Integration. September 28, 2017
|
|
- Noel Tate
- 6 years ago
- Views:
Transcription
1 Integrtion September 8, 7 Introduction We hve lerned in previous chpter on how to do the differentition. It is conventionl in mthemtics tht we re supposed to lern bout the integrtion s well. As you my hve lredy know, integrtion is the inverse of differentition, or sometimes clled ntiderivtive. In fct, by relising the reltionship between these two opertions, hs lid out the foundtion of wht we now clled Clculus which ws invented by Newton nd Leibniz independently in 7th century. Our min gol for this chpter is to lern bout the integrtion of the trnscendentl functions. First we will strt on reviewing the integrtion of simple integrnd(the eqution tht we wnt to integrte, the indefinite nd the definite integrl. Next we will lern the three techniques of integrtion, nd then we will pply these techniques on integrting trnscendentl functions ccordingly. Review of Integrtion. Bsic Integrl Formul Exmple : Evlute the following indefinite integrls. x x x 3 x
2 sin x cos x e x x x 8 x 4 5 x 3 x 5 4 sec x cosec x ( + tn x sec x sin x cos x sin x cos x. Integrls with Sum, Difference nd Fctor Exmple : Evlute the following indefinite integrls. (3x + 8. ( 4 sech x tnh x
3 (x 4 + x 6 (3x x 4 x 3 (4 + tn x (sin x cos x (x + (3x 8. + e x sec x sec x.3 Definite Integrl Exmple 3: Evlute the following integrls.. 3. (3x + 7 (x + (x + 3 x 3 + x 3 3 Techniques of Integrtion 3. Integrtion by Substitution Method of substitution is strightforwrd from chin rule, nd this method is wht you should consider first before other methods. We usully choose the substitution for the inside function nd then mnipulte some lgebr for wht s left on the outside function. Exmple 4: Evlute the following integrl by using the proper substitution. (4x +. c x + b 3
4 e 4x e 3x+5 (x 9 5 sin 5x cos(x + 3 tn(3 x 9. x + 3 x + 3x +. x x 4 x + 6. x 3 + 4x. x + cos x 3x + 6 sin x 3. tn x sin x cos x sec x 4x(x 3 6 sin x cos 4 x e x (e x 3 9. ln x x 4
5 x ln x x sec (x + 3 tn(x + 3 xe x x(x + 5 x 3x x x x 3 cos x 4 e x sin e x e tn x sec x 5x cos(5 x + x + e x x x 4 sin (6x + 7 Sometimes the substitution is given to us. Hve in mind tht the substitution will help us to simplify the integrnd, not mking it more difficult to integrte, which is the sole reson on why we re lerning to find the pproprite substitution in using this technique. Exmple 5 Evlute the following integrls by using the given substitution. 3x 3x with substitution u = 3x. 5
6 . 4x with substitution x = sin θ. x 3. 4 x with substitution x = sin θ. 3. Integrtion by Prts Integrtion by prts is powerful method to evlute integrls when the integrnd is in the form of product of lgebric nd trnscendentl, such s x ln x, xe x, e x cos x. Generl formul for integrtion by prts is u dv = uv v du Tips If the integrl is in the form of x n ln x, tke u = ln x nd dv = x n. If the integrl is in the form of e x sin bx or e x cos bx, tke u = sin bx nd dv = e x. Or, we cn use the substitution the other wy round by tking u = e x nd dv = sin x. If the integrl is in the form of x n e x, x n sin x or use the substitution u = x n nd dv = e x, sin x, cos x. x n cos x, Exmple 6 Evlute the following integrl using the following integrls. x ln x. ln x 3. xe x xe x x sin x 6
7 6. 7. π x cos x x cos x π x sin x e x cos x x ln x 3.3 Tbulr Method When integrting by prts hve mny repeted integrtions nd differentitions, tbulr method is very useful trick since it s mking the integrtion by prt procedure more net. Exmple 7 Evlute the integrl using the tbulr method. x 3 e x.. x sin 3x. 3. x sec x. 4. x (x e 3x cos x 6. cos 5x sin 4x 3.4 Integrtion by Prtil Frction By using this technique, we rewrite the integrl which initlly in product nd quotient form, in terms of frction. In prctice the integrl is in the form of polynomil of rtionl functions, where the denomintor usully cn be fctorised. Though pretty much most of polynomil of degree cn be fctorised, ber in mind tht if the 7
8 fctor on the numertor results to complicted lgebr, you re dvised to use other method ie substitution or inverse trigonometric nd inverse hyperbolic. If the numertor hs the sme order of degree or more with the denomintor, it becomes n improper frction. For this prticulr cse, you re required to use long division of polynomils. Exmple 8 Evlute the integrl by using prtil frctions. 3x + x + 3x +. x 3 3x + x x + 7 (x x + x + 6 x 3 + x + x x x + x 3 x x x + 3x + x 5 5x (x + 4 Integrls of Hyperbolic Functions Integrls of hyperbolic functions re similr to the integrls of trigonometric functions. You my refer to the formul to find the ntiderivtive of the hyperbolic functions. As for the more complicted integrls, we cn use the three methods described erlier. Exmple. Evlute the following integrls sinh x b 5 cosh 3x c sinh x 8
9 d e f g cosh x 3 sech 4x cosh (5x + cosh x + 3 sinh x. By using the definitions show tht Hence show tht cosh x + sinh x = cosh x sinh x. cosh x + sinh x = ( e. 3. By using the tbulr method, evlute the following integrl x 3 cosh x b x sinh 3x c e 3x cosh x d sin x sinh 3x 5 Integrtion Involving Inverse Hyperbolic Functions nd Inverse Trigonometric Functions When integrtion involving inverse hyperbolic nd inverse trigonometric functions re in the form of x sinh - x, x cosh - x, tn - x, they cn be integrted using integrtion by prt(ibp sin - x, u dv = uv v du. It is solvble if we choose the inverse functions s u in the IBP formul, since the inverse function cnnot be integrted directly, nd cn only undergo differentition. Another form is when the integrnd is in the form of frction. Usully when we re given n integrl in frction form, we first try the method of substitution. If it is 9
