Mathematics for Engineers Part II (ISE) Version 1.1/

Size: px
Start display at page:

Download "Mathematics for Engineers Part II (ISE) Version 1.1/"

Transcription

1 Mthemtics or Egieers Prt II (ISE Versio /4-6- Curves i Prmetric descriptio o curves We exted the theory o derivtives d itegrls to uctios whose rge re vectors i isted o rel umers Deiitio : A curve C i is the grph o uctio : I rom itervl I [;] ito such tht ech t i I hs the imge ( ( t ( ( t, ( t,, ( t The compoets i : I IR,t (t, i,,,, i o ( t ( ( t, ( t,, ( t re cotiuous uctios o the itervl I Equtio ( is clled the prmetric equtio o the curve C The poits ( d ( re clled the edpoits o the curve Remrk : I we idetiy ech poit ( (t ( (t, (t, (t [ (t, (t, (t] ( ( t ( ( t, ( t,, ( t with its positio vector [ ( t, ( t,, ( t ], the we c uderstd s vectorvlued uctio Fig Prmetric curve i Exmple : (i Suppose r > is rel umer, the the grph o the curve C give y the prmetric equtio : [ ;] IR, t ( r cost,r sit is circle i o rdius r (Fig (ii Let : IR IR,t (r cost,r sit, ct, r >, c, e the prmetric equtio o curve i The grph o the curve trced y (t s t vries is clled circulr helix (Fig -

2 Mthemtics or Egieers Prt II (ISE Versio /4-6- (iii Suppose g: I is rel vlued cotiuous uctio o the itervl I The grph o this uctio g is suset o d c e uderstood s curve i with the prmetric equtio ( : IR,t ( t,g( t I Fig Circle i Deiitio : Let I e ope itervl d (,,, : I IR The curve C give y is clled dieretile i ech compoet i : I IR,t (t, i,,,, i is dieretile t t I The vector ( t [ ( t, ( t,, ( t ] is clled tget vector o the curve t the poit (t Fig circulr helix Remrk : (i I ( t [,,], the uit tget vector to is give y the ormul ( T ( t ( t ( t [ ( t,, ( t ] ( ( t + + ( ( t (ii A curve C give y : I is ot ecessrily the grph o ijective uctio I, or t t, ( t ( t d (d,d,,d, the d is poit o itersectio o C At poit o itersectio there re i geerl two dieret tget vectors For exmple, the curve C deied y : with the prmetric equtio t t,t t (Fig 4 hs poit o itersectio t d (,, sice ( ( ( (- (, Fig 4 Poit o itersectio d - The tget vector t (t is give y ( t [ t, t ] Thus ( [,] ( [,] -

3 Mthemtics or Egieers Prt II (ISE Versio /4-6- Deiitio : Suppose : I is cotiuously dieretile prmeteriztio o curve C The curve C is clled regulr i ( t (,, or ll t I A prmeter vlue t I or t,, is clled sigulr which ( ( Exmple : We cosider Neil s prol give y : IR ( t,t IR, t The grph o the curve give prmetriclly y is the set { IR } x,y x / ( IR ( x,y ±, we get ( (, Thus the oly sigulr poit o Neil s prol is t t Sice ( t ( t, t Fig 5 Neil s prol Deiitio 4: Suppose : I d g: I re cotiuously dieretile prmeteriztios o the regulr curves C d C g I ( t g( t or some t I, the the gle o itersectio ϑ etwee C d C g is deied s t I (4 ( t g ( t ( t g ( t cosϑ, ϑ π Remrk : The gle o itersectio ϑ etwee C d C g is the gle etwee the two tget vectors t the poit o itersectio Fig 6 Agle o itersectio -

4 Mthemtics or Egieers Prt II (ISE Versio /4-6- Legth o curves Suppose [;] is itervl d C give y :[;] is smooth (ie cotiuously dieretile curve which does ot itersect itsel, tht is, dieret vlues o t [;] determie dieret poits o the grph o Cosider prtitio o [;] give y Let ti ti ti d let ( t < t < t < < t < t t i e the poit o the grph o determied y t i I ( t ( is the legth o the lie segmet ( t ( t i t i roke lie show i Fig is, the the legth L p o the i i ( t ( t k k L p ( t,,t ( tk ( tk tk k k tk Thereore we get or the legth o curve: L ( t,,t ( t dt lim Lp Propositio : I smooth curve C is give prmetriclly y Fig Lie segmets :[;], ( t ( ( t, ( t,, ( t d i C does ot itersect itsel, except possily t the edpoits, the the legth L o C is ( L (t dt Note! ( t ( ( t + + ( ( t Exmple : (i Suppose s > is rel umer We cosider the circulr rc give y : [,s] IR, ( t ( cost,sit, Sice ( t ( sit,cost we get ( t si t+ cos t Hece the rc legth is give y -4

5 Mthemtics or Egieers Prt II (ISE Versio /4-6- s ( t dt dt s L I prticulr the circumerece o the uit circle equls π s (ii We cosider the cycloid give prmetriclly y ( t ( t sit, cost : IR IR, The cycloid is the locus o poit o the rim o circle o rdius r rollig log stright lie (Fig It ws studied d med y Glileo i 599 Fig cycloid We compute the legth L o the prt o the cycloid elogig to the prmeter vlues t π, tht is the rc ABC i Fig Sice ( t ( cost,sit we get y the dditio theorems: Fig ( t ( cost cost 4 si t + si t Thereore ( t Hece we get: t t si si or t π t L si dt 4 sixdx 8 Remrkle: The legth o the rc ABC is iteger -5

