Multiplicative Versions of Infinitesimal Calculus
|
|
|
- Michael Goodman
- 6 years ago
- Views:
Transcription
1 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 D ( + ( ) ) lim ( + ( ) D) D i i d Cll these lter two itegrls multigrls o Type I d II. (Note: ulike orml products, these products re ot discrete ut cotiuous over itervl). Cosider ech i tur. Multigrls (Type I) By stdrd opertios ( ) e ( l( ( )) ) By ot tkig limits, iite product pproimtio c e otied. For emple, let () rom to. The the Type I multigrl o rom to is: e ( l( ) ) / e This c e pproimted y the sequece [( )( )] ( ) [( )( )( )] ( ) [( )( )( )( )] ( ) [( )( )...( )] ( ) etc 999 Which teds to /e (use Stirlig s Formul to support this).
2 pi / For ()t() i rdis rom to pi/, t( ) e Ad [ t( p / 6)t( p / 6) ] ( ) [ t( p /8)t( p / 8)t(3 p / 8) ] ( )... etc. 3 The ove pproimtios o the multigrl c e likeed to the mid-poit-rule whe pproimtig stdrd itegrls. Like stdrd itegrls, multiplictive logs o the Trpezoidl Rule d Simpso s Rule c e oud, like: Simpso s Product: ( )» ì[ ( ) ( )] ü D í{ [ ( + D ) ( + 3 D)...] 4 } ý ( ) 3 î {[ ( + D)...] } þ Cosider the ollowig pproimtios: Y e ( l( ) e ( l() - + ) 4 / e Multiplictive Alog o. Mid-poit Rule Trpezoidl Rule Simpso s Rule.5 [()()]^(/).44. / [(.5)(.75)]^(/) [()()]^(/4) [.5]^(/) /3 [(7/6)(9/6)(/6)]^(/3) [()()]^(/6) [(4/3)(5/3]^(/3) [()()]^((/)(/3)) [.5]^((4/3)(/)) /4 [(9/8)(/8)(3/8) (5/8)]^(/4) [()()]^(/8) [(5/4)(6/4)(7/4)]^(/4) [()()]^((/4)(/3)) [(.5)(.75)]^((4/3)(/4)) [.5]^((/3)(/4)) Like stdrd clculus you c deie multiplictive log o the derivtive ( the m-derivtive), costruct multiplictive versio o the Fudmetl Theorem o Clculus, costruct multiplictive log o Mcluri s Series, etc.
3 The m-derivtive or Type I multigrls is: ( ) ( ) e ( ) I ( ) The Fudmetl Theorem is: ( ) ( ) ( ) e (( ) ) I ( ) ( ) Compre with the Fudmetl Theorem o Stdrd Clculus: ( ) ( ) - ( ) Smll progrms c e writte to pproimte the ove results y iite products or those who dout. Type I multigrls id pplictio i the re o popultio dymics. With stochstic irth- d deth- rtes, the covetiol pproch is to use mes (ie: epecttios). Without migrtio, me popultios E(P) remi costt i me irth-rtes E() me deth-rtes E(d) uder the stochstic recursive equtio P ( d) P. But, while mthemticlly correct, this result is misledig. I certi circumstces, simultios show tht me irth-rtes c sigiictly eceed me deth-rtes yet MOST popultio trils declie, eve though the me popultio o my trils stys costt. True. Let G() ( p( ) ) where X the rdom vrile o (+-d) d p() is its proility desity uctio. It c e show tht the MODE o popultios (P ) teds to {G() }P s. I geerl G() is < E()E(+-d). Thus whe E()E(d), the mode o P s eve though E(P ) P. Thus the stochstic recursive equtios (where r# is rdom umer etwee d ) P ( r #) P + P ( r # ) P + P ( r # +.544) P + re ll costt (i the log-term mode) ulike P ( r #) P + P ( r # +.5) P +
4 which re costt i the log-term me ut ted to zero i the mode. (Try simultig usig Ecel i you do t elieve). Now cosider Multigrls (TypeII) Cosider ( + ) which is the limit o the sequece: æ ö ç +.5 è ø æ öæ 3 ö ç + ç è 4 øè 4 ø æ öæ 3 öæ 5 ö ç + ç + ç è 6 3 øè 6 3 øè 6 3 ø æ öæ 3 öæ 5 öæ 7 ö ç + ç + ç + ç etc. è 8 4 øè 8 4 øè 8 4 øè 8 4 ø which teds to sqr(e) This is due to the o-stdrd itegrl (which is ot o the orm ( ) ) l( + ).5 d thus ( + ) e ( l( + )) e ( ) e I geerl, ( + ( ) ) e ( ( ) ) provided ( ( ) ) or Î N ³. Fuctios () which il the lter coditio pper to e ew. For istce, ()/ ils this test rom to s ( ) p / 6 (look t the limit deiitio o the itegrl uder equl suitervls to see this). But or most other uctios ( ( ) ) or Î N ³ For Type II multigrls, the m-derivtive is: ( ) II ( ) ( )