10 not working, the we try the method of prtil frction. It is not necessrily in tht order, it cn be interchnge. However if both methods re not pplicble, then it must hve involved inverse hyperbolic or inverse trigonometric. inverse hyperbolic/trigonometric is in the form of Ax + B or Ax + B or Generlly the integrtion tht will result to x Ax + B. (5. Another form is Ax + Bx + C or Ax + Bx + C or x Ax + Bx + C, (5. tht requires completing the squre to get the form in Eqution 5.. Here, x is ny function of x. Once we hve the form s in Eqution 5., we just need to mtch it with the formul given by using the method of substitution to get the ntiderivtive tht involve the inverse trnscendentl function. It would be useful if we fmilirize ourselves with the technique of completing the squre tht will be used frequently in trnsforming the integrnd. Completing the squre is trnsforming the qudrtic eqution into x + bx + c =, (x + d + e =, where d = b b nd e = c 4. Prctice : Rewrite this expression by completing the squre. x + 4x +. x 6x 3. x 8x x + 7x + 3 Another pproch is to mke trigonometric substitution, which is originlly how the formul mterilized to the generl form s we cn use now. The form nd
11 respectively the substitution re given s : if x, substitute x = sin θ or x = cos θ (5.3 if + x, substitute x = sinh θ or x = tn θ (5.4 if x, substitute x = cosh θ or x = sec θ, (5.5 where is constnt. 5. Integrtion Involving Inverse Trigonometric Functions In previous chpter we hve lerned how to do the differentition of the inverse function using the formul. Hence to find the integrl, the ide is the sme. Wht we need to do is to mnipulte the integrnd(provided tht we re sure tht the integrl will result to inverse trigonometric or inverse hyperbolic, so tht it mtches the formul. Recll tht the differentition of inverse trigonometric is given s d [sin- x] = x d [cos- x] = x d [tn- x] = + x d [cot- x] = + x d [sec- x] = x x d [cosec- x] = x x.
12 Therefore the integrl formul is the ntiderivtive of those nd given s = x sin- x + C, x < = x cos- x + C, x < + x = tn- x + C + x = cot- x + C x x = sec- x + C, x > x x = cosec- x + C, x > Generlly, the integrl for the inverse trigonometric will look like this x, nd t first glnce, we know it looks like the differentition eqution of we let x = u, then = du. Hence x = du ( x u = sin- + C. d [sin- x]. If The sme procedure pplied on the pproprite integrnd similr to the form of differentition of tn - x nd sec - x. The generl formul for this type of integrnds re given s ( x x = sin- + C, x < + x = ( x tn- + C x x = ( x sec- + C, x >, where the integrtion formul tht ssocites with cos - x, cot - x nd cosec - x re just the negtive terms of the bove integrnds respectively. Exmple Evlute the following integrls.. 5 x. 9 + x
13 x x 4 9x 4 + 9x x 4x x x x + 4x + 5 x + x + x / x x x + (x x x 3 sin - x cos - x x tn - x 5. Integrtion Involving Inverse Hyperbolic Functions The procedure of solving the integrtion involving inverse hyperbolic functions is similr to the procedure of solving the integrting involving inverse trigonometric 3
14 functions. The generl formul for the integrnd tht will result to the inverse hyperbolic functions re given s ( x + x = sinh- x + C, > ( x x = cosh- x + C, x > x = ( x tnh- + C, if x < x = ( x coth- + C, if x > ( x x x = sech- + C, if < x < ( x x + x = cosech- + C, if < x < Exmple Evlute the following integrl.. x + x e x. 6 e x x x 6, x > 4 4 x, x < x 4x x x 4x + 5 4
15 . 5 3 x x.. (x + x + x + x x tnh - x 6 Conclusion In this topic, you re required to know how to integrte the integrls tht involve trigonometric, inverse trigonometric, hyperbolic nd inverse hyperbolic functions. Wht you should know first is to mster ll the techniques of integrtion which re Integrtion by Substitution Integrtion by Prts( nd Tbulr Method Integrtion by Prtil Frctions Once you hve fmilir yourselves with ll these techniques, the next step is to recognised the pttern of which technique you should use given n integrl. There is no trick or shortcut to it. It is dvised for you to do s mny integrtion prctises s you cn, then you will be ble to recognised the pttern. However it is suggested tht (but don t tke the following s strict rule tht you hve to follow for you to lwys try the method of substitution first. If it s not working ie the resulting substitution is not simple lgebric eqution, then you cn try prtil frction (if the integrl is in frction, nd followed by integrtion by prts. If you were given n integrtion in the form of frction with the denomintor is in squre root form, try the method of substitution first, if it isn t working, see if the integrtion of inverse trigonometric or hyperbolic works. If you were given n integrtion in form of frction of polynomil, check the order of the power. If the order of the numertor is the sme or greter thn the denomintor, then use long division first. 5
Integration. October 25, 2016
Integrtion October 5, 6 Introduction We hve lerned in previous chpter on how to do the differentition. It is conventionl in mthemtics tht we re supposed to lern bout the integrtion s well. As you my hve
More informationImproper Integrals. October 4, 2017
Improper Integrls October 4, 7 Introduction We hve seen how to clculte definite integrl when the it is rel number. However, there re times when we re interested to compute the integrl sy for emple 3. Here
More informationMath 142, Exam 1 Information.
