3.10"iNumbers" A"Practice"Understanding"Task"
|
|
|
- Alban Walters
- 6 years ago
- Views:
Transcription
1 iNumbers APracticeUnderstandingTask Inordertofindsolutionstoallquadraticequations,wehavehadto extendthenumbersystemtoincludecomplexnumbers. 2013www.flickr.com/photos/milongadas Dothefollowingforeachoftheproblemsbelow: Choosethebestwordtocompleteeachconjecture. Afteryouhavemadeaconjecture,createatleastfourexamplestoshowwhyyourconjecture istrue. IfyoufindacounterOexample,changeyourconjecturetofityourwork. Conjecture#1:Thesumoftwointegersis[always,sometime,never]aninteger. Conjecture#2:Thesumoftworationalnumbersis[always,sometimes,never]arationalnumber. Conjecture#3:Thesumoftwoirrationalnumbersis[always,sometimes,never]anirrational number. 2013MathematicsVisionProject MVP InpartnershipwiththeUtahStateOfficeofEducation LicensedundertheCreativeCommonsAttribution4NonCommercial4ShareAlike3.0Unportedlicense.
2 63 Conjecture#4:Thesumoftworealnumbersis[always,sometimes,never]arealnumber. Conjecture#5:Thesumoftwocomplexnumbersis[always,sometimes,never]acomplexnumber. Conjecture#6:Theproductoftwointegersis[always,sometime,never]aninteger. Conjecture#7:Thequotientoftwointegersis[always,sometime,never]aninteger. Conjecture#8:Theproductoftworationalnumbersis[always,sometimes,never]arational number. Conjecture#9:Thequotientoftworationalnumbersis[always,sometimes,never]arational number Conjecture#10:Theproductoftwoirrationalnumbersis[always,sometimes,never]anirrational number. 2013MathematicsVisionProject MVP InpartnershipwiththeUtahStateOfficeofEducation LicensedundertheCreativeCommonsAttribution4NonCommercial4ShareAlike3.0Unportedlicense.
3 64 Conjecture#11:Theproductoftworealnumbersis[always,sometimes,never]arealnumber. Conjecture#12:Theproductoftwocomplexnumbersis[always,sometimes,never]acomplex number. 13.Theratioofthecircumferenceofacircletoitsdiameterisgivenbytheirrationalnumberπ.Can thediameterofacircleandthecircumferenceofthesamecirclebothberationalnumbers?explain whyorwhynot. TheArithmeticofPolynomials InthetaskToBeDetermined...wedefinedpolynomialstobeexpressionsofthefollowingform: a 0x n + a 1 x n 1 + a 2 x n 2 + a n 3 x 3 + a n 2 x 2 + a n 1 x + a n wherealloftheexponentsarepositiveintegersandallofthecoefficientsa 0...a nareconstants. Dothefollowingforeachoftheproblemsbelow: Choosethebestwordtocompleteeachconjecture. Afteryouhavemadeaconjecture,createatleastfourexamplestoshowwhyyourconjecture istrue. IfyoufindacounterOexample,changeyourconjecturetofityourwork. Conjecture#P1:Thesumoftwopolynomialsis[always,sometime,never]apolynomial. 2013MathematicsVisionProject MVP InpartnershipwiththeUtahStateOfficeofEducation LicensedundertheCreativeCommonsAttribution4NonCommercial4ShareAlike3.0Unportedlicense.
4 65 Conjecture#P2:Thedifferenceoftwopolynomialsis[always,sometime,never]apolynomial. Conjecture#P3:Theproductoftwopolynomialsis[always,sometime,never]apolynomial. 2013MathematicsVisionProject MVP InpartnershipwiththeUtahStateOfficeofEducation LicensedundertheCreativeCommonsAttribution4NonCommercial4ShareAlike3.0Unportedlicense.
5 3.10iNumbers TeacherNotes APracticeUnderstandingTask Purpose:Thepurposeofthistaskistopracticeworkingwiththearithmeticofirrationaland complexnumbersandtomakeconjecturesastowhichofthesetsofintegers,rationalnumbers, irrationalnumbers,realnumbersorcomplexnumbersareclosedundertheoperationsofaddition, subtractionandmultiplication;thatis,thesumorproductofanytwonumbersfromthesetalways producesanothernumberinthatset.studentsalsoexperimentwiththeclosureofthesetof polynomialfunctionsundertheoperationsofaddition,subtractionandmultiplication.fromthis perspective,thesetofintegersbehaveinthesamewayasthesetofpolynomials.oncepolynomial divisionhasbeenintroducedinmathematicsiii,itcanbeshownthatneithersetisclosedunderthe operationofdivision dividingtwointegersresultsinarationalnumberanddividingtwo polynomialsresultsinarationalfunction. CoreStandardsFocus: N.RN.3Explainwhythesumorproductoftworationalnumbersisrational;thatthesumofa rationalnumberandanirrationalnumberisirrational;andthattheproductofanonzerorational numberandanirrationalnumberisirrational. NoteforMathematicsII:ConnectN.RN.3tophysicalsituations,e.g.,findingtheperimeterofasquare ofarea2. N.CN.1Knowthereisacomplexnumberisuchthati 2 = 1,andeverycomplexnumberhastheform a+biwithaandbreal. N.CN.2Usetherelationi 2 = 1andthecommutative,associative,anddistributivepropertiestoadd, subtract,andmultiplycomplexnumbers. NoteforMathematicsII:Limittomultiplicationsthatinvolvei 2 asthehighestpowerofi. A.APR.1Understandthatpolynomialsformasystemanalogoustotheintegers,namely,theyare closedundertheoperationsofaddition,subtraction,andmultiplication;add,subtract,andmultiply polynomials. NoteforMathematicsII:Focusonpolynomialexpressionsthatsimplifytoformsthatarelinearor quadraticinapositiveintegerpowerofx. RelatedStandards: 2013MathematicsVisionProject MVP InpartnershipwiththeUtahStateOfficeofEducation LicensedundertheCreativeCommonsAttribution4NonCommercial4ShareAlike3.0Unportedlicense.
6 LaunchWholeClass): ExaminetheVenndiagramofthecomplexnumbersystemgivenatthebeginningofthetask.Ask studentstosuggestaquadraticequationthatwouldnothaveasolutionifwelimitedthesetof acceptablesolutionstoonlynaturalnumbersforexample, x 2 + 3x = 0wouldnotbesolvable). Whatifwelimitedthesetofacceptablesolutionstoonlyintegers?Then2x +1)3x 2) = 0 would notbesolvable.)whatifwelimitedthesetofacceptablesolutionstorationalnumbers?real numbers?pointouttostudentsthatwehaveexpandedournumbersystemtoallowustosolvea greatervarietyofequations. Letstudentsknowthattheirworktodayistomakeconjecturesaboutoperationswithinthesesets ofnumbers.canwealwaysdoarithmeticwithininasetofnumberswithouthavingtogooutside thesettogetananswer?mathematiciansrefertothispropertyofasetofnumbersasclosure. ExploreSmallGroup): Asstudentsexploretheconjectures,makesuretheyaretryingoutarangeofpossibilities,andtrying tosearchforcounteroexamplestothe always conjectures.remindstudentsoftheworkofthe previoustaskastheyconsideroperationswithirrationalandcomplexnumbers. Monitorstudents workwiththearithmeticofpolynomials:aretheywritingonlypolynomial expressionsintheirexamples?howdotheyaddtwopolynomialexpressions?dotheyonlyaddlike terms?howdotheymultiplypolynomialexpressionsthatcontainmultipleterms?dothey correctlyusethedistributiveproperty?howdotheymultiplyvariablesraisedtopowers? DiscussWholeClass): Asneeded,shareconjecturesandsupportingevidencewiththewholeclass.Discussthequestion aboutthedefinitionofπ = C d.sinceπisanirrationalnumber,thisratioimpliesthatthe circumferenceanddiameterofacirclecannotbothberationalnumbers.includingπinthe discussionofirrationalnumbersallowsyoutopointoutthatthereismoretothesetofirrational numbersthanjustradicals. Alsodiscussthealgebraicworkwithpolynomialexpressionstoresolveanymisconceptionsyoumay haveobservedduringthiswork. AlignedReady,Set,Go:SolvingQuadraticandOtherEquations MathematicsVisionProject MVP InpartnershipwiththeUtahStateOfficeofEducation LicensedundertheCreativeCommonsAttribution4NonCommercial4ShareAlike3.0Unportedlicense.
7 SolvingQuadraticandOtherEquations MATHEMATICSVISIONPROJECT MV P InpartnershipwiththeUtahStateOfficeofEducation LicensedundertheCreativeCommonsAttribution4NonCommercial4ShareAlike3.0Unportedlicense Ready,'Set,'Go' ' Ready' Topic:Attributesofquadraticsandotherfunctions 1.Summarizewhatyouhavelearnedaboutquadratic functionstothispoint.inadditiontoyourwrittenexplanationprovidegraphs,tablesandexamples toillustratewhatyouknow. 2.Inpriorworkyouhavelearnedagreatdealaboutbothlinearandexponentialfunctions. Compareandcontrastlinearandexponentialfunctionswithquadraticfunctions.Whatsimilarities ifanyarethereandwhatdifferencesaretherebetweenlinear,exponentialandquadratic functions? Name: 2013www.flickr.com/photos/milongadas 66
8 SolvingQuadraticandOtherEquations 3.10 Set' Topic:Operationsondifferenttypesofnumbers 3.TheNaturalnumbers,N,arejustthatthenumbersthatcomenaturallyorthecountingnumbers. Asanychildfirstlearnsnumberstheylearn1,2,3, WhatoperationsontheNaturalnumbers wouldcausetheneedforothertypesofnumbers?whatoperationonnaturalnumberscreatea needforintegersorrationalnumbersandsoforth.giveexamplesandexplain.) In'each'of'the'problems'below'use'the'given'items'to'determine'whether'or'not'it'is'possible' always,'sometimes'or'never'to'create'a'new'element*'that'is'in'the'desired'set.' 4.UsingtheoperationofadditionandelementsfromtheIntegers,Z,[always,sometime,never]an elementoftheirrationalnumbers,q,willbecreated.explain. 5.Considertheequation =c,where Nand N,cwillbeanInteger,Z[always, sometimes,never].explain. 6.Considertheequation =c,where Zand Z,thenis Z[sometimes,always,never]. Explain. *Thenumbersinanygivensetofnumbersmaybereferredtoaselementsoftheset.For' example,therationalnumberset,q,containselementsornumbersthatcanbewrittenintheform, whereaandbareintegervaluesb 0) MATHEMATICSVISIONPROJECT MV P InpartnershipwiththeUtahStateOfficeofEducation LicensedundertheCreativeCommonsAttribution4NonCommercial4ShareAlike3.0Unportedlicense
9 SolvingQuadraticandOtherEquations UsingtheoperationofsubtractionandelementsfromtheIrrationals,Q,anelementofthe Irrationalnumbers,Q,willbecreated[always,sometime,never].Explain. 8.IftwoComplexnumbers,C,aresubtractedtheresultwill[always,sometimes,never]bea Complexnumber,C.Explain. Go' Topic:SolvingalltypesofQuadraticEquations,SimplifyingRadicals Make'a'prediction'as'to'the'nature'of'the'solutions'for'each'quadratic'Real,'Complex,'Integer,' etc.)'then'solve'each'of'the'quadratic'equations'below'using'an'appropriate'and'efficient' method.'give'the'solutions'and'compare'to'your'prediction.' 9.45x 2 +3x+2=0 10.x 2 +3x+2=0 68 Prediction: Solutions: Prediction: Solutions: 11.x 2 +3x412=0 12.4x 2 419x45=0 Prediction: Solutions: Prediction: Solutions: 2013MATHEMATICSVISIONPROJECT MV P InpartnershipwiththeUtahStateOfficeofEducation LicensedundertheCreativeCommonsAttribution4NonCommercial4ShareAlike3.0Unportedlicense
10 SolvingQuadraticandOtherEquations 3.10 Simplify'each'of'the'radical'expressions.'Use'rational'exponents'if'desired.' Fill'in'the'table'so'each'expression'is'written'in'radical'form'and'with'rational'exponents.' RadicalForm ExponentialForm x 23 y a 9 b MATHEMATICSVISIONPROJECT MV P InpartnershipwiththeUtahStateOfficeofEducation LicensedundertheCreativeCommonsAttribution4NonCommercial4ShareAlike3.0Unportedlicense
7.2"Circle"Dilations" A"Develop"Understanding"Task"
9 7.2CircleDilations ADevelopUnderstandingTask Thestatement allcirclesaresimilar mayseemintuitivelyobvious,sinceall circleshavethesameshapeeventhoughtheymaybedifferentsizes. However,wecanlearnalotaboutthepropertiesofcirclesbyworkingonthe
3.8"To"Be"Determined"."."." A"Develop"Understanding"Task"
49 3.8ToBeDetermined... ADevelopUnderstandingTask IsraelandMiriamareworkingtogetheronahomework assignment.theyneedtowritetheequationsofquadratic functionsfromtheinformationgiveninatableoragraph.at first,thisworkseemedreallyeasy.however,astheycontinued
7.4*Composing*and*Decomposing* A"Develop"Understanding"Task"
23 7.4ComposingandDecomposing A"Develop"Understanding"Task" Asthedaygetwarmer,youandyourfriendsdecidetocooloffby takingarideontheturbulent"waters"dive.asyouarewaitingin lineyourtourguideexplainsthemathematicsbehinddesigning
3.2"Half"Interested" A"Solidify"Understanding"Task"
3.2HalfInterested ASolidifyUnderstandingTask CarlosandClarita,theMartineztwins,haveruna summerbusinesseveryyearforthepastfiveyears. Theirfirstbusiness,aneighborhoodlemonadestand, earnedasmallprofitthattheirfatherinsistedthey
7.1*Function*Family*Reunion* A"Solidify"Understanding"Task"
2 7.1FunctionFamilyReunion A"Solidify"Understanding"Task" Duringthepastfewyearsofmathclassesyouhavestudieda varietyoffunctions:linear,exponential,quadratic, polynomial,rational,radical,absolutevalue,logarithmicand
4.7$Graphing$Rational$Functions$$ A"Practice!Understanding+Task!
35 4.7$Graphing$Rational$Functions$$ A"PracticeUnderstanding+Task PartI:SeeingStructure Foreachfunction,determineintercepts,domain,asymptotes,andcompleteasignline.Usethis informationtosketchthegraph. 1.
3.7"Perfecting"My"Quads" A"Practice"Understanding"Task"
42 3.7PerfectingMyQuads APracticeUnderstandingTask CarlosandClarita,TiaandTehani,andtheirbest friendzacarealldiscussingtheirfavoritemethods forsolvingquadraticequationsoftheform ax 2 + bx + c = 0.Eachstudentthinksaboutthe
x(t)*=*2t"+*1* y(t)*=*3t* *4**
47 7.8HParametrically8DefinedCurves A"Solidify"and"Practice"Understanding"Task" " Aparametriccurveisdefinedbyasetofequationsthatgivethe coordinatesofpointsonthecurveintermsofanothervariablecalleda parameter.
Secondary Mathematics III: An Integrated Approach Module 5 Modeling with Geometry
1 : An Integrated Approach Module 5 Modeling with Geometry By The Mathematics Vision Project: Scott Hendrickson, Joleigh Honey, Barbara Kuehl, Travis Lemon, Janet Sutorius www.mathematicsvisionproject.org
4.5$Watch$Your$ Behavior $ A"Develop!Understanding+Task!
23 4.5$Watch$Your$ Behavior $ A"DevelopUnderstