( ).. 4. m m 5. Defiitio of logrithm:
More informationIndices and Logarithms
the Further Mthemtics etwork www.fmetwork.org.uk V 7 SUMMARY SHEET AS Core Idices d Logrithms The mi ides re AQA Ed MEI OCR Surds C C C C Lws of idices C C C C Zero, egtive d frctiol idices C C C C Bsic
More informationImportant Facts You Need To Know/Review:
Importt Fcts You Need To Kow/Review: Clculus: If fuctio is cotiuous o itervl I, the its grph is coected o I If f is cotiuous, d lim g Emple: lim eists, the lim lim f g f g d lim cos cos lim 3 si lim, t
More informationMTH213 Calculus. Trigonometry: Unit Circle ( ) ( ) ( )
MTH3 Clculus Formuls from Geometry: Trigle A = h Pythgore: + = c Prllelogrm A = h Trpezoi A = h ( + ) Circle A = π r C = π r = π Trigoometry: Uit Circle 0, π 3,, 3 35, 3π 4,, 50, 5π 6, 3, 90 π 0, Coe V
More informationReview Handout For Math 2280
Review Hout For Mth 80 si(α ± β siαcos β ± cos α si β si θ 1 [1 cos(θ] cos si α cos β 1 [si(α + β + si(α β] si cos α cos β 1 [cos(α + β + cos(α β] si x +siy si ( ( x+y cos x y Trigoometric Ietites cos(α
More informationPhysicsAndMathsTutor.com
PhysicsAdMthsTutor.com PhysicsAdMthsTutor.com Jue 009 4. Give tht y rsih ( ), > 0, () fid d y d, givig your swer s simplified frctio. () Leve lk () Hece, or otherwise, fid 4 d, 4 [ ( )] givig your swer
More informationThings I Should Know In Calculus Class
Thigs I Should Kow I Clculus Clss Qudrtic Formul = 4 ± c Pythgore Idetities si cos t sec cot csc + = + = + = Agle sum d differece formuls ( ) ( ) si ± y = si cos y± cos si y cos ± y = cos cos ym si si
More informationAP Calculus Formulas Matawan Regional High School Calculus BC only material has a box around it.
AP Clcls Formls Mtw Regiol High School Clcls BC oly mteril hs bo ro it.. floor fctio (ef) Gretest iteger tht is less th or eql to.. (grph) 3. ceilig fctio (ef) Lest iteger tht is greter th or eql to. 4.
More informationNational Quali cations AHEXEMPLAR PAPER ONLY
Ntiol Quli ctios AHEXEMPLAR PAPER ONLY EP/AH/0 Mthemtics Dte Not pplicble Durtio hours Totl mrks 00 Attempt ALL questios. You my use clcultor. Full credit will be give oly to solutios which coti pproprite
More informationNational Quali cations SPECIMEN ONLY
AH Ntiol Quli ctios SPECIMEN ONLY SQ/AH/0 Mthemtics Dte Not pplicble Durtio hours Totl mrks 00 Attempt ALL questios. You my use clcultor. Full credit will be give oly to solutios which coti pproprite workig.
More informationExponential and Logarithmic Functions (4.1, 4.2, 4.4, 4.6)
WQ017 MAT16B Lecture : Mrch 8, 017 Aoucemets W -4p Wellm 115-4p Wellm 115 Q4 ue F T 3/1 10:30-1:30 FINAL Expoetil Logrithmic Fuctios (4.1, 4., 4.4, 4.6) Properties of Expoets Let b be positive rel umbers.
More informationFor students entering Honors Precalculus Summer Packet
Hoors PreClculus Summer Review For studets eterig Hoors Preclculus Summer Pcket The prolems i this pcket re desiged to help ou review topics from previous mthemtics courses tht re importt to our success
More informationMathematical Notation Math Calculus & Analytic Geometry I
Mthemticl Nottio Mth - Clculus & Alytic Geometry I Nme : Use Wor or WorPerect to recrete the ollowig ocumets. Ech rticle is worth poits c e prite give to the istructor or emile to the istructor t jmes@richl.eu.
More informationStudents must always use correct mathematical notation, not calculator notation. the set of positive integers and zero, {0,1, 2, 3,...
Appedices Of the vrious ottios i use, the IB hs chose to dopt system of ottio bsed o the recommedtios of the Itertiol Orgiztio for Stdrdiztio (ISO). This ottio is used i the emitio ppers for this course
More informationMathematical Notation Math Calculus & Analytic Geometry I
Mthemticl Nottio Mth - Clculus & Alytic Geometry I Use Wor or WorPerect to recrete the ollowig ocumets. Ech rticle is worth poits shoul e emile to the istructor t jmes@richl.eu. Type your me t the top
More informationSM2H. Unit 2 Polynomials, Exponents, Radicals & Complex Numbers Notes. 3.1 Number Theory
SMH Uit Polyomils, Epoets, Rdicls & Comple Numbers Notes.1 Number Theory .1 Addig, Subtrctig, d Multiplyig Polyomils Notes Moomil: A epressio tht is umber, vrible, or umbers d vribles multiplied together.
More informationDifferentiation Formulas
AP CALCULUS BC Fil Notes Trigoometric Formuls si θ + cos θ = + t θ = sec θ 3 + cot θ = csc θ 4 si( θ ) = siθ 5 cos( θ ) = cosθ 6 t( θ ) = tθ 7 si( A + B) = si Acos B + si B cos A 8 si( A B) = si Acos B
More informationNorthwest High School s Algebra 2
Northwest High School s Algebr Summer Review Pcket 0 DUE August 8, 0 Studet Nme This pcket hs bee desiged to help ou review vrious mthemticl topics tht will be ecessr for our success i Algebr. Istructios:
More informationCITY UNIVERSITY LONDON
CITY UNIVERSITY LONDON Eg (Hos) Degree i Civil Egieerig Eg (Hos) Degree i Civil Egieerig with Surveyig Eg (Hos) Degree i Civil Egieerig with Architecture PART EXAMINATION SOLUTIONS ENGINEERING MATHEMATICS
More information0 otherwise. sin( nx)sin( kx) 0 otherwise. cos( nx) sin( kx) dx 0 for all integers n, k.
. Computtio of Fourier Series I this sectio, we compute the Fourier coefficiets, f ( x) cos( x) b si( x) d b, i the Fourier series To do this, we eed the followig result o the orthogolity of the trigoometric
More informationFOURIER SERIES PART I: DEFINITIONS AND EXAMPLES. To a 2π-periodic function f(x) we will associate a trigonometric series. a n cos(nx) + b n sin(nx),
FOURIER SERIES PART I: DEFINITIONS AND EXAMPLES To -periodic fuctio f() we will ssocite trigoometric series + cos() + b si(), or i terms of the epoetil e i, series of the form c e i. Z For most of the
More informationCalculus Summary Sheet
Clculus Summry Sheet Limits Trigoometric Limits: siθ lim θ 0 θ = 1, lim 1 cosθ = 0 θ 0 θ Squeeze Theorem: If f(x) g(x) h(x) if lim f(x) = lim h(x) = L, the lim g(x) = L x x x Ietermite Forms: 0 0,,, 0,
More informationINTEGRATION TECHNIQUES (TRIG, LOG, EXP FUNCTIONS)
Mthemtics Revisio Guides Itegrtig Trig, Log d Ep Fuctios Pge of MK HOME TUITION Mthemtics Revisio Guides Level: AS / A Level AQA : C Edecel: C OCR: C OCR MEI: C INTEGRATION TECHNIQUES (TRIG, LOG, EXP FUNCTIONS)
More informationContent: Essential Calculus, Early Transcendentals, James Stewart, 2007 Chapter 1: Functions and Limits., in a set B.