Mth 14, Exm 1 Informtion. 9/14/10, LC 41, 9:30-10:45. Exm 1 will be bsed on: Sections 7.1-7.5. The corresponding ssigned homework problems (see http://www.mth.sc.edu/ boyln/sccourses/14f10/14.html) At
More informationIntroduction to Integration
Introduction to Integrtion Definite integrls of piecewise constnt functions A constnt function is function of the form Integrtion is two things t the sme time: A form of summtion. The opposite of differentition.
More informationUnit #9 : Definite Integral Properties, Fundamental Theorem of Calculus
Unit #9 : Definite Integrl Properties, Fundmentl Theorem of Clculus Gols: Identify properties of definite integrls Define odd nd even functions, nd reltionship to integrl vlues Introduce the Fundmentl
More informationThe Fundamental Theorem of Calculus
MATH 6 The Fundmentl Theorem of Clculus The Fundmentl Theorem of Clculus (FTC) gives method of finding the signed re etween the grph of f nd the x-xis on the intervl [, ]. The theorem is: FTC: If f is
More informationa < a+ x < a+2 x < < a+n x = b, n A i n f(x i ) x. i=1 i=1
Mth 33 Volume Stewrt 5.2 Geometry of integrls. In this section, we will lern how to compute volumes using integrls defined by slice nlysis. First, we recll from Clculus I how to compute res. Given the
More information10.5 Graphing Quadratic Functions
0.5 Grphing Qudrtic Functions Now tht we cn solve qudrtic equtions, we wnt to lern how to grph the function ssocited with the qudrtic eqution. We cll this the qudrtic function. Grphs of Qudrtic Functions
More informationSolutions to Math 41 Final Exam December 12, 2011
Solutions to Mth Finl Em December,. ( points) Find ech of the following its, with justifiction. If there is n infinite it, then eplin whether it is or. ( ) / ln() () (5 points) First we compute the it:
More informationSubtracting Fractions
Lerning Enhncement Tem Model Answers: Adding nd Subtrcting Frctions Adding nd Subtrcting Frctions study guide. When the frctions both hve the sme denomintor (bottom) you cn do them using just simple dding
More information9 4. CISC - Curriculum & Instruction Steering Committee. California County Superintendents Educational Services Association
9. CISC - Curriculum & Instruction Steering Committee The Winning EQUATION A HIGH QUALITY MATHEMATICS PROFESSIONAL DEVELOPMENT PROGRAM FOR TEACHERS IN GRADES THROUGH ALGEBRA II STRAND: NUMBER SENSE: Rtionl
More informationRational Numbers---Adding Fractions With Like Denominators.
Rtionl Numbers---Adding Frctions With Like Denomintors. A. In Words: To dd frctions with like denomintors, dd the numertors nd write the sum over the sme denomintor. B. In Symbols: For frctions c nd b
More informationsuch that the S i cover S, or equivalently S
MATH 55 Triple Integrls Fll 16 1. Definition Given solid in spce, prtition of consists of finite set of solis = { 1,, n } such tht the i cover, or equivlently n i. Furthermore, for ech i, intersects i
More informationMA1008. Calculus and Linear Algebra for Engineers. Course Notes for Section B. Stephen Wills. Department of Mathematics. University College Cork
MA1008 Clculus nd Liner Algebr for Engineers Course Notes for Section B Stephen Wills Deprtment of Mthemtics University College Cork s.wills@ucc.ie http://euclid.ucc.ie/pges/stff/wills/teching/m1008/ma1008.html
More information1. SEQUENCES INVOLVING EXPONENTIAL GROWTH (GEOMETRIC SEQUENCES)
Numbers nd Opertions, Algebr, nd Functions 45. SEQUENCES INVOLVING EXPONENTIAL GROWTH (GEOMETRIC SEQUENCES) In sequence of terms involving eponentil growth, which the testing service lso clls geometric
More information)
Chpter Five /SOLUTIONS Since the speed ws between nd mph during this five minute period, the fuel efficienc during this period is between 5 mpg nd 8 mpg. So the fuel used during this period is between
More informationLecture 7: Integration Techniques
Lecture 7: Integrtion Techniques Antiderivtives nd Indefinite Integrls. In differentil clculus, we were interested in the derivtive of given rel-vlued function, whether it ws lgeric, eponentil or logrithmic.