5 Ad the Fudmetl Theorem is: ( ) ( ) ( + II ( ) ) ( + ) ( ) ( ) Higher order m-derivtives c e lso used, like: ì ( ) ü í ý ( ) ( ) ( ) II ( ) ( ) ( ) ( + II ( ) ) ( ( ) ) î þ + - ( ) ( ) II ( ) ì ( ) ü ( ) ( ) í ý î ( ) þ Ad so o. I geerl, the Fudmetl Theorem ecomes more complicted or higher order m-derivtives, ulike (sy) polyomils with stdrd clculus. For istce, II ( ) Ad thus é ù ( ) ( ) æ ( ) ö æ ( ) ö - - ç + ç ( ) ( ) ç ( ) ç ( ) è ø è ø é ( ) ( ) ù - ( ) ( ) é ù ì ( ) ( ) æ ( ) ö æ ( ) ö ü ( ) ( ) - - ç + ç é ù ( ) ( ) ( ) ( ) ç ç è ø è ø - ( ) ( ) ( ( ) ) + í + II é ( ) ( ) ù ý - ( ) ( ) ( ) ( ) é ù - ( ) ( ) î þ For emple, let ()l(+) the '()/(+), ''()-/(+)^, '''()/(+)^3 d ()l(), ()l(3), etc. The II II ( ) ( )
6 ( + ( ) ) ì é /( + ) 3 /( + ) æ - /( + ) ö ææ/( + ) ö öù ü ( ) + ( ) - ç ( ) + ç /( ) l( ) /( ) ç l( ) + + è + ø èè + ø ø í + ý é - /( + ) æ/( + ) öù ( ) - /( + ) ç l( + ) î è ø þ é ù l { ( + ) + l( + ) - } ( ) + l( + ) + () () é () () ù - () () () () - () () (/3)(( + / l(3))/( + / l())) Approimtig usig N suitervls gives: N pproimtio Whcko! Like stdrd clculus you c chge vriles i the stdrd wy: ( ) ì du ( + ( ) ) ( + u ) ( ) ( - ( u )) rom ílet u() the du () Þ u() Þ u() - îdu/ ()du/( ( ( u))) Ad thus, or emple: ( + ) ( + (( - ) + )( - ) ) Product d Quotiet Rules or Type I d II multigrls re:
7 Type I Derivtive ( ) e ( ) ( ) Product Rule ( g) g Quotiet Rule ( ) g g Type II ( ) ( ) ( g) + g ( ) - g g Surprisigly Type II multigrls hve the sme sort o Mcluri s Product s Type I. It is () æ () ö æ () ö e ç ç ()! è () ø 3! è () ø 3 ( ) () ( ) Ad the two types o multigrl c e relted y ( ) ( + l( ( )) ) or cceptle (). Other Types o Multigrl With type II multigrls, prolems rise or uctios like ()/ due to the ct tht ( ( ) ) ¹ or Î ³. But sometimes relted multigrls c e evluted usig certi thet uctios. For istce, 3 5 ( + ( ) ) 5( + ( ) )( + ( ) )... cosh( p )» d ( - ( ) 4) - cosh( p )» d é ì 3 ü ù ( ) 3 íg ý cosh( p 3) ( ( ) 3) (4 3 ) î þ p d the like. G( ) However, these type III multigrls hve certi uusul properties like
8 ( + ( ) )) ( + ( ) )) k k ( + ( ) )) ( + ( ) ))... etc. So tke cre whe plyig roud with. Type IV Multigrls Surprisigly the multigrl e e e.5 ( + ( )) e ( )»... eists! ( e -) ( e -) ( e 3-) This is thks to the o-stdrd stdrd itegrls o e ( ) ( e -) k e ( k / ) ( ) ( e k - ) These type o multigrls re more restricted (i rge) th type I d II, ut c still e used to derive certi stochstic limits such s ì ü e mod íå( r # i) ý» î i þ ( e -) ì ü æ e e e.5 ö mod í ( + ( r # ) ) ý e ç »... i î i þ è ( e -) ( e -) ( e 3 -) ø Where mod is the mode d r# is rdom umer etwee d. Uswered Questios. How my types o multigrls re there? Do they ll hve m-derivtives, Fudmetl Theorems, logs o Simpso s Rule, Mcluri Series, etc?. Wht do multigrls do i the comple ple? Aswers plese. Hppy multigrtig! All commets welcome. Plese sed to: everythiglows@hotmil.com e
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
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) =
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
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:
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
Math1242 Project I (TI 84) Name:
Mth4 Project I (TI 84) Nme: Riem Sums d Defiite Itegrls The re uder the grph of positive fuctio is give y the defiite itegrl of the fuctio. The defiite itegrl c e pproimted y the followig sums: Left Riem
Simpson s 1/3 rd Rule of Integration
Simpso s 1/3 rd Rule o Itegrtio Mjor: All Egieerig Mjors Authors: Autr Kw, Chrlie Brker Trsormig Numericl Methods Eductio or STEM Udergrdutes 1/10/010 1 Simpso s 1/3 rd Rule o Itegrtio Wht is Itegrtio?