Review Sheet: Chpter Cotet: Essetil Clculus, Erly Trscedetls, Jmes Stewrt, 007 Chpter : Fuctios d Limits Cocepts, Defiitios, Lws, Theorems: A fuctio, f, is rule tht ssigs to ech elemet i set A ectly oe
More informationMath 3B Midterm Review
Mth 3B Midterm Review Writte by Victori Kl vtkl@mth.ucsb.edu SH 643u Office Hours: R 11:00 m - 1:00 pm Lst updted /15/015 Here re some short otes o Sectios 7.1-7.8 i your ebook. The best idictio of wht
More informationDefinition Integral. over[ ab, ] the sum of the form. 2. Definite Integral
Defiite Itegrl Defiitio Itegrl. Riem Sum Let f e futio efie over the lose itervl with = < < < = e ritrr prtitio i suitervl. We lle the Riem Sum of the futio f over[, ] the sum of the form ( ξ ) S = f Δ
More informationAdd Maths Formulae List: Form 4 (Update 18/9/08)
Add Mths Formule List: Form 4 (Updte 8/9/08) 0 Fuctios Asolute Vlue Fuctio f ( ) f( ), if f( ) 0 f( ), if f( ) < 0 Iverse Fuctio If y f( ), the Rememer: Oject the vlue of Imge the vlue of y or f() f()
More informationHarold s Calculus Notes Cheat Sheet 15 December 2015
Hrol s Clculus Notes Chet Sheet 5 Decemer 05 AP Clculus Limits Defiitio of Limit Let f e fuctio efie o ope itervl cotiig c let L e rel umer. The sttemet: lim x f(x) = L mes tht for ech ε > 0 there exists
More information334 MATHS SERIES DSE MATHS PREVIEW VERSION B SAMPLE TEST & FULL SOLUTION
MATHS SERIES DSE MATHS PREVIEW VERSION B SAMPLE TEST & FULL SOLUTION TEST SAMPLE TEST III - P APER Questio Distributio INSTRUCTIONS:. Attempt ALL questios.. Uless otherwise specified, ll worig must be
More informationLinford 1. Kyle Linford. Math 211. Honors Project. Theorems to Analyze: Theorem 2.4 The Limit of a Function Involving a Radical (A4)
Liford 1 Kyle Liford Mth 211 Hoors Project Theorems to Alyze: Theorem 2.4 The Limit of Fuctio Ivolvig Rdicl (A4) Theorem 2.8 The Squeeze Theorem (A5) Theorem 2.9 The Limit of Si(x)/x = 1 (p. 85) Theorem
More informationBITSAT MATHEMATICS PAPER. If log 0.0( ) log 0.( ) the elogs to the itervl (, ] () (, ] [,+ ). The poit of itersectio of the lie joiig the poits i j k d i+ j+ k with the ple through the poits i+ j k, i
More informationFACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING. Lectures
FACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING Lectures MODULE 0 FURTHER CALCULUS II. Sequeces d series. Rolle s theorem d me vlue theorems 3. Tlor s d Mcluri s theorems 4. L Hopitl
More information1 Tangent Line Problem
October 9, 018 MAT18 Week Justi Ko 1 Tget Lie Problem Questio: Give the grph of fuctio f, wht is the slope of the curve t the poit, f? Our strteg is to pproimte the slope b limit of sect lies betwee poits,
More informationis continuous at x 2 and g(x) 2. Oil spilled from a ruptured tanker spreads in a circle whose area increases at a
. Cosider two fuctios f () d g () defied o itervl I cotiig. f () is cotiuous t d g() is discotiuous t. Which of the followig is true bout fuctios f g d f g, the sum d the product of f d g, respectively?
More informationSurds, Indices, and Logarithms Radical
MAT 6 Surds, Idices, d Logrithms Rdicl Defiitio of the Rdicl For ll rel, y > 0, d ll itegers > 0, y if d oly if y where is the ide is the rdicl is the rdicd. Surds A umber which c be epressed s frctio
More informationWeek 13 Notes: 1) Riemann Sum. Aim: Compute Area Under a Graph. Suppose we want to find out the area of a graph, like the one on the right:
Week 1 Notes: 1) Riem Sum Aim: Compute Are Uder Grph Suppose we wt to fid out the re of grph, like the oe o the right: We wt to kow the re of the red re. Here re some wys to pproximte the re: We cut the
More informationy udv uv y v du 7.1 INTEGRATION BY PARTS
7. INTEGRATION BY PARTS Ever differetitio rule hs correspodig itegrtio rule. For istce, the Substitutio Rule for itegrtio correspods to the Chi Rule for differetitio. The rule tht correspods to the Product
More informationUnit 1. Extending the Number System. 2 Jordan School District
Uit Etedig the Number System Jord School District Uit Cluster (N.RN. & N.RN.): Etedig Properties of Epoets Cluster : Etedig properties of epoets.. Defie rtiol epoets d eted the properties of iteger epoets
More informationA.P. Calculus Formulas Hanford High School, Richland, Washington revised 8/25/08
A.P. Clls Formls 008-009 Hfor High Shool, Rihl, Wshigto revise 8/5/08. floor ftio (ef) Gretest iteger tht is less th or eql to.. (grph) 3. eilig ftio (ef) Lest iteger tht is greter th or eql to. 4. (grph)
More informationThomas Whitham Sixth Form
Thoms Whithm ith Form Pure Mthemtis Uit lger Trigoometr Geometr lulus lger equees The ifiite sequee of umers U U U... U... is si to e () overget if U L fiite limit s () iverget to if U s Emple The sequee...