More information12-B FRACTIONS AND DECIMALS
-B Frctions nd Decimls. () If ll four integers were negtive, their product would be positive, nd so could not equl one of them. If ll four integers were positive, their product would be much greter thn
More informationSection 10.4 Hyperbolas
66 Section 10.4 Hyperbols Objective : Definition of hyperbol & hyperbols centered t (0, 0). The third type of conic we will study is the hyperbol. It is defined in the sme mnner tht we defined the prbol
More informationWebAssign Lesson 1-3a Substitution Part 1 (Homework)
WeAssign Lesson -3 Sustitution Prt (Homework) Current Score : / 3 Due : Fridy, June 7 04 :00 AM MDT Jimos Skriletz Mth 75, section 3, Summer 04 Instructor: Jimos Skriletz. /.5 points Suppose you hve the
More informationIf f(x, y) is a surface that lies above r(t), we can think about the area between the surface and the curve.
Line Integrls The ide of line integrl is very similr to tht of single integrls. If the function f(x) is bove the x-xis on the intervl [, b], then the integrl of f(x) over [, b] is the re under f over the
More information6.2 Volumes of Revolution: The Disk Method
mth ppliction: volumes by disks: volume prt ii 6 6 Volumes of Revolution: The Disk Method One of the simplest pplictions of integrtion (Theorem 6) nd the ccumultion process is to determine so-clled volumes
More informationPointwise convergence need not behave well with respect to standard properties such as continuity.
Chpter 3 Uniform Convergence Lecture 9 Sequences of functions re of gret importnce in mny res of pure nd pplied mthemtics, nd their properties cn often be studied in the context of metric spces, s in Exmples
More informationMath 464 Fall 2012 Notes on Marginal and Conditional Densities October 18, 2012
Mth 464 Fll 2012 Notes on Mrginl nd Conditionl Densities klin@mth.rizon.edu October 18, 2012 Mrginl densities. Suppose you hve 3 continuous rndom vribles X, Y, nd Z, with joint density f(x,y,z. The mrginl
More informationSIMPLIFYING ALGEBRA PASSPORT.
SIMPLIFYING ALGEBRA PASSPORT www.mthletics.com.u This booklet is ll bout turning complex problems into something simple. You will be ble to do something like this! ( 9- # + 4 ' ) ' ( 9- + 7-) ' ' Give
More informationP(r)dr = probability of generating a random number in the interval dr near r. For this probability idea to make sense we must have
Rndom Numers nd Monte Crlo Methods Rndom Numer Methods The integrtion methods discussed so fr ll re sed upon mking polynomil pproximtions to the integrnd. Another clss of numericl methods relies upon using
More informationSection 3.1: Sequences and Series
Section.: Sequences d Series Sequences Let s strt out with the definition of sequence: sequence: ordered list of numbers, often with definite pttern Recll tht in set, order doesn t mtter so this is one
More informationHyperbolas. Definition of Hyperbola
CHAT Pre-Clculus Hyperols The third type of conic is clled hyperol. For n ellipse, the sum of the distnces from the foci nd point on the ellipse is fixed numer. For hyperol, the difference of the distnces
More informationLecture Overview. Knowledge-based systems in Bioinformatics, 1MB602. Procedural abstraction. The sum procedure. Integration as a procedure
Lecture Overview Knowledge-bsed systems in Bioinformtics, MB6 Scheme lecture Procedurl bstrction Higher order procedures Procedures s rguments Procedures s returned vlues Locl vribles Dt bstrction Compound
More informationChapter Spline Method of Interpolation More Examples Electrical Engineering
Chpter. Spline Method of Interpoltion More Exmples Electricl Engineering Exmple Thermistors re used to mesure the temperture of bodies. Thermistors re bsed on mterils chnge in resistnce with temperture.
More informationMath 17 - Review. Review for Chapter 12
Mth 17 - eview Ying Wu eview for hpter 12 1. Given prmetric plnr curve x = f(t), y = g(t), where t b, how to eliminte the prmeter? (Use substitutions, or use trigonometry identities, etc). How to prmeterize
More information1 The Definite Integral
The Definite Integrl Definition. Let f be function defined on the intervl [, b] where
More informationUnit 5 Vocabulary. A function is a special relationship where each input has a single output.
MODULE 3 Terms Definition Picture/Exmple/Nottion 1 Function Nottion Function nottion is n efficient nd effective wy to write functions of ll types. This nottion llows you to identify the input vlue with
More informationCOMP 423 lecture 11 Jan. 28, 2008
COMP 423 lecture 11 Jn. 28, 2008 Up to now, we hve looked t how some symols in n lphet occur more frequently thn others nd how we cn sve its y using code such tht the codewords for more frequently occuring
More informationGraphing Conic Sections
Grphing Conic Sections Definition of Circle Set of ll points in plne tht re n equl distnce, clled the rdius, from fixed point in tht plne, clled the center. Grphing Circle (x h) 2 + (y k) 2 = r 2 where
More information50 AMC LECTURES Lecture 2 Analytic Geometry Distance and Lines. can be calculated by the following formula:
5 AMC LECTURES Lecture Anlytic Geometry Distnce nd Lines BASIC KNOWLEDGE. Distnce formul The distnce (d) between two points P ( x, y) nd P ( x, y) cn be clculted by the following formul: d ( x y () x )
More informationx )Scales are the reciprocal of each other. e
9. Reciprocls A Complete Slide Rule Mnul - eville W Young Chpter 9 Further Applictions of the LL scles The LL (e x ) scles nd the corresponding LL 0 (e -x or Exmple : 0.244 4.. Set the hir line over 4.
More information9.1 apply the distance and midpoint formulas
9.1 pply the distnce nd midpoint formuls DISTANCE FORMULA MIDPOINT FORMULA To find the midpoint between two points x, y nd x y 1 1,, we Exmple 1: Find the distnce between the two points. Then, find the
More informationIterated Integrals. f (x; y) dy dx. p(x) To evaluate a type I integral, we rst evaluate the inner integral Z q(x) f (x; y) dy.
Iterted Integrls Type I Integrls In this section, we begin the study of integrls over regions in the plne. To do so, however, requires tht we exmine the importnt ide of iterted integrls, in which inde
More informationThe Basic Properties of the Integral
The Bsic Properties of the Integrl When we compute the derivtive of complicted function, like + sin, we usull use differentition rules, like d [f()+g()] d f()+ d g(), to reduce the computtion d d d to
More informationRay surface intersections
Ry surfce intersections Some primitives Finite primitives: polygons spheres, cylinders, cones prts of generl qudrics Infinite primitives: plnes infinite cylinders nd cones generl qudrics A finite primitive
More information8.2 Areas in the Plane
39 Chpter 8 Applictions of Definite Integrls 8. Ares in the Plne Wht ou will lern out... Are Between Curves Are Enclosed Intersecting Curves Boundries with Chnging Functions Integrting with Respect to
More informationThe Math Learning Center PO Box 12929, Salem, Oregon Math Learning Center
Resource Overview Quntile Mesure: Skill or Concept: 80Q Multiply two frctions or frction nd whole numer. (QT N ) Excerpted from: The Mth Lerning Center PO Box 99, Slem, Oregon 9709 099 www.mthlerningcenter.org
More informationa(e, x) = x. Diagrammatically, this is encoded as the following commutative diagrams / X
4. Mon, Sept. 30 Lst time, we defined the quotient topology coming from continuous surjection q : X! Y. Recll tht q is quotient mp (nd Y hs the quotient topology) if V Y is open precisely when q (V ) X
More informationMatrices and Systems of Equations
Mtrices Mtrices nd Sstems of Equtions A mtri is rectngulr rr of rel numbers. CHAT Pre-Clculus Section 8. m m m............ n n n mn We will use the double subscript nottion for ech element of the mtri.
More informationThis notebook investigates the properties of non-integer differential operators using Fourier analysis.
Frctionl erivtives.nb Frctionl erivtives by Fourier ecomposition by Eric Thrne 4/9/6 This notebook investigtes the properties of non-integer differentil opertors using Fourier nlysis. In[]:=
More informationB. Definition: The volume of a solid of known integrable cross-section area A(x) from x = a
Mth 176 Clculus Sec. 6.: Volume I. Volume By Slicing A. Introduction We will e trying to find the volume of solid shped using the sum of cross section res times width. We will e driving towrd developing
More informationStudy Guide for Exam 3
Mth 05 Elementry Algebr Fll 00 Study Guide for Em Em is scheduled for Thursdy, November 8 th nd ill cover chpters 5 nd. You my use "5" note crd (both sides) nd scientific clcultor. You re epected to no
More informationEssential Question What are some of the characteristics of the graph of a rational function?
8. TEXAS ESSENTIAL KNOWLEDGE AND SKILLS A..A A..G A..H A..K Grphing Rtionl Functions Essentil Question Wht re some of the chrcteristics of the grph of rtionl function? The prent function for rtionl functions
More information4452 Mathematical Modeling Lecture 4: Lagrange Multipliers
Mth Modeling Lecture 4: Lgrnge Multipliers Pge 4452 Mthemticl Modeling Lecture 4: Lgrnge Multipliers Lgrnge multipliers re high powered mthemticl technique to find the mximum nd minimum of multidimensionl
More informationAre You Ready for Algebra 3/Trigonometry? Summer Packet **Required for all Algebra 3/Trig CP and Honors students**
Are You Red for Algebr /Trigonometr? Summer Pcket **Required for ll Algebr /Trig CP nd Honors students** Pge of The Algebr /Trigonometr course prepres students for Clculus nd college science courses. In
More informationf[a] x + f[a + x] x + f[a +2 x] x + + f[b x] x
Bsic Integrtion This chpter contins the fundmentl theory of integrtion. We begin with some problems to motivte the min ide: pproximtion by sum of slices. The chpter confronts this squrely, nd Chpter 3
More informationStained Glass Design. Teaching Goals:
Stined Glss Design Time required 45-90 minutes Teching Gols: 1. Students pply grphic methods to design vrious shpes on the plne.. Students pply geometric trnsformtions of grphs of functions in order to
More informationAngle properties of lines and polygons
chievement Stndrd 91031 pply geometric resoning in solving problems Copy correctly Up to 3% of workbook Copying or scnning from ES workbooks is subject to the NZ Copyright ct which limits copying to 3%
More informationCOMBINATORIAL PATTERN MATCHING
COMBINATORIAL PATTERN MATCHING Genomic Repets Exmple of repets: ATGGTCTAGGTCCTAGTGGTC Motivtion to find them: Genomic rerrngements re often ssocited with repets Trce evolutionry secrets Mny tumors re chrcterized
More informationClass-XI Mathematics Conic Sections Chapter-11 Chapter Notes Key Concepts
Clss-XI Mthemtics Conic Sections Chpter-11 Chpter Notes Key Concepts 1. Let be fixed verticl line nd m be nother line intersecting it t fixed point V nd inclined to it t nd ngle On rotting the line m round
More informationIntroduction Transformation formulae Polar graphs Standard curves Polar equations Test GRAPHS INU0114/514 (MATHS 1)
POLAR EQUATIONS AND GRAPHS GEOMETRY INU4/54 (MATHS ) Dr Adrin Jnnett MIMA CMth FRAS Polr equtions nd grphs / 6 Adrin Jnnett Objectives The purpose of this presenttion is to cover the following topics:
More informationCHAPTER 8 Quasi-interpolation methods
CHAPTER 8 Qusi-interpoltion methods In Chpter 5 we considered number of methods for computing spline pproximtions. The strting point for the pproximtion methods is dt set tht is usully discrete nd in the
More informationBefore We Begin. Introduction to Spatial Domain Filtering. Introduction to Digital Image Processing. Overview (1): Administrative Details (1):
Overview (): Before We Begin Administrtive detils Review some questions to consider Winter 2006 Imge Enhncement in the Sptil Domin: Bsics of Sptil Filtering, Smoothing Sptil Filters, Order Sttistics Filters
More informationCS481: Bioinformatics Algorithms
CS481: Bioinformtics Algorithms Cn Alkn EA509 clkn@cs.ilkent.edu.tr http://www.cs.ilkent.edu.tr/~clkn/teching/cs481/ EXACT STRING MATCHING Fingerprint ide Assume: We cn compute fingerprint f(p) of P in
More informationZZ - Advanced Math Review 2017
ZZ - Advnced Mth Review Mtrix Multipliction Given! nd! find the sum of the elements of the product BA First, rewrite the mtrices in the correct order to multiply The product is BA hs order x since B is
More informationMATH 25 CLASS 5 NOTES, SEP
MATH 25 CLASS 5 NOTES, SEP 30 2011 Contents 1. A brief diversion: reltively prime numbers 1 2. Lest common multiples 3 3. Finding ll solutions to x + by = c 4 Quick links to definitions/theorems Euclid
More informationDouble Integrals. MATH 375 Numerical Analysis. J. Robert Buchanan. Fall Department of Mathematics. J. Robert Buchanan Double Integrals
Double Integrls MATH 375 Numericl Anlysis J. Robert Buchnn Deprtment of Mthemtics Fll 2013 J. Robert Buchnn Double Integrls Objectives Now tht we hve discussed severl methods for pproximting definite integrls
More informationINTRODUCTION TO SIMPLICIAL COMPLEXES
INTRODUCTION TO SIMPLICIAL COMPLEXES CASEY KELLEHER AND ALESSANDRA PANTANO 0.1. Introduction. In this ctivity set we re going to introduce notion from Algebric Topology clled simplicil homology. The min
More informationYoplait with Areas and Volumes
Yoplit with Ares nd Volumes Yoplit yogurt comes in two differently shped continers. One is truncted cone nd the other is n ellipticl cylinder (see photos below). In this exercise, you will determine the
More informationCS311H: Discrete Mathematics. Graph Theory IV. A Non-planar Graph. Regions of a Planar Graph. Euler s Formula. Instructor: Işıl Dillig