6 Mthemtics or Egieers Prt II (ISE Versio /4-6- Remrk : There re iiitely my possile prmeteriztios o give curve C Cosider the prt o the uit prol prmetriclly give y :[;], ( t ( t,t (Fig 4 For exmple, ll uctios o the type t t [ ] ( α : ; α IR,α t,, α >, α α Fig 4 Uit prol hve the sme grph Tht is, i you oly look t the poits o the grph i you cot decide which prmeteriztio is resposile or this curve Now rises importt questio: Does the legth o curve C deped o the prmetric equtio? The swer is: O course ot! We will prove this i the ollowig Deiitio :, is dieretile strictly mootoe icresig uctio; ϑ, the we cll ϑ vlid trsormtio I ϑ:[α;β] [;], t ϑ( t ie ( t > Propositio : The legth o curve C is prmeteriztio-ivrit or vlid trsormtios Tht is: I : [;], ( τ ( ( τ, ( τ,, ( τ, is prmeteriztio o curve C d ϑ:[α;β] [;], t ϑ ( t τ, is vlid trsormtio, the ϑ : [ ; ], ( t ( ϑ( t ( ( ϑ( t, ( ϑ( t,, ( ( t is prmeteriztio o C d ϑ ϑ L C ( d t ( τ ϑ t dτ Proo: It is esy to see y the chi rule tht ( t ( ( ϑ( t ( ( ϑ( t ϑ ( t,, ( ϑ( t ( t ϑ ϑ ϑ : Thereore, sice ( t > ( t ( ( ϑ( t ( ϑ ( t + + ( ( ϑ( t ( ϑ ( t ( ϑ( t ( t ϑ ϑ -6

7 Mthemtics or Egieers Prt II (ISE Versio /4-6- We c coclude y direct sustitutio: L C ( ( ( ϑ t dt ϑ t ϑ ( t dt ( τ dτ Motio I this sectio we itroduce cocepts tht re eeded to study the motio o poit or prticle i To lyze the ehviour o poit, it is ecessry to kow its positio ( x ( t,,x ( t t every time t I we set the time depedig positio ( ( t ( ( t, ( t,, ( t, the s t vries, the edpoit o the positio vector [ ( t, ( t,, ( t ] trces the pth C o the prticle From the precedig sectio, the tget lie to C t the poit P t (t is prllel to ( ( t [ ( t, ( t,, ( t ] d the mgitude o this vector is ( ( t ( ( t + + ( ( t Suppose C is smooth curve i prmetriclly give y :[t ;t ], ( t ( ( t, ( t,, ( t with the strtig poit P correspodig to t t The the rc legth s σ(t o C rom P to P t (t is σ d (4 ( t ( t t The rc legth s σ(t is rel vlued uctio d its esy to see tht σ is σ t t > or regulr curve (this is cosequece o the dieretile with ( ( me vlue theorem or itegrls I other words: ( t is the rte o chge o rc legth with respect to time For this reso we reer to ( t s the speed o the prticle The tget vector ( t, whose mgitude is the speed, is deied s the -7

8 Mthemtics or Egieers Prt II (ISE Versio /4-6- velocity o the prticle t the time t The vector ( t prticle t the time t is clled the ccelertio o the Remrk : (i Velocity d ccelertio deped o prmeteriztio I ϑ:[α;β] [;] is vlid trsormtio d ϑ ( t ( ϑ( t the ( t ( ϑ( t ϑ ( t ϑ Note: Prmeteriztio determies the speed o the prticle, ut ot the shpe o the curve (ii The rc legth σ d, σ:[;] [;L], ( t ( t is dieretile strictly mootoe icresig uctio Thus the iverse uctio σ - : [;L] [;], σ - (s t, is lso dieretile d strictly mootoe icresig Thereore σ - is vlid trsormtio Exmple : (i For the helix i Exmple (ii prmetriclly give y : [ ; ] IR, t (r cost,r sit, ct, r, >, we get ( t ( rsit,rcost,c ( t r c + Thereore the rc legth s is give y t ( t r + c d t r c s + Hece t r s + c, d ormul or the helix with the rc legth s s prmeter is : [; r + c ], ( s s s cs ϕ r cos,r si, r + c r + c r + c -8

9 Mthemtics or Egieers Prt II (ISE Versio /4-6- (ii The positio o prticle movig i the ple is give y r : [;], ( t ( t,t r The velocity d the ccelertio t the time t re ( t [, t] d r ( t [,] r I prticulr, t t ( [,] d r ( [,] r Fig These vectors d the pth o the prticle re sketched i Figure The speed o the prticle t the poit P(, is ( r Suppose α > is rel umer, the prmeteriztio o the sme curve i Figure is give y t 4 t γ: [;α], ( t, I this cse the trsormtio is deied y ϑ: [;α] [;], ϑ(t t/α The velocity d the ccelertio t the time t re ow 8 t ( t, d ( t, 8 γ Usig this prmeteriztio, the prticle reches the poit P(, ter t α/ Hece the velocity d ccelertio t P(, re 4 8, d γ, Thereore the speed t the poit P(, is ow

10 Mthemtics or Egieers Prt II (ISE Versio / Curvture For my pplictios ivolvig vector-vlued uctios it is coveiet to employ uit tget vectors to curves We cosider curves i prmetriclly give y (4 ( t ( ( t, ( t,, ( t From previous sectios we kow tht ( t [ ( t, ( t,, ( t ] t the poit P t ( ( t, ( t,, ( t I ( t [,,,] is give y (4 ( t ( t ( t T is tget vector to C the uit tget vector to C Propositio 4: Suppose smooth curve C is give y : [;], ( t ( ( t, ( t,, ( t I ( t is costt, the the vector ( t [ ( t, ( t,, ( t ] vector o ( t or every t [;] Proo: By hypothesis or rel umer c Thereore d dt is orthogol to the positio ( t ( t ( t c ( ( t ( t ( t ( t + ( t ( t ( t ( t Hece ( t ( t, tht is ( t d ( t re orthogol Sice i (4 T ( t, it ollows with Propositio 4 tht ( t T ( t or every t We deie (4 N ( t T T which gives uit vector orthogol to the uit tget vector T ( t ( t ( t T is orthogol to Deiitio 4: is clled the uit orml vector to the curve C with The vector N ( t T ( t T ( t prmeteriztio : [; ], ( t ( ( t, ( t,, ( t, ( t (,,, -

11 Mthemtics or Egieers Prt II (ISE Versio /4-6- simpliyig leds to N ( T ( Fig 4 Uit tget d uit orml vector Exmple 4: Let C e the curve prmetriclly give y ( t ( t,t, t 4 t + d ( t (, t It is esy to veriy / ( 4 t + T t Applyig (4 d 4 t + The T ( t ( t, T tht ( ( 4 t + Note: ( t N ( t ( t t N T 4 t + ( t (, t Thus T (t d N (t re, ideed, orthogol As we metioed eore there re iiitely my wys to represet curve prmetriclly Sometimes it is coveiet to use rc legth s prmeter Suppose ϕ is the rc legth prmeteriztio o curve C with give ϕ s g σ s g t pplies: ( ( prmeteriztio g Sice ( ( d d ϕ ds ds (44 ( s g ( t σ ( s ( s Now ( s d thus ( s ds d ( ( ( s ( s d ds : Thereore we get (rememer ( t g ( t (45 ( s Hece with (44 ds d ( t g ( t d ds (46 ϕ ( s ϕ( s g g ( t ( t, s σ(t Thus ϕ ( s d thereore ϕ ( s [ ϕ ( s, ϕ ( s,, ( s ] is uit tget vector ϕ -

12 Mthemtics or Egieers Prt II (ISE Versio /4-6- Note: Prmeter trsormtio to rc legth prmeter leds to prmeteriztio o curve with costt speed ϕ ( s Deiitio 4: Let C e curve i give y ( s ( ϕ ( s, ϕ ( s,, ϕ ( s prmeter Let T ( s ϕ ( s poit P s ( ϕ ( s, ϕ ( s,, ( s is deied s ϕ (47 K ( s T ( s ϕ where s is the rc legth e the uit tget vector The curvture K(s t the We give ow ormuls tht c e used to id the curvture o curves i d o spce curves i Propositio 4: Let C e curve i give prmetriclly y ( t ( ( t, ( t, where the compoet uctios d re two times cotiuously dieretile The the curvture K(t t P t ( ( t, ( t is (48 K( t ( t ( t ( t ( t ( ( t + ( ( t I prticulr the grph o twice cotiuously dieretile uctio : I hs the curvture (49 K( x ( x + ( ( x Proo: Suppose ϕ is trsormtio o to rc legth prmeter; ie: ( ( s ( s ϕ With (44 d (45 pplies: (4 ( ( t ϕ s, ( t ( t, d ϕ ( s ( t Sice ( t ( k t k ( t ollows: ( t k ( t ( ( t ( t ( ( s + -

13 Mthemtics or Egieers Prt II (ISE Versio /4-6- (4 ( ( s ( t ( ( ( ( ( ( ( k t 4 k t s t t k Thereore we get with (4 d (4: (4 ( s Now or : (4 ϕ ( Hece ϕ ( t ( t ( t kk 4 k ( t ( t ( ( t ( t ( t ( t [ ( t, ( t ] s 4 (44 K( t ϕ ( ( t ( t 4 ( t ( t ( t ( t ( t ( t ( t ( t ( t ( t Remrk 4: I the curvture K t poit P o curve C is ot zero, the the circle o rdius /K whose cetre lies o the cocve side o C d which hs the sme tget lie t P s C is clled the circle o curvture or P Exmple 4: Let C e the curve prmetriclly give y ( t ( t, t K ( t, t The the curvture is t 6 t t 6 t / 4 [( t + ( t ] / ( 4 t + 9t I t /, the P, 4 8 K 96 5 The poit correspodig to t / hs coordites d the rdius o curvture ρ t tht poit is 5 ρ K / ( 96 Propositio 4: Let curve C e give y ( t ( ( t, ( t, ( t, where the compoet uctios re two times cotiuously dieretile The curvture K(t t the poit P ( ( t, ( t, ( t o C is -

14 Mthemtics or Egieers Prt II (ISE Versio /4-6- (48 K( t ( t ( t ( t Exmple 4: We compute the curvture o the twisted cuic x t, y t, z t t the poit P(t,t,t t t,t, t the we get: I we set ( ( K 4 ( ( 9 t + 9 t + t 4 ( + 4 t + 9 t / / Note: Formul (48 is oly vlid i, sice we use the vector product -4

15 Mthemtics or Egieers Prt II (ISE Versio / Ple curves d polr coordites I curve i is ot give i prmetric represettio, the polr coordites my e very helpul (5 Polr coordites: The rectgulr coordites (x,y d polr coordites (r,ϕ o poit P(x,y re relted s ollows: (i x r cos ϕ, y r siϕ (ii y t ϕ, x r + x y, x rccos r π rccos ϕ x r udeied, i,y,y < r y The prmetric equtio i polr coordites o curve i is give y (5 ( ϕ r ( ϕ( cosϕ, siϕ, r P(x,y with the polr gle ϕ, α ϕ β, s prmeter d r rel vlued uctio o ϕ ϕ Fig 5 Polr coordites Fig 5 Descrtes le x Exmple 5: The set o the solutios o the equtio (5 x + y xy i two ukows determies curve i kow s Descrtes le (Fig 5 We use polr coordites to id prmetric represettio or the curve deied y equtio (5 By settig x r cosϕ d y r siϕ we get: r d thereore ϕ + r ϕ r ϕ ϕ cos si cos si cosϕ siϕ r ( ϕ cos ϕ + si ϕ -5

16 Mthemtics or Egieers Prt II (ISE Versio /4-6- Hece prmetric equtio o Descrtes le is give y cosϕ siϕ, ϕ π cos ϕ + si ϕ (54 ( ϕ ( cosϕ, siϕ Asymptote o the curve is the lie y -x - It is lso possile to get prmetric equtio or Descrtes le which is ot i polr orm; mely: (55 g ( t, + + t t t t, t \{-} I Figure 5 ud 54 you c see two spshots rom the geertio o the curve Propositio 5: Suppose C is curve i give y prmetric equtio i polr coordites; ie: ( ϕ r ( ϕ( cosϕ, siϕ, α ϕ β The the rc legth L o C is Fig 5 Grph o g(t, -8 t (56 L ( r ( + ( r ( ϕ ϕ d ϕ Proo: Sice ( ϕ ( r ( ϕ cosϕ r ( ϕ siϕ,r ( ϕ siϕ r ( ϕ cosϕ + it is esy to veriy tht ( ϕ ( r ( ϕ + ( r ( ϕ Fig 54 Grph o g(t, - t d thus (56 ollows immeditely Exmple 5: We cosider the curve C give y the prmetric equtio i polr coordites ϕ ϕ ϕ ϕ ϕ (57 ( ( cos,si, >, < This curve is clled the Archimede spirl (Figure 55 d 56-6

17 Mthemtics or Egieers Prt II (ISE Versio /4-6- The rc legth ter oe circultio is π L ϕ + d ϕ Fig 55 Archimede spirl < ϕ π d thereore ( l( + 4 L + Propositio 5: Suppose C is smooth curve i give prmetriclly y ( t ( ( t, ( t, t I Fig 56 Archimede spirl < ϕ Let ech stright lie rom origi cut the curve oly oce the the re o the regio eclosed y tht curve d the stright lies rom origi to the edpoits o the curve, the so clled sector re (Figure 58, is give y A dt (58 ( ( t ( t ( t ( t I the Curve C hs piecewise smooth prmeteriztio i polr coordites r r(ϕ, α ϕ β, the pplies: Fig 57 Archimede spirl < ϕ (59 A r ( ϕ d ϕ Exmple 5: (i The cotet o the ellipse x cos(t, y si(t, t π is Fig 58 Sector re A ( cos t+ si t dt A -7

18 Mthemtics or Egieers Prt II (ISE Versio /4-6- (ii The re eclosed y the Archimede spirl ter oe circultio (Figure 55 is give y 4 ( ϕ d A ϕ More geerlly th the sector re ormul is the ollowig result Propositio 5: The re eclosed y piecewise cotiuously dieretile closed curve C with t t, t, t, which does ot itersect itsel is prmeteriztio ( ( ( ( A dt (5 ( ( t ( t ( t ( t I this cse (59 is lso vlid or polr coordites prmeteriztio Exmple 54: (i The steroid i the ollowig Figure 59 is prmetriclly give y ( t ( Rcos t+,rsi t+, t Fig 59 Asteroid, R The re eclosed y the steroid is -8

19 Mthemtics or Egieers Prt II (ISE Versio /4-6- A ( Rcos t+ Rsi t cost+ ( Rsi t+ Rcos t sit dt Thus 8 A R The steroid is the locus o poit o the rim o circle o rdius r R/4 rollig log the ier lie o the circle o rdius R (Figure 5 Sice R 4r pplies, A R 6r 8 Fig 5 Geertig steroid (ii A prmeteriztio o the lemiscte (Fig 5 is cost ( t (,sit, t + si t The right loop o the lemiscte i polr coordites is give y g( ϕ cos ϕ ( cosϕ,siϕ, ϕ 4 4 Thereore the re o oe loop o the lemiscte is Fig 5 Lemiscte / 4 / 4 A r ( d cos( d ϕ ϕ ϕ ϕ / 4 / 4 Thereore the re eclosed y the lemiscte is A The lemiscte is lso kow s Eight Curve -9

20 Mthemtics or Egieers Prt II (ISE Versio / Ple curves d solids o revolutio I the ollowig we cosider solids o revolutio i geerted y ple curves with prmeteriztio [ ; ] IR, ( t ( x( t,y( t : I y ( t throughout [;], we c use rgumets similr to tht give i the previous sectios to compute the re o the surce geerted y revolvig give curve C out the x-xis respectively the y-xis i x ( t (see Figure 6 d 6 Fig 6 Revolvig C out the x-xis Propositio 6: Let smooth curve C give y [ ; ] IR, ( t ( x( t,y( t : d suppose C does ot itersect itsel, except possily t the poits ( d ( I y ( t, or t [;], the the re S o the surce o revolutio otied y revolvig C out the x-xis is (6 S y( t ( x ( t + ( y ( t x dt Fig 6 Revolvig C out the y-xis I x ( t, or t [;], the the re S o the surce o revolutio otied y revolvig C out the y-xis is (6 S x( t ( x ( t + ( y ( t y dt Remrk 6: Suppose the curve C is prmetriclly give y : [ ; ] IR, ( t ( x( t,y( t prmetric dieretil o rc legth is deied s dx dt dy dt (6 ds ( dx + ( dy + dt ( x ( t + ( y ( t dt The -

21 Mthemtics or Egieers Prt II (ISE Versio /4-6- Usig (6 we c write (6 d (6 s (64 S y( t ds d x( t x y S ds Exmple 6: (i Veriy tht the surce re o sphere o rdius r is 4πr Solutio: I C is the upper hl o the circle x + y r, the the sphericl surce my e otied e revolvig C out the x-xis A prmeteriztio o C is give y : [ ; ] IR, ( t ( r cos t,r si t π Applyig ormul (6 d usig the idetity si ( t + cos ( t π, we get: ( r si t r si t + r cos t dt r sit dt 4 S x r (ii We compute the re o the surce show i Figure 6 The geertig curve C is prmetriclly give y Usig ormul (6 we hve: [ ; ] IR, ( t ( t,cos t : π π S y π t + si t dt 5 85 Remrk 6: The ormuls (6 d (6 c e exteded to the cse i which y(t or x(t is egtive or some t [;] y replcig the uctio y(t tht precedes ds y y(t (or x(t y x(t I the meridi o solid o revolutio is prmetriclly give s ple curve C, the we c esily compute the Volume o this solid geerted y revolvig C out the x- xis or the y-xis usig the ollowig Propositio 6: Let smooth curve C give y [ ; ] IR, ( t ( x( t,y( t : -

22 Mthemtics or Egieers Prt II (ISE Versio /4-6- d suppose C is the grph o Crtesi uctio with respect to the xis o revolutio, the pplies: (i The volume V x o the solid o revolutio otied y revolvig C out the x-xis is (65 V ( y( t x ( t x dt (ii The volume V y o the solid o revolutio otied y revolvig C out the y-xis is (66 V ( x( t y ( t y dt Corollry 6: Suppose the cotiuously dieretile Crtesi uctio g : [ ; ] IR, y g( x the meridi o ple curve C The : [ ; ] IR, ( x ( x,g( x prmeteriztio o C d with Propositio 6 we hve: (67 ( ( x g x V dx d (68 V x g ( x y dx is, is Exmple 6: The upper hl o the steroid (see Figure 59 or 5 is prmetriclly give y ( t ( Rcos t,rsi t, t, R > The volume o the solid geerted y revolvig this curve out the x-xis is Fig 6 Asteroid revolved out the x-xis V x π R R π si si 7 6 t R cos t cos t dt t ( si t dt Thus we get with the ollowig Lemm 6: -

23 Mthemtics or Egieers Prt II (ISE Versio /4-6- V x R 9 R R 8 5 π si π 7 t dt R si t dt R [ cost] R π si π 5 t dt sit dt Computig the lst trigoometric itegrls we used Lemm 6: Let m d e positive itegers, the pplies: m (i si ( x cos ( x (ii si ( x si dx m ( x cos ( x ( + m m + + m + ( x cos( x + si dx dx si ( x si m ( x cos ( x dx -

BC Calculus Review Sheet

BC 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 information

FOURIER 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 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 information

0 otherwise. sin( nx)sin( kx) 0 otherwise. cos( nx) sin( kx) dx 0 for all integers n, k.

0 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 information

Week 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 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 information

Module 4. Signal Representation and Baseband Processing. Version 2 ECE IIT, Kharagpur

Module 4. Signal Representation and Baseband Processing. Version 2 ECE IIT, Kharagpur Module 4 Sigl Represettio d Bsed Processig Versio ECE IIT, Khrgpur Lesso 5 Orthogolity Versio ECE IIT, Khrgpur Ater redig this lesso, you will ler out Bsic cocept o orthogolity d orthoormlity; Strum -

More information

HIGHER 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 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 information

[Q. Booklet Number]

[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 information

General properties of definite integrals

General properties of definite integrals Roerto s Notes o Itegrl Clculus Chpter 4: Defiite itegrls d the FTC Sectio Geerl properties of defiite itegrls Wht you eed to kow lredy: Wht defiite Riem itegrl is. Wht you c ler here: Some key properties

More information

PhysicsAndMathsTutor.com

PhysicsAndMathsTutor.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 information

National Quali cations AHEXEMPLAR PAPER ONLY

National 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 information

Review of the Riemann Integral

Review of the Riemann Integral Chpter 1 Review of the Riem Itegrl This chpter provides quick review of the bsic properties of the Riem itegrl. 1.0 Itegrls d Riem Sums Defiitio 1.0.1. Let [, b] be fiite, closed itervl. A prtitio P of

More information

The Reimann Integral is a formal limit definition of a definite integral

The Reimann Integral is a formal limit definition of a definite integral MATH 136 The Reim Itegrl The Reim Itegrl is forml limit defiitio of defiite itegrl cotiuous fuctio f. The costructio is s follows: f ( x) dx for Reim Itegrl: Prtitio [, ] ito suitervls ech hvig the equl

More information

EVALUATING DEFINITE INTEGRALS

EVALUATING DEFINITE INTEGRALS Chpter 4 EVALUATING DEFINITE INTEGRALS If the defiite itegrl represets re betwee curve d the x-xis, d if you c fid the re by recogizig the shpe of the regio, the you c evlute the defiite itegrl. Those

More information

Important Facts You Need To Know/Review:

Important 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 information

BITSAT 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 information

8.1 Arc Length. What is the length of a curve? How can we approximate it? We could do it following the pattern we ve used before

8.1 Arc Length. What is the length of a curve? How can we approximate it? We could do it following the pattern we ve used before 8.1 Arc Legth Wht is the legth of curve? How c we pproximte it? We could do it followig the ptter we ve used efore Use sequece of icresigly short segmets to pproximte the curve: As the segmets get smller

More information

Limit of a function:

Limit 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 information

Linford 1. Kyle Linford. Math 211. Honors Project. Theorems to Analyze: Theorem 2.4 The Limit of a Function Involving a Radical (A4)

Linford 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 information

1.3 Continuous Functions and Riemann Sums

1.3 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 information

B. 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

B. 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 information

Theorem 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

Theorem 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 information

is continuous at x 2 and g(x) 2. Oil spilled from a ruptured tanker spreads in a circle whose area increases at a

is 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 information

Fourier Series and Applications

Fourier Series and Applications 9/7/9 Fourier Series d Applictios Fuctios epsio is doe to uderstd the better i powers o etc. My iportt probles ivolvig prtil dieretil equtios c be solved provided give uctio c be epressed s iiite su o

More information

Vectors. Vectors in Plane ( 2

Vectors. Vectors in Plane ( 2 Vectors Vectors i Ple ( ) The ide bout vector is to represet directiol force Tht mes tht every vector should hve two compoets directio (directiol slope) d mgitude (the legth) I the ple we preset vector

More information

Multiplication and Translation Operators on the Fock Spaces for the q-modified Bessel Function *

Multiplication and Translation Operators on the Fock Spaces for the q-modified Bessel Function * Advces i Pure Mthemtics 0-7 doi:0436/pm04039 Pulished Olie July 0 (http://wwwscirporg/jourl/pm) Multiplictio d Trsltio Opertors o the Fock Spces or the -Modiied Bessel Fuctio * Astrct Fethi Solti Higher

More information

MA123, Chapter 9: Computing some integrals (pp )

MA123, 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 information

The Definite Riemann Integral

The Definite Riemann Integral These otes closely follow the presettio of the mteril give i Jmes Stewrt s textook Clculus, Cocepts d Cotexts (d editio). These otes re iteded primrily for i-clss presettio d should ot e regrded s sustitute

More information

Pre-Calculus - Chapter 3 Sections Notes

Pre-Calculus - Chapter 3 Sections Notes Pre-Clculus - Chpter 3 Sectios 3.1-3.4- Notes Properties o Epoets (Review) 1. ( )( ) = + 2. ( ) =, (c) = 3. 0 = 1 4. - = 1/( ) 5. 6. c Epoetil Fuctios (Sectio 3.1) Deiitio o Epoetil Fuctios The uctio deied

More information

( ) dx ; f ( x ) is height and Δx is

( ) 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

334 MATHS SERIES DSE MATHS PREVIEW VERSION B SAMPLE TEST & FULL SOLUTION

334 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 information

4. When is the particle speeding up? Why? 5. When is the particle slowing down? Why?

4. When is the particle speeding up? Why? 5. When is the particle slowing down? Why? AB CALCULUS: 5.3 Positio vs Distce Velocity vs. Speed Accelertio All the questios which follow refer to the grph t the right.. Whe is the prticle movig t costt speed?. Whe is the prticle movig to the right?

More information

Approximate Integration

Approximate 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 information

Objective Mathematics

Objective 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

n 2 + 3n + 1 4n = n2 + 3n + 1 n n 2 = n + 1

n 2 + 3n + 1 4n = n2 + 3n + 1 n n 2 = n + 1 Ifiite Series Some Tests for Divergece d Covergece Divergece Test: If lim u or if the limit does ot exist, the series diverget. + 3 + 4 + 3 EXAMPLE: Show tht the series diverges. = u = + 3 + 4 + 3 + 3

More information

UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION

UNIVERSITY 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 information

Graphing Review Part 3: Polynomials

Graphing 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 information

GRAPHING 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. 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 information

Approximations of Definite Integrals

Approximations 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 information

Things I Should Know In Calculus Class

Things 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 information

Second Mean Value Theorem for Integrals By Ng Tze Beng. The Second Mean Value Theorem for Integrals (SMVT) Statement of the Theorem

Second Mean Value Theorem for Integrals By Ng Tze Beng. The Second Mean Value Theorem for Integrals (SMVT) Statement of the Theorem Secod Me Vlue Theorem for Itegrls By Ng Tze Beg This rticle is out the Secod Me Vlue Theorem for Itegrls. This theorem, first proved y Hoso i its most geerlity d with extesio y ixo, is very useful d lmost

More information

Add Maths Formulae List: Form 4 (Update 18/9/08)

Add 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 information

F x = 2x λy 2 z 3 = 0 (1) F y = 2y λ2xyz 3 = 0 (2) F z = 2z λ3xy 2 z 2 = 0 (3) F λ = (xy 2 z 3 2) = 0. (4) 2z 3xy 2 z 2. 2x y 2 z 3 = 2y 2xyz 3 = ) 2

F x = 2x λy 2 z 3 = 0 (1) F y = 2y λ2xyz 3 = 0 (2) F z = 2z λ3xy 2 z 2 = 0 (3) F λ = (xy 2 z 3 2) = 0. (4) 2z 3xy 2 z 2. 2x y 2 z 3 = 2y 2xyz 3 = ) 2 0 微甲 07- 班期中考解答和評分標準 5%) Fid the poits o the surfce xy z = tht re closest to the origi d lso the shortest distce betwee the surfce d the origi Solutio Cosider the Lgrge fuctio F x, y, z, λ) = x + y + z

More information

1. (25 points) Use the limit definition of the definite integral and the sum formulas to compute. [1 x + x2

1. (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 information

Mathematical Notation Math Calculus & Analytic Geometry I

Mathematical 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 information

1 Tangent Line Problem

1 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 information

BC Calculus Path to a Five Problems

BC 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 information

Sequence and Series of Functions

Sequence and Series of Functions 6 Sequece d Series of Fuctios 6. Sequece of Fuctios 6.. Poitwise Covergece d Uiform Covergece Let J be itervl i R. Defiitio 6. For ech N, suppose fuctio f : J R is give. The we sy tht sequece (f ) of fuctios

More information

Convergence rates of approximate sums of Riemann integrals

Convergence rates of approximate sums of Riemann integrals Covergece rtes of pproximte sums of Riem itegrls Hiroyuki Tski Grdute School of Pure d Applied Sciece, Uiversity of Tsuku Tsuku Irki 5-857 Jp tski@mth.tsuku.c.jp Keywords : covergece rte; Riem sum; Riem

More information

MATH 104 FINAL SOLUTIONS. 1. (2 points each) Mark each of the following as True or False. No justification is required. y n = x 1 + x x n n

MATH 104 FINAL SOLUTIONS. 1. (2 points each) Mark each of the following as True or False. No justification is required. y n = x 1 + x x n n MATH 04 FINAL SOLUTIONS. ( poits ech) Mrk ech of the followig s True or Flse. No justifictio is required. ) A ubouded sequece c hve o Cuchy subsequece. Flse b) A ifiite uio of Dedekid cuts is Dedekid cut.

More information

Name of the Student:

Name 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 information

BRAIN TEASURES INDEFINITE INTEGRATION+DEFINITE INTEGRATION EXERCISE I

BRAIN TEASURES INDEFINITE INTEGRATION+DEFINITE INTEGRATION EXERCISE I EXERCISE I t Q. d Q. 6 6 cos si Q. Q.6 d d Q. d Q. Itegrte cos t d by the substitutio z = + e d e Q.7 cos. l cos si d d Q. cos si si si si b cos Q.9 d Q. si b cos Q. si( ) si( ) d ( ) Q. d cot d d Q. (si

More information

Integral Operator Defined by k th Hadamard Product

Integral Operator Defined by k th Hadamard Product ITB Sci Vol 4 A No 35-5 35 Itegrl Opertor Deied by th Hdmrd Product Msli Drus & Rbh W Ibrhim School o Mthemticl Scieces Fculty o sciece d Techology Uiversiti Kebgs Mlysi Bgi 436 Selgor Drul Ehs Mlysi Emil:

More information

Mathematics Extension 2

Mathematics Extension 2 05 Bored of Studies Tril Emitios Mthemtics Etesio Writte by 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 information

MAS221 Analysis, Semester 2 Exercises

MAS221 Analysis, Semester 2 Exercises MAS22 Alysis, Semester 2 Exercises Srh Whitehouse (Exercises lbelled * my be more demdig.) Chpter Problems: Revisio Questio () Stte the defiitio of covergece of sequece of rel umbers, ( ), to limit. (b)

More information

MTH 146 Class 16 Notes

MTH 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 information

Definition Integral. over[ ab, ] the sum of the form. 2. Definite Integral

Definition 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 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!

 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 information

Math 104: Final exam solutions

Math 104: Final exam solutions Mth 14: Fil exm solutios 1. Suppose tht (s ) is icresig sequece with coverget subsequece. Prove tht (s ) is coverget sequece. Aswer: Let the coverget subsequece be (s k ) tht coverges to limit s. The there

More information

HIGHER SCHOOL CERTIFICATE EXAMINATION MATHEMATICS 4 UNIT (ADDITIONAL) Time allowed Three hours (Plus 5 minutes reading time)

HIGHER SCHOOL CERTIFICATE EXAMINATION MATHEMATICS 4 UNIT (ADDITIONAL) Time allowed Three hours (Plus 5 minutes reading time) HIGHER SCHOOL CERTIFICATE EXAMINATION 998 MATHEMATICS 4 UNIT (ADDITIONAL) Time llowed Three hours (Plus 5 miutes redig time) DIRECTIONS TO CANDIDATES Attempt ALL questios ALL questios re of equl vlue All

More information

Orthogonal functions - Function Approximation

Orthogonal functions - Function Approximation Orthogol uctios - Fuctio Approimtio - he Problem - Fourier Series - Chebyshev Polyomils he Problem we re tryig to pproimte uctio by other uctio g which cosists o sum over orthogol uctios Φ weighted by

More information

Calculus Summary Sheet

Calculus 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 information

Definite Integral. The Left and Right Sums

Definite Integral. The Left and Right Sums Clculus Li Vs Defiite Itegrl. The Left d Right Sums The defiite itegrl rises from the questio of fidig the re betwee give curve d x-xis o itervl. The re uder curve c be esily clculted if the curve is give

More information

Mathematical Notation Math Calculus & Analytic Geometry I

Mathematical 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 information

POWER SERIES R. E. SHOWALTER

POWER 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 information

UNIVERSITY OF BRISTOL. Examination for the Degrees of B.Sc. and M.Sci. (Level C/4) ANALYSIS 1B, SOLUTIONS MATH (Paper Code MATH-10006)

UNIVERSITY OF BRISTOL. Examination for the Degrees of B.Sc. and M.Sci. (Level C/4) ANALYSIS 1B, SOLUTIONS MATH (Paper Code MATH-10006) UNIVERSITY OF BRISTOL Exmitio for the Degrees of B.Sc. d M.Sci. (Level C/4) ANALYSIS B, SOLUTIONS MATH 6 (Pper Code MATH-6) My/Jue 25, hours 3 miutes This pper cotis two sectios, A d B. Plese use seprte

More information

Numerical Integration

Numerical Integration Numericl tegrtio Newto-Cotes Numericl tegrtio Scheme Replce complicted uctio or tulted dt with some pproimtig uctio tht is esy to itegrte d d 3-7 Roerto Muscedere The itegrtio o some uctios c e very esy

More information

National Quali cations SPECIMEN ONLY

National 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 information

INTEGRATION TECHNIQUES (TRIG, LOG, EXP FUNCTIONS)

INTEGRATION 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 information

Students must always use correct mathematical notation, not calculator notation. the set of positive integers and zero, {0,1, 2, 3,...

Students 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 information

Riemann Integral and Bounded function. Ng Tze Beng

Riemann Integral and Bounded function. Ng Tze Beng Riem Itegrl d Bouded fuctio. Ng Tze Beg I geerlistio of re uder grph of fuctio, it is ormlly ssumed tht the fuctio uder cosidertio e ouded. For ouded fuctio, the rge of the fuctio is ouded d hece y suset

More information

BC Calculus Review Sheet. converges. Use the integral: L 1

BC Calculus Review Sheet. converges. Use the integral: L 1 BC Clculus Review Sheet Whe yu see the wrds.. Fid the re f the uuded regi represeted y the itegrl (smetimes f ( ) clled hriztl imprper itegrl).. Fid the re f differet uuded regi uder f() frm (,], where

More information

DEPARTMENT OF ELECTRICAL &ELECTRONICS ENGINEERING SIGNALS AND SYSTEMS. Assoc. Prof. Dr. Burak Kelleci. Spring 2018

DEPARTMENT OF ELECTRICAL &ELECTRONICS ENGINEERING SIGNALS AND SYSTEMS. Assoc. Prof. Dr. Burak Kelleci. Spring 2018 DEPARTMENT OF ELECTRICAL &ELECTRONICS ENGINEERING SIGNALS AND SYSTEMS Assoc. Prof. Dr. Bur Kelleci Sprig 8 OUTLINE The Z-Trsform The Regio of covergece for the Z-trsform The Iverse Z-Trsform Geometric

More information

Chapter 7 Infinite Series

Chapter 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 information

Elementary Linear Algebra

Elementary Linear Algebra Elemetry Lier Alger Ato & Rorres, th Editio Lecture Set Chpter : Systems of Lier Equtios & Mtrices Chpter Cotets Itroductio to System of Lier Equtios Gussi Elimitio Mtrices d Mtri Opertios Iverses; Rules

More information

f(t)dt 2δ f(x) f(t)dt 0 and b f(t)dt = 0 gives F (b) = 0. Since F is increasing, this means that

f(t)dt 2δ f(x) f(t)dt 0 and b f(t)dt = 0 gives F (b) = 0. Since F is increasing, this means that Uiversity of Illiois t Ur-Chmpig Fll 6 Mth 444 Group E3 Itegrtio : correctio of the exercises.. ( Assume tht f : [, ] R is cotiuous fuctio such tht f(x for ll x (,, d f(tdt =. Show tht f(x = for ll x [,

More information

Area, Volume, Rotations, Newton s Method

Area, Volume, Rotations, Newton s Method Are, Volume, Rottio, Newto Method Are etwee curve d the i A ( ) d Are etwee curve d the y i A ( y) yd yc Are etwee curve A ( ) g( ) d where ( ) i the "top" d g( ) i the "ottom" yd Are etwee curve A ( y)

More information

Qn Suggested Solution Marking Scheme 1 y. G1 Shape with at least 2 [2]

Qn 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 information

INFINITE SERIES. ,... having infinite number of terms is called infinite sequence and its indicated sum, i.e., a 1

INFINITE SERIES. ,... having infinite number of terms is called infinite sequence and its indicated sum, i.e., a 1 Appedix A.. Itroductio As discussed i the Chpter 9 o Sequeces d Series, sequece,,...,,... hvig ifiite umber of terms is clled ifiite sequece d its idicted sum, i.e., + + +... + +... is clled ifite series

More information

lecture 16: Introduction to Least Squares Approximation

lecture 16: Introduction to Least Squares Approximation 97 lecture 16: Itroductio to Lest Squres Approximtio.4 Lest squres pproximtio The miimx criterio is ituitive objective for pproximtig fuctio. However, i my cses it is more ppelig (for both computtio d

More information

Riemann Integral Oct 31, such that

Riemann Integral Oct 31, such that Riem Itegrl Ot 31, 2007 Itegrtio of Step Futios A prtitio P of [, ] is olletio {x k } k=0 suh tht = x 0 < x 1 < < x 1 < x =. More suitly, prtitio is fiite suset of [, ] otiig d. It is helpful to thik of

More information

Homework 2 solutions

Homework 2 solutions Sectio 2.1: Ex 1,3,6,11; AP 1 Sectio 2.2: Ex 3,4,9,12,14 Homework 2 solutios 1. Determie i ech uctio hs uique ixed poit o the speciied itervl. gx = 1 x 2 /4 o [0,1]. g x = -x/2, so g is cotiuous d decresig

More information

Convergence rates of approximate sums of Riemann integrals

Convergence rates of approximate sums of Riemann integrals Jourl of Approximtio Theory 6 (9 477 49 www.elsevier.com/locte/jt Covergece rtes of pproximte sums of Riem itegrls Hiroyuki Tski Grdute School of Pure d Applied Sciece, Uiversity of Tsukub, Tsukub Ibrki

More information

Time: 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).

Time: 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 information

Trapezoidal Rule of Integration

Trapezoidal Rule of Integration Trpezoidl Rule o Itegrtio Mjor: All Egieerig Mjors Authors: Autr Kw, Chrlie Brker Trsormig Numericl Methods Eductio or STEM Udergrdutes /0/200 Trpezoidl Rule o Itegrtio Wht is Itegrtio Itegrtio: The process

More information

Trapezoidal Rule of Integration

Trapezoidal Rule of Integration Trpezoidl Rule o Itegrtio Civil Egieerig Mjors Authors: Autr Kw, Chrlie Brker http://umericlmethods.eg.us.edu Trsormig Numericl Methods Eductio or STEM Udergrdutes /0/00 http://umericlmethods.eg.us.edu

More information

Interpolation. 1. What is interpolation?

Interpolation. 1. What is interpolation? Iterpoltio. Wht is iterpoltio? A uctio is ote give ol t discrete poits such s:.... How does oe id the vlue o t other vlue o? Well cotiuous uctio m e used to represet the + dt vlues with pssig through the

More information

Multiplicative Versions of Infinitesimal Calculus

Multiplicative Versions of Infinitesimal Calculus Multiplictive Versios o Iiitesiml Clculus Wht hppes whe you replce the summtio o stdrd itegrl clculus with multiplictio? Compre the revited deiitio o stdrd itegrl D å ( ) lim ( ) D i With ( ) lim ( ) D

More information

UNIVERSITY 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. (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 information

10.5 Power Series. In this section, we are going to start talking about power series. A power series is a series of the form

10.5 Power Series. In this section, we are going to start talking about power series. A power series is a series of the form 0.5 Power Series I the lst three sectios, we ve spet most of tht time tlkig bout how to determie if series is coverget or ot. Now it is time to strt lookig t some specific kids of series d we will evetully

More information

Remarks: (a) The Dirac delta is the function zero on the domain R {0}.

Remarks: (a) The Dirac delta is the function zero on the domain R {0}. Sectio Objective(s): The Dirc s Delt. Mi Properties. Applictios. The Impulse Respose Fuctio. 4.4.. The Dirc Delt. 4.4. Geerlized Sources Defiitio 4.4.. The Dirc delt geerlized fuctio is the limit δ(t)

More information

The Discrete-Time Fourier Transform (DTFT)

The Discrete-Time Fourier Transform (DTFT) EEL: Discrete-Time Sigals ad Systems The Discrete-Time Fourier Trasorm (DTFT) The Discrete-Time Fourier Trasorm (DTFT). Itroductio I these otes, we itroduce the discrete-time Fourier trasorm (DTFT) ad

More information

2.1.1 Definition The Z-transform of a sequence x [n] is simply defined as (2.1) X re x k re x k r

2.1.1 Definition The Z-transform of a sequence x [n] is simply defined as (2.1) X re x k re x k r Z-Trsforms. INTRODUCTION TO Z-TRANSFORM The Z-trsform is coveiet d vluble tool for represetig, lyig d desigig discrete-time sigls d systems. It plys similr role i discrete-time systems to tht which Lplce

More information

Taylor 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

Taylor 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 information

(1) Functions A relationship between two variables that assigns to each element in the domain exactly one element in the range.

(1) Functions A relationship between two variables that assigns to each element in the domain exactly one element in the range. -. ALGEBRA () Fuctios A reltioship etwee two vriles tht ssigs to ech elemet i the domi ectly oe elemet i the rge. () Fctorig Aother ottio for fuctio of is f e.g. Domi: The domi of fuctio Rge: The rge of

More information

Content: Essential Calculus, Early Transcendentals, James Stewart, 2007 Chapter 1: Functions and Limits., in a set B.

Content: 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 information

Handout #2. Introduction to Matrix: Matrix operations & Geometric meaning

Handout #2. Introduction to Matrix: Matrix operations & Geometric meaning Hdout # Title: FAE Course: Eco 8/ Sprig/5 Istructor: Dr I-Mig Chiu Itroductio to Mtrix: Mtrix opertios & Geometric meig Mtrix: rectgulr rry of umers eclosed i pretheses or squre rckets It is covetiolly

More information

ELLIPSE. 1. If the latus rectum of an ellipse be equal to half of its minor axis, then its eccentricity is [Karnataka CET 2000]

ELLIPSE. 1. If the latus rectum of an ellipse be equal to half of its minor axis, then its eccentricity is [Karnataka CET 2000] ELLIPSE. If the ltus rectum of ellipse e equl to hlf of its mior is, the its eccetricit is [Krtk CET 000] / / / d /. The legth of the ltus rectum of the ellipse is [MNR 7, 0, ] / / / d 0/. Eccetricit of

More information

Infinite Series Sequences: terms nth term Listing Terms of a Sequence 2 n recursively defined n+1 Pattern Recognition for Sequences Ex:

Infinite 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 information

Numerical Methods. Lecture 5. Numerical integration. dr hab. inż. Katarzyna Zakrzewska, prof. AGH. Numerical Methods lecture 5 1

Numerical Methods. Lecture 5. Numerical integration. dr hab. inż. Katarzyna Zakrzewska, prof. AGH. Numerical Methods lecture 5 1 Numeril Methods Leture 5. Numeril itegrtio dr h. iż. Ktrzy Zkrzewsk, pro. AGH Numeril Methods leture 5 Outlie Trpezoidl rule Multi-segmet trpezoidl rule Rihrdso etrpoltio Romerg's method Simpso's rule

More information

( a n ) converges or diverges.

( a n ) converges or diverges. Chpter Ifiite Series Pge of Sectio E Rtio Test Chpter : Ifiite Series By the ed of this sectio you will be ble to uderstd the proof of the rtio test test series for covergece by pplyig the rtio test pprecite

More information