More informationHIGHER SCHOOL CERTIFICATE EXAMINATION MATHEMATICS 3 UNIT (ADDITIONAL) AND 3/4 UNIT (COMMON) Time allowed Two hours (Plus 5 minutes reading time)
HIGHER SCHOOL CERTIFICATE EXAMINATION 999 MATHEMATICS 3 UNIT (ADDITIONAL) AND 3/ UNIT (COMMON) Time llowed Two hours (Plus 5 miutes redig time) DIRECTIONS TO CANDIDATES Attempt ALL questios. ALL questios
More informationChapter 7 Infinite Series
MA Ifiite Series Asst.Prof.Dr.Supree Liswdi Chpter 7 Ifiite Series Sectio 7. Sequece A sequece c be thought of s list of umbers writte i defiite order:,,...,,... 2 The umber is clled the first term, 2
More informationA.P. Calculus Formulas. 1. floor function (def) Greatest integer that is less than or equal to x.
A.P. Clls Formls. floor ftio (ef) Gretest iteger tht is less th or eql to.. (grph). eilig ftio (ef) Lest iteger tht is greter th or eql to. 4. (grph) 5. 6. g h g h Pge 7. f ( ) (grph) - - - - - - 8. Chge
More informationIn an algebraic expression of the form (1), like terms are terms with the same power of the variables (in this case
Chpter : Algebr: A. Bckgroud lgebr: A. Like ters: I lgebric expressio of the for: () x b y c z x y o z d x... p x.. we cosider x, y, z to be vribles d, b, c, d,,, o,.. to be costts. I lgebric expressio
More informationQn Suggested Solution Marking Scheme 1 y. G1 Shape with at least 2 [2]
Mrkig Scheme for HCI 8 Prelim Pper Q Suggested Solutio Mrkig Scheme y G Shpe with t lest [] fetures correct y = f'( ) G ll fetures correct SR: The mimum poit could be i the first or secod qudrt. -itercept
More informationGraphing Review Part 3: Polynomials
Grphig Review Prt : Polomils Prbols Recll, tht the grph of f ( ) is prbol. It is eve fuctio, hece it is smmetric bout the bout the -is. This mes tht f ( ) f ( ). Its grph is show below. The poit ( 0,0)
More informationSCHAUM'S outlines. Mathematical Handbook of Formulas and Tables
SCHAUM'S outlies Mthemticl Hdbook of Formuls d Tbles This pge itetiolly left blk SCHAUM'S outlies Mthemticl Hdbook of Formuls d Tbles Third Editio Murry R. Spiegel, PhD Former Professor d Chirm Mthemtics
More information( ) dx ; f ( x ) is height and Δx is
Mth : 6.3 Defiite Itegrls from Riem Sums We just sw tht the exct re ouded y cotiuous fuctio f d the x xis o the itervl x, ws give s A = lim A exct RAM, where is the umer of rectgles i the Rectgulr Approximtio
More information(200 terms) equals Let f (x) = 1 + x + x 2 + +x 100 = x101 1
SECTION 5. PGE 78.. DMS: CLCULUS.... 5. 6. CHPTE 5. Sectio 5. pge 78 i + + + INTEGTION Sums d Sigm Nottio j j + + + + + i + + + + i j i i + + + j j + 5 + + j + + 9 + + 7. 5 + 6 + 7 + 8 + 9 9 i i5 8. +
More information[Q. Booklet Number]
6 [Q. Booklet Numer] KOLKATA WB- B-J J E E - 9 MATHEMATICS QUESTIONS & ANSWERS. If C is the reflecto of A (, ) i -is d B is the reflectio of C i y-is, the AB is As : Hits : A (,); C (, ) ; B (, ) y A (,
More informationCalculus Definitions, Theorems
Sectio 1.1 A Itrouctio To Limits Clculus Defiitios, Theorems f(c + ) f(c) m sec = = c + - c f(c + ) f(c) Sectio 1. Properties of Limits Theorem Some Bsic Limits Let b c be rel umbers let be positive iteger.
More information[ 20 ] 1. Inequality exists only between two real numbers (not complex numbers). 2. If a be any real number then one and only one of there hold.
[ 0 ]. Iequlity eists oly betwee two rel umbers (ot comple umbers).. If be y rel umber the oe d oly oe of there hold.. If, b 0 the b 0, b 0.. (i) b if b 0 (ii) (iii) (iv) b if b b if either b or b b if
More informationCalculus II Homework: The Integral Test and Estimation of Sums Page 1
Clculus II Homework: The Itegrl Test d Estimtio of Sums Pge Questios Emple (The p series) Get upper d lower bouds o the sum for the p series i= /ip with p = 2 if the th prtil sum is used to estimte the
More informationChapter Real Numbers
Chpter. - Rel Numbers Itegers: coutig umbers, zero, d the egtive of the coutig umbers. ex: {,-3, -, -,,,, 3, } Rtiol Numbers: quotiets of two itegers with ozero deomitor; termitig or repetig decimls. ex:
More informationAdvanced Higher Grade
Prelim Emitio / (Assessig Uits & ) MATHEMATICS Avce Higher Gre Time llowe - hors Re Crefll. Fll creit will be give ol where the soltio cotis pproprite workig.. Clcltors m be se i this pper.. Aswers obtie
More informationA GENERAL METHOD FOR SOLVING ORDINARY DIFFERENTIAL EQUATIONS: THE FROBENIUS (OR SERIES) METHOD
Diol Bgoo () A GENERAL METHOD FOR SOLVING ORDINARY DIFFERENTIAL EQUATIONS: THE FROBENIUS (OR SERIES) METHOD I. Itroductio The first seprtio of vribles (see pplictios to Newto s equtios) is ver useful method
More information82A Engineering Mathematics
Clss Notes 9: Power Series /) 8A Egieerig Mthetics Secod Order Differetil Equtios Series Solutio Solutio Ato Differetil Equtio =, Hoogeeous =gt), No-hoogeeous Solutio: = c + p Hoogeeous No-hoogeeous Fudetl
More information1.3 Continuous Functions and Riemann Sums
mth riem sums, prt 0 Cotiuous Fuctios d Riem Sums I Exmple we sw tht lim Lower() = lim Upper() for the fuctio!! f (x) = + x o [0, ] This is o ccidet It is exmple of the followig theorem THEOREM Let f be
More informationLimit of a function:
- Limit of fuctio: We sy tht f ( ) eists d is equl with (rel) umer L if f( ) gets s close s we wt to L if is close eough to (This defiitio c e geerlized for L y syig tht f( ) ecomes s lrge (or s lrge egtive
More informationMathematics Extension 2
04 Bored of Studies Tril Emitios Mthemtics Etesio Writte b Crrotsticks & Trebl Geerl Istructios Totl Mrks 00 Redig time 5 miutes. Workig time 3 hours. Write usig blck or blue pe. Blck pe is preferred.
More informationLEVEL I. ,... if it is known that a 1
LEVEL I Fid the sum of first terms of the AP, if it is kow tht + 5 + 0 + 5 + 0 + = 5 The iterior gles of polygo re i rithmetic progressio The smllest gle is 0 d the commo differece is 5 Fid the umber of
More informationTaylor Polynomials. The Tangent Line. (a, f (a)) and has the same slope as the curve y = f (x) at that point. It is the best
Tylor Polyomils Let f () = e d let p() = 1 + + 1 + 1 6 3 Without usig clcultor, evlute f (1) d p(1) Ok, I m still witig With little effort it is possible to evlute p(1) = 1 + 1 + 1 (144) + 6 1 (178) =
More informationThe Exponential Function
The Epoetil Fuctio Defiitio: A epoetil fuctio with bse is defied s P for some costt P where 0 d. The most frequetly used bse for epoetil fuctio is the fmous umber e.788... E.: It hs bee foud tht oyge cosumptio
More informationApproximations of Definite Integrals
Approximtios of Defiite Itegrls So fr we hve relied o tiderivtives to evlute res uder curves, work doe by vrible force, volumes of revolutio, etc. More precisely, wheever we hve hd to evlute defiite itegrl
More informationName: A2RCC Midterm Review Unit 1: Functions and Relations Know your parent functions!
Nme: ARCC Midterm Review Uit 1: Fuctios d Reltios Kow your pret fuctios! 1. The ccompyig grph shows the mout of rdio-ctivity over time. Defiitio of fuctio. Defiitio of 1-1. Which digrm represets oe-to-oe
More informationUNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION. (2014 Admn. onwards) III Semester. B.Sc. Mathematics CORE COURSE CALCULUS AND ANALYTICAL GEOMETRY
UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION (0 Adm. owrds) III Semester B.Sc. Mthemtics CORE COURSE CALCULUS AND ANALYTICAL GEOMETRY Questio Bk & Aswer Key. l l () =... 0.00 b) 0 c). l d =... c
More informationEXERCISE a a a 5. + a 15 NEETIIT.COM
- Dowlod our droid App. Sigle choice Type Questios EXERCISE -. The first term of A.P. of cosecutive iteger is p +. The sum of (p + ) terms of this series c be expressed s () (p + ) () (p + ) (p + ) ()
More informationB. Examples 1. Finite Sums finite sums are an example of Riemann Sums in which each subinterval has the same length and the same x i
Mth 06 Clculus Sec. 5.: The Defiite Itegrl I. Riem Sums A. Def : Give y=f(x):. Let f e defied o closed itervl[,].. Prtitio [,] ito suitervls[x (i-),x i ] of legth Δx i = x i -x (i-). Let P deote the prtitio
More informationMA123, Chapter 9: Computing some integrals (pp )
MA13, Chpter 9: Computig some itegrls (pp. 189-05) Dte: Chpter Gols: Uderstd how to use bsic summtio formuls to evlute more complex sums. Uderstd how to compute its of rtiol fuctios t ifiity. Uderstd how
More informationPOWER SERIES R. E. SHOWALTER
POWER SERIES R. E. SHOWALTER. sequeces We deote by lim = tht the limit of the sequece { } is the umber. By this we me tht for y ε > 0 there is iteger N such tht < ε for ll itegers N. This mkes precise
More informationREVISION SHEET FP1 (AQA) ALGEBRA. E.g., if 2x
The mi ides re: The reltioships betwee roots d coefficiets i polyomil (qudrtic) equtios Fidig polyomil equtios with roots relted to tht of give oe the Further Mthemtics etwork wwwfmetworkorguk V 7 REVISION
More information1. (25 points) Use the limit definition of the definite integral and the sum formulas to compute. [1 x + x2
Mth 3, Clculus II Fil Exm Solutios. (5 poits) Use the limit defiitio of the defiite itegrl d the sum formuls to compute 3 x + x. Check your swer by usig the Fudmetl Theorem of Clculus. Solutio: The limit
More informationTheorem 5.3 (Continued) The Fundamental Theorem of Calculus, Part 2: ab,, then. f x dx F x F b F a. a a. f x dx= f x x
Chpter 6 Applictios Itegrtio Sectio 6. Regio Betwee Curves Recll: Theorem 5.3 (Cotiued) The Fudmetl Theorem of Clculus, Prt :,,, the If f is cotiuous t ever poit of [ ] d F is tiderivtive of f o [ ] (
More informationStudent Success Center Elementary Algebra Study Guide for the ACCUPLACER (CPT)
Studet Success Ceter Elemetry Algebr Study Guide for the ACCUPLACER (CPT) The followig smple questios re similr to the formt d cotet of questios o the Accuplcer Elemetry Algebr test. Reviewig these smples
More informationALGEBRA. Set of Equations. have no solution 1 b1. Dependent system has infinitely many solutions
Qudrtic Equtios ALGEBRA Remider theorem: If f() is divided b( ), the remider is f(). Fctor theorem: If ( ) is fctor of f(), the f() = 0. Ivolutio d Evlutio ( + b) = + b + b ( b) = + b b ( + b) 3 = 3 +
More informationMathematical Notations and Symbols xi. Contents: 1. Symbols. 2. Functions. 3. Set Notations. 4. Vectors and Matrices. 5. Constants and Numbers
Mthemticl Nottios d Symbols i MATHEMATICAL NOTATIONS AND SYMBOLS Cotets:. Symbols. Fuctios 3. Set Nottios 4. Vectors d Mtrices 5. Costts d Numbers ii Mthemticl Nottios d Symbols SYMBOLS = {,,3,... } set
More information n. A Very Interesting Example + + = d. + x3. + 5x4. math 131 power series, part ii 7. One of the first power series we examined was. 2!
mth power series, prt ii 7 A Very Iterestig Emple Oe of the first power series we emied ws! + +! + + +!! + I Emple 58 we used the rtio test to show tht the itervl of covergece ws (, ) Sice the series coverges
More informationTime: 2 hours IIT-JEE 2006-MA-1. Section A (Single Option Correct) + is (A) 0 (B) 1 (C) 1 (D) 2. lim (sin x) + x 0. = 1 (using L Hospital s rule).