CS311H: Discrete Mthemtics Grph Theory IV Instructor: Işıl Dillig Instructor: Işıl Dillig, CS311H: Discrete Mthemtics Grph Theory IV 1/25 A Non-plnr Grph Regions of Plnr Grph The plnr representtion of
More information5/9/17. Lesson 51 - FTC PART 2. Review FTC, PART 1. statement as the Integral Evaluation Theorem as it tells us HOW to evaluate the definite integral
Lesson - FTC PART 2 Review! We hve seen definition/formul for definite integrl s n b A() = lim f ( i )Δ = f ()d = F() = F(b) F() n i=! where F () = f() (or F() is the ntiderivtive of f() b! And hve seen
More informationEXPONENTIAL & POWER GRAPHS
Eponentil & Power Grphs EXPONENTIAL & POWER GRAPHS www.mthletics.com.u Eponentil EXPONENTIAL & Power & Grphs POWER GRAPHS These re grphs which result from equtions tht re not liner or qudrtic. The eponentil
More informationCOMPUTER SCIENCE 123. Foundations of Computer Science. 6. Tuples
COMPUTER SCIENCE 123 Foundtions of Computer Science 6. Tuples Summry: This lecture introduces tuples in Hskell. Reference: Thompson Sections 5.1 2 R.L. While, 2000 3 Tuples Most dt comes with structure
More informationPhysicsAndMathsTutor.com
M Centres of Mss - Rigid bodies nd composites. Figure A continer is formed by removing right circulr solid cone of height l from uniform solid right circulr cylinder of height 6l. The centre O of the plne
More informationMath 35 Review Sheet, Spring 2014
Mth 35 Review heet, pring 2014 For the finl exm, do ny 12 of the 15 questions in 3 hours. They re worth 8 points ech, mking 96, with 4 more points for netness! Put ll your work nd nswers in the provided
More information6.3 Definite Integrals and Antiderivatives
Section 6. Definite Integrls nd Antiderivtives 8 6. Definite Integrls nd Antiderivtives Wht ou will lern out... Properties of Definite Integrls Averge Vlue of Function Men Vlue Theorem for Definite Integrls
More information6.3 Volumes. Just as area is always positive, so is volume and our attitudes towards finding it.
6.3 Volumes Just s re is lwys positive, so is volume nd our ttitudes towrds finding it. Let s review how to find the volume of regulr geometric prism, tht is, 3-dimensionl oject with two regulr fces seprted
More informationRevisit: Limits at Infinity
Revisit: Limits t Infinity Limits t Infinity: Wewrite to men the following: f () =L, or f ()! L s! + Conceptul Mening: Thevlueoff () willbesclosetol s we like when is su Forml Definition: Forny"> 0(nomtterhowsmll)thereeistsnM
More informationRATIONAL EQUATION: APPLICATIONS & PROBLEM SOLVING
RATIONAL EQUATION: APPLICATIONS & PROBLEM SOLVING When finding the LCD of problem involving the ddition or subtrction of frctions, it my be necessry to fctor some denomintors to discover some restricted
More informationName Date Class. cot. tan. cos. 1 cot 2 csc 2
Fundmentl Trigonometric Identities To prove trigonometric identit, use the fundmentl identities to mke one side of the eqution resemle the other side. Reciprocl nd Rtio Identities csc sec sin cos Negtive-Angle
More informationLecture 5: Spatial Analysis Algorithms
Lecture 5: Sptil Algorithms GEOG 49: Advnced GIS Sptil Anlsis Algorithms Bsis of much of GIS nlsis tod Mnipultion of mp coordintes Bsed on Eucliden coordinte geometr http://stronom.swin.edu.u/~pbourke/geometr/
More informationarxiv: v2 [math.ho] 4 Jun 2012
Volumes of olids of Revolution. Unified pproch Jorge Mrtín-Morles nd ntonio M. Oller-Mrcén jorge@unizr.es, oller@unizr.es rxiv:5.v [mth.ho] Jun Centro Universitrio de l Defens - IUM. cdemi Generl Militr,
More informationSimplifying Algebra. Simplifying Algebra. Curriculum Ready.
Simplifying Alger Curriculum Redy www.mthletics.com This ooklet is ll out turning complex prolems into something simple. You will e le to do something like this! ( 9- # + 4 ' ) ' ( 9- + 7-) ' ' Give this
More informationFig.25: the Role of LEX
The Lnguge for Specifying Lexicl Anlyzer We shll now study how to uild lexicl nlyzer from specifiction of tokens in the form of list of regulr expressions The discussion centers round the design of n existing
More informationMATH 2530: WORKSHEET 7. x 2 y dz dy dx =
MATH 253: WORKSHT 7 () Wrm-up: () Review: polr coordintes, integrls involving polr coordintes, triple Riemnn sums, triple integrls, the pplictions of triple integrls (especilly to volume), nd cylindricl
More information5 Regular 4-Sided Composition
Xilinx-Lv User Guide 5 Regulr 4-Sided Composition This tutoril shows how regulr circuits with 4-sided elements cn be described in Lv. The type of regulr circuits tht re discussed in this tutoril re those
More informationQuestions About Numbers. Number Systems and Arithmetic. Introduction to Binary Numbers. Negative Numbers?
Questions About Numbers Number Systems nd Arithmetic or Computers go to elementry school How do you represent negtive numbers? frctions? relly lrge numbers? relly smll numbers? How do you do rithmetic?
More informationECE 468/573 Midterm 1 September 28, 2012
ECE 468/573 Midterm 1 September 28, 2012 Nme:! Purdue emil:! Plese sign the following: I ffirm tht the nswers given on this test re mine nd mine lone. I did not receive help from ny person or mteril (other
More informationLecture 10 Evolutionary Computation: Evolution strategies and genetic programming
Lecture 10 Evolutionry Computtion: Evolution strtegies nd genetic progrmming Evolution strtegies Genetic progrmming Summry Negnevitsky, Person Eduction, 2011 1 Evolution Strtegies Another pproch to simulting
More informationIntroduction. Chapter 4: Complex Integration. Introduction (Cont d)
Introduction Chpter 4: Complex Integrtion Li, Yongzho Stte Key Lbortory of Integrted Services Networks, Xidin University October 10, 2010 The two-dimensionl nture of the complex plne required us to generlize
More informationMTH 146 Conics Supplement
105- Review of Conics MTH 146 Conics Supplement In this section we review conics If ou ne more detils thn re present in the notes, r through section 105 of the ook Definition: A prol is the set of points
More informationWhat are suffix trees?
Suffix Trees 1 Wht re suffix trees? Allow lgorithm designers to store very lrge mount of informtion out strings while still keeping within liner spce Allow users to serch for new strings in the originl
More informationLIMITS AND CONTINUITY
LIMITS AND CONTINUITY Joe McBride/Stone/Gett Imges Air resistnce prevents the velocit of skdiver from incresing indefinitel. The velocit pproches it, clled the terminl velocit. The development of clculus
More informationMath 4 Review for Quarter 2 Cumulative Test
Mth 4 Review for Qurter 2 Cumultive Test Nme: I. Right Tringle Trigonometry (3.1-3.3) Key Fcts Pythgoren Theorem - In right tringle, 2 + b 2 = c 2 where c is the hypotenuse s shown below. c b Trigonometric
More informationIf you are at the university, either physically or via the VPN, you can download the chapters of this book as PDFs.
Lecture 5 Wlks, Trils, Pths nd Connectedness Reding: Some of the mteril in this lecture comes from Section 1.2 of Dieter Jungnickel (2008), Grphs, Networks nd Algorithms, 3rd edition, which is ville online
More informationONU Calculus I Math 1631
ONU Clculus I Mth 1631 2013-2014 Syllus Mrs. Trudy Thompson tthompson@lcchs.edu Text: Clculus 8 th Edition, Anton, Bivens nd Dvis Prerequisites: C or etter in Pre-Clc nd techer s permission This course
More informationEXPONENT RULES Add Multiply Subtraction Flip
Algebr II Finl Em Review Nme Chpter 7 REVIEW: EXPONENT RULES Add Multiply Subtrction Flip Simplify the epression using the properties of eponents. Assume ll vribles re positive. 4 4. 8 8.. 4. 5. 9 9 5
More informationArea & Volume. Chapter 6.1 & 6.2 September 25, y = 1! x 2. Back to Area:
Bck to Are: Are & Volume Chpter 6. & 6. Septemer 5, 6 We cn clculte the re etween the x-xis nd continuous function f on the intervl [,] using the definite integrl:! f x = lim$ f x * i )%x n i= Where fx
More informationIntroduction to Algebra
INTRODUCTORY ALGEBRA Mini-Leture 1.1 Introdution to Alger Evlute lgeri expressions y sustitution. Trnslte phrses to lgeri expressions. 1. Evlute the expressions when =, =, nd = 6. ) d) 5 10. Trnslte eh
More informationDr. D.M. Akbar Hussain
Dr. D.M. Akr Hussin Lexicl Anlysis. Bsic Ide: Red the source code nd generte tokens, it is similr wht humns will do to red in; just tking on the input nd reking it down in pieces. Ech token is sequence
More informationWhat do all those bits mean now? Number Systems and Arithmetic. Introduction to Binary Numbers. Questions About Numbers
Wht do ll those bits men now? bits (...) Number Systems nd Arithmetic or Computers go to elementry school instruction R-formt I-formt... integer dt number text chrs... floting point signed unsigned single
More informationChapter 2 Sensitivity Analysis: Differential Calculus of Models
Chpter 2 Sensitivity Anlysis: Differentil Clculus of Models Abstrct Models in remote sensing nd in science nd engineering, in generl re, essentilly, functions of discrete model input prmeters, nd/or functionls
More informationCS143 Handout 07 Summer 2011 June 24 th, 2011 Written Set 1: Lexical Analysis
CS143 Hndout 07 Summer 2011 June 24 th, 2011 Written Set 1: Lexicl Anlysis In this first written ssignment, you'll get the chnce to ply round with the vrious constructions tht come up when doing lexicl
More information1.1 Lines AP Calculus
. Lines AP Clculus. LINES Notecrds from Section.: Rules for Rounding Round or Truncte ll finl nswers to 3 deciml plces. Do NOT round before ou rech our finl nswer. Much of Clculus focuses on the concept
More information