the midpoint of the ith subinterval, and n is even for
Mth4 Project I (TI 89) Nme: Riem Sums d Defiite Itegrls The re uder the grph of positive fuctio is give y the defiite itegrl of the fuctio. The defiite itegrl c e pproimted y the followig sums: Left Riem
Crushed Notes on MATH132: Calculus
Mth 13, Fll 011 Siyg Yg s Outlie Crushed Notes o MATH13: Clculus The otes elow re crushed d my ot e ect This is oly my ow cocise overview of the clss mterils The otes I put elow should ot e used to justify
Mathacle. PSet Stats, Concepts In Statistics Level Number Name: Date:
APPENDEX I. THE RAW ALGEBRA IN STATISTICS A I-1. THE INEQUALITY Exmple A I-1.1. Solve ech iequlity. Write the solutio i the itervl ottio..) 2 p - 6 p -8.) 2x- 3 < 5 Solutio:.). - 4 p -8 p³ 2 or pî[2, +
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
1 Section 8.1: Sequences. 2 Section 8.2: Innite Series. 1.1 Limit Rules. 1.2 Common Sequence Limits. 2.1 Denition. 2.
Clculus II d Alytic Geometry Sectio 8.: Sequeces. Limit Rules Give coverget sequeces f g; fb g with lim = A; lim b = B. Sum Rule: lim( + b ) = A + B, Dierece Rule: lim( b ) = A B, roduct Rule: lim b =
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 [ ] (
Calculus 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
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
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 [,
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,
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
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
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
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
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
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
Section 6.3: Geometric Sequences
40 Chpter 6 Sectio 6.: Geometric Sequeces My jobs offer ul cost-of-livig icrese to keep slries cosistet with ifltio. Suppose, for exmple, recet college grdute fids positio s sles mger erig ul slry of $6,000.
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
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
REVIEW OF CHAPTER 5 MATH 114 (SECTION C1): ELEMENTARY CALCULUS
REVIEW OF CHAPTER 5 MATH 4 (SECTION C): EEMENTARY CACUUS.. Are.. Are d Estimtig with Fiite Sums Emple. Approimte the re of the shded regio R tht is lies ove the -is, elow the grph of =, d etwee the verticl
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
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
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)
( ) 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
f ( x) ( ) dx =
Defiite Itegrls & Numeric Itegrtio Show ll work. Clcultor permitted o, 6,, d Multiple Choice. (Clcultor Permitted) If the midpoits of equl-width rectgles is used to pproximte the re eclosed etwee the x-xis
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
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?
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
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
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
AP Calculus Notes: Unit 6 Definite Integrals. Syllabus Objective: 3.4 The student will approximate a definite integral using rectangles.
AP Clculus Notes: Uit 6 Defiite Itegrls Sllus Ojective:.4 The studet will pproimte defiite itegrl usig rectgles. Recll: If cr is trvelig t costt rte (cruise cotrol), the its distce trveled is equl to rte
5.3. The Definite Integral. Limits of Riemann Sums
. The Defiite Itegrl 4. The Defiite Itegrl I Sectio. we ivestigted the limit of fiite sum for fuctio defied over closed itervl [, ] usig suitervls of equl width (or legth), s - d>. I this sectio we cosider
Limits 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
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
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
FACULTY 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
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)
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
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
Options: Calculus. O C.1 PG #2, 3b, 4, 5ace O C.2 PG.24 #1 O D PG.28 #2, 3, 4, 5, 7 O E PG #1, 3, 4, 5 O F PG.
O C. PG.-3 #, 3b, 4, 5ce O C. PG.4 # Optios: Clculus O D PG.8 #, 3, 4, 5, 7 O E PG.3-33 #, 3, 4, 5 O F PG.36-37 #, 3 O G. PG.4 #c, 3c O G. PG.43 #, O H PG.49 #, 4, 5, 6, 7, 8, 9, 0 O I. PG.53-54 #5, 8
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.
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
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
( ) = A n + B ( ) + Bn
MATH 080 Test 3-SOLUTIONS Fll 04. Determie if the series is coverget or diverget. If it is coverget, fid its sum.. (7 poits) = + 3 + This is coverget geometric series where r = d
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
Topic 4 Fourier Series. Today
Topic 4 Fourier Series Toy Wves with repetig uctios Sigl geertor Clssicl guitr Pio Ech istrumet is plyig sigle ote mile C 6Hz) st hrmoic hrmoic 3 r hrmoic 4 th hrmoic 6Hz 5Hz 783Hz 44Hz A sigle ote will
f(x) is a function of x and it is defined on the set R of real numbers. If then f(x) is continuous at x=x 0, where x 0 R.
MATHEMATICAL PRELIMINARIES Limit Cotiuity Coverget squece Series Dieretible uctios Itegrble uctios Summtio deiitio o itegrl Me vlue theorem Me vlue theorem or itegrls Tylor's theorem Computer represettio
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
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
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,
Feedback & Assessment of Your Success. 1 Calculus AP U5 Integration (AP) Name: Antiderivatives & Indefinite Integration (AP) Journal #1 3days
Clculus AP U5 Itegrtio (AP) Nme: Big ide Clculus is etire rch of mthemtics. Clculus is uilt o two mjor complemetry ides. The first is differetil clculus, which is cocered with the istteous rte of chge.
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
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 -
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
( ) k ( ) 1 T n 1 x = xk. Geometric series obtained directly from the definition. = 1 1 x. See also Scalars 9.1 ADV-1: lim n.
Sclrs-9.0-ADV- Algebric Tricks d Where Tylor Polyomils Come From 207.04.07 A.docx Pge of Algebric tricks ivolvig x. You c use lgebric tricks to simplify workig with the Tylor polyomils of certi fuctios..
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
Fig. 1. I a. V ag I c. I n. V cg. Z n Z Y. I b. V bg
ymmetricl Compoets equece impedces Although the followig focuses o lods, the results pply eqully well to lies, or lies d lods. Red these otes together with sectios.6 d.9 of text. Cosider the -coected lced
Project 3: Using Identities to Rewrite Expressions
MAT 5 Projet 3: Usig Idetities to Rewrite Expressios Wldis I lger, equtios tht desrie properties or ptters re ofte lled idetities. Idetities desrie expressio e repled with equl or equivlet expressio tht
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
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()
* 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
9.5. Alternating series. Absolute convergence and conditional convergence
Chpter 9: Ifiite Series I this Chpter we will be studyig ifiite series, which is just other me for ifiite sums. You hve studied some of these i the pst whe you looked t ifiite geometric sums of the form:
Relation of BSTs to Quicksort, Analysis of Random BST. Lecture 9
Reltio o BSTs to Quicsort, Alysis o Rdom BST Lecture 9 Biry-serch-tree sort T Crete empty BST or i = to do TREE-INSERT(T, A[i]) Perorm iorder tree wl o T. Emple: 3 A = [3 8 2 6 7 5] 8 Tree-wl time = O(),
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
10. 3 The Integral and Comparison Test, Estimating Sums
0. The Itegrl d Comriso Test, Estimtig Sums I geerl, it is hrd to fid the ect sum of series. We were le to ccomlish this for geometric series d for telescoig series sice i ech of those cses we could fid
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
Frequency-domain Characteristics of Discrete-time LTI Systems
requecy-domi Chrcteristics of Discrete-time LTI Systems Prof. Siripog Potisuk LTI System descriptio Previous bsis fuctio: uit smple or DT impulse The iput sequece is represeted s lier combitio of shifted
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
1.1 The FTC and Riemann Sums. An Application of Definite Integrals: Net Distance Travelled
mth 3 more o the fudmetl theorem of clculus The FTC d Riem Sums A Applictio of Defiite Itegrls: Net Distce Trvelled I the ext few sectios (d the ext few chpters) we will see severl importt pplictios of
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
THE 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
Math 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
Chapter 5. The Riemann Integral. 5.1 The Riemann integral Partitions and lower and upper integrals. Note: 1.5 lectures
Chpter 5 The Riem Itegrl 5.1 The Riem itegrl Note: 1.5 lectures We ow get to the fudmetl cocept of itegrtio. There is ofte cofusio mog studets of clculus betwee itegrl d tiderivtive. The itegrl is (iformlly)
We saw in Section 5.1 that a limit of the form. 2 DEFINITION OF A DEFINITE INTEGRAL If f is a function defined for a x b,
3 6 6 CHAPTER 5 INTEGRALS CAS 5. Fid the ect re uder the cosie curve cos from to, where. (Use computer lger sstem oth to evlute the sum d compute the it.) I prticulr, wht is the re if? 6. () Let A e the
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
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
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
SM2H. 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.