IIT-JEE 6-MA- FIITJEE Solutios to IITJEE 6 Mthemtics Time: hours Note: Questio umber to crries (, -) mrks ech, to crries (5, -) mrks ech, to crries (5, -) mrks ech d to crries (6, ) mrks ech.. For >, lim
More informationApproximate Integration
Study Sheet (7.7) Approimte Itegrtio I this sectio, we will ler: How to fid pproimte vlues of defiite itegrls. There re two situtios i which it is impossile to fid the ect vlue of defiite itegrl. Situtio:
More informationUNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION
School Of Distce Eductio Questio Bk UNIVERSITY OF ALIUT SHOOL OF DISTANE EDUATION B.Sc MATHEMATIS (ORE OURSE SIXTH SEMESTER ( Admissio OMPLEX ANALYSIS Module- I ( A lytic fuctio with costt modulus is :
More informationMTH 146 Class 16 Notes
MTH 46 Clss 6 Notes 0.4- Cotiued Motivtio: We ow cosider the rc legth of polr curve. Suppose we wish to fid the legth of polr curve curve i terms of prmetric equtios s: r f where b. We c view the cos si
More information0 dx C. k dx kx C. e dx e C. u C. sec u. sec. u u 1. Ch 05 Summary Sheet Basic Integration Rules = Antiderivatives Pattern Recognition Related Rules:
Ch 05 Smmry Sheet Bic Itegrtio Rle = Atierivtive Ptter Recogitio Relte Rle: ( ) kf ( ) kf ( ) k f ( ) Cott oly! ( ) g ( ) f ( ) g ( ) f ( ) g( ) f ( ) g( ) Bic Atierivtive: C 0 0 C k k k k C 1 1 1 Mlt.
More informationSet 1 Paper 2. 1 Pearson Education Asia Limited 2014
. C. A. C. B 5. C 6. A 7. D 8. B 9. C 0. C. D. B. A. B 5. C 6. C 7. C 8. B 9. C 0. A. A. C. B. A 5. C 6. C 7. B 8. D 9. B 0. C. B. B. D. D 5. D 6. C 7. B 8. B 9. A 0. D. D. B. A. C 5. C Set Pper Set Pper
More information* power rule: * fraction raised to negative exponent: * expanded power rule:
Mth 15 Iteredite Alger Stud Guide for E 3 (Chpters 7, 8, d 9) You use 3 5 ote crd (oth sides) d scietific clcultor. You re epected to kow (or hve writte o our ote crd) foruls ou eed. Thik out rules d procedures
More informationINTEGRAL SOLUTIONS OF THE TERNARY CUBIC EQUATION
Itertiol Reserch Jourl of Egieerig d Techology IRJET) e-issn: 9-006 Volume: 04 Issue: Mr -017 www.irjet.et p-issn: 9-007 INTEGRL SOLUTIONS OF THE TERNRY CUBIC EQUTION y ) 4y y ) 97z G.Jki 1, C.Sry,* ssistt
More informationName of the Student:
Egieerig Mthemtics 5 NAME OF THE SUBJECT : Mthemtics I SUBJECT CODE : MA65 MATERIAL NAME : Additiol Prolems MATERIAL CODE : HGAUM REGULATION : R UPDATED ON : M-Jue 5 (Sc the ove QR code for the direct
More informationTHE NATIONAL UNIVERSITY OF IRELAND, CORK COLÁISTE NA hollscoile, CORCAIGH UNIVERSITY COLLEGE, CORK SUMMER EXAMINATION 2005 FIRST ENGINEERING
OLLSCOIL NA héireann, CORCAIGH THE NATIONAL UNIVERSITY OF IRELAND, CORK COLÁISTE NA hollscoile, CORCAIGH UNIVERSITY COLLEGE, CORK SUMMER EXAMINATION 2005 FIRST ENGINEERING MATHEMATICS MA008 Clculus d Lier
More informationBC Calculus Review Sheet
BC Clculus Review Sheet Whe you see the words. 1. Fid the re of the ubouded regio represeted by the itegrl (sometimes 1 f ( ) clled horizotl improper itegrl). This is wht you thik of doig.... Fid the re
More informationMath 153: Lecture Notes For Chapter 1
Mth : Lecture Notes For Chpter Sectio.: Rel Nubers Additio d subtrctios : Se Sigs: Add Eples: = - - = - Diff. Sigs: Subtrct d put the sig of the uber with lrger bsolute vlue Eples: - = - = - Multiplictio
More informationLimits and an Introduction to Calculus
Nme Chpter Limits d Itroductio to Clculus Sectio. Itroductio to Limits Objective: I this lesso ou lered how to estimte limits d use properties d opertios of limits. I. The Limit Cocept d Defiitio of Limit
More informationInfinite Series Sequences: terms nth term Listing Terms of a Sequence 2 n recursively defined n+1 Pattern Recognition for Sequences Ex:
Ifiite Series Sequeces: A sequece i defied s fuctio whose domi is the set of positive itegers. Usully it s esier to deote sequece i subscript form rther th fuctio ottio.,, 3, re the terms of the sequece
More informationf(bx) dx = f dx = dx l dx f(0) log b x a + l log b a 2ɛ log b a.
Eercise 5 For y < A < B, we hve B A f fb B d = = A B A f d f d For y ɛ >, there re N > δ >, such tht d The for y < A < δ d B > N, we hve ba f d f A bb f d l By ba A A B A bb ba fb d f d = ba < m{, b}δ
More informationMTH112 Trigonometry 2 2 2, 2. 5π 6. cscθ = 1 sinθ = r y. secθ = 1 cosθ = r x. cotθ = 1 tanθ = cosθ. central angle time. = θ t.
MTH Trigoometry,, 5, 50 5 0 y 90 0, 5 0,, 80 0 0 0 (, 0) x, 7, 0 5 5 0, 00 5 5 0 7,,, Defiitios: siθ = opp. hyp. = y r cosθ = adj. hyp. = x r taθ = opp. adj. = siθ cosθ = y x cscθ = siθ = r y secθ = cosθ
More informationGRAPHING LINEAR EQUATIONS. Linear Equations. x l ( 3,1 ) _x-axis. Origin ( 0, 0 ) Slope = change in y change in x. Equation for l 1.
GRAPHING LINEAR EQUATIONS Qudrt II Qudrt I ORDERED PAIR: The first umer i the ordered pir is the -coordite d the secod umer i the ordered pir is the y-coordite. (, ) Origi ( 0, 0 ) _-is Lier Equtios Qudrt
More informationChapter Real Numbers
Chpter. - Rel Numbers Itegers: coutig umbers, zero, d the egtive of the coutig umbers. ex: {,-3, -, -, 0,,, 3, } Rtiol Numbers: quotiets of two itegers with ozero deomitor; termitig or repetig decimls.
More informationSet 3 Paper 2. Set 3 Paper 2. 1 Pearson Education Asia Limited 2017
Set Pper Set Pper. D. A.. D. 6. 7. B 8. D 9. B 0. A. B. D. B.. B 6. B 7. D 8. A 9. B 0. A. D. B.. A. 6. A 7. 8. 9. B 0. D.. A. D. D. A 6. 7. A 8. B 9. D 0. D. A. B.. A. D Sectio A. D ( ) 6. A b b b ( b)
More informationdenominator, think trig! Memorize the following two formulas; you will use them often!
7. Bsic Itegrtio Rules Some itegrls re esier to evlute th others. The three problems give i Emple, for istce, hve very similr itegrds. I fct, they oly differ by the power of i the umertor. Eve smll chges
More informationBC Calculus Path to a Five Problems
BC Clculus Pth to Five Problems # Topic Completed U -Substitutio Rule Itegrtio by Prts 3 Prtil Frctios 4 Improper Itegrls 5 Arc Legth 6 Euler s Method 7 Logistic Growth 8 Vectors & Prmetrics 9 Polr Grphig
More informationSolutions to Problem Set 7
8.0 Clculus Jso Strr Due by :00pm shrp Fll 005 Fridy, Dec., 005 Solutios to Problem Set 7 Lte homework policy. Lte work will be ccepted oly with medicl ote or for other Istitute pproved reso. Coopertio
More informationObjective Mathematics
. o o o o {cos 4 cos 9 si cos 65 } si 7º () cos 6º si 8º. If x R oe of these, the mximum vlue of the expressio si x si x.cos x c cos x ( c) is : () c c c c c c. If ( cos )cos cos ; 0, the vlue of 4. The
More information