Why study large deviations? The problem of estimating buer overow frequency The performance of many systems is limited by events which have a small pr
Why study lrge devitios? The problem of estimtig buer overow frequecy The performce of my systems is ited by evets which hve smll probbility of occurrig, but which hve severe cosequeces whe they occur.
The Elementary Arithmetic Operators of Continued Fraction
Americ-Eursi Jourl of Scietific Reserch 0 (5: 5-63, 05 ISSN 88-6785 IDOSI Pulictios, 05 DOI: 0.589/idosi.ejsr.05.0.5.697 The Elemetry Arithmetic Opertors of Cotiued Frctio S. Mugssi d F. Mistiri Deprtmet
Numerical Solutions of Fredholm Integral Equations Using Bernstein Polynomials
Numericl Solutios of Fredholm Itegrl Equtios Usig erstei Polyomils A. Shiri, M. S. Islm Istitute of Nturl Scieces, Uited Itertiol Uiversity, Dhk-, gldesh Deprtmet of Mthemtics, Uiversity of Dhk, Dhk-,
Review of Sections
Review of Sectios.-.6 Mrch 24, 204 Abstrct This is the set of otes tht reviews the mi ides from Chpter coverig sequeces d series. The specific sectios tht we covered re s follows:.: Sequces..2: Series,
where i is the index of summation, a i is the ith term of the sum, and the upper and lower bounds of summation are n and 1.
Chpter Itegrtio. Are Use sigm ottio to write d evlute sum. Uderstd the cocept o re. Approimte the re o ple regio. Fid the re o ple regio usig limits. Sigm Nottio I the precedig sectio, ou studied tidieretitio.
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
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)
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
Logarithmic Scales: the most common example of these are ph, sound and earthquake intensity.
Numercy Itroductio to Logrithms Logrithms re commoly credited to Scottish mthemtici med Joh Npier who costructed tle of vlues tht llowed multiplictios to e performed y dditio of the vlues from the tle.
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
Topic 9 - Taylor and MacLaurin Series
Topic 9 - Taylor ad MacLauri Series A. Taylors Theorem. The use o power series is very commo i uctioal aalysis i act may useul ad commoly used uctios ca be writte as a power series ad this remarkable result
under the curve in the first quadrant.
NOTES 5: INTEGRALS Nme: Dte: Perod: LESSON 5. AREAS AND DISTANCES Are uder the curve Are uder f( ), ove the -s, o the dom., Prctce Prolems:. f ( ). Fd the re uder the fucto, ove the - s, etwee,.. f ( )
We will begin by supplying the proof to (a).
(The solutios of problem re mostly from Jeffrey Mudrock s HWs) Problem 1. There re three sttemet from Exmple 5.4 i the textbook for which we will supply proofs. The sttemets re the followig: () The spce
Synopsis Grade 12 Math Part II
Syopsis Grde 1 Mth Prt II Chpter 7: Itegrls d Itegrtio is the iverse process of differetitio. If f ( ) g ( ), the we c write g( ) = f () + C. This is clled the geerl or the idefiite itegrl d C is clled
Mathematical Notation Math Calculus for Business and Social Science
Mthemticl Nottio Mth 190 - Clculus for Busiess d Socil Sciece Use Word or WordPerfect to recrete the followig documets. Ech rticle is worth 10 poits d should e emiled to the istructor t jmes@richld.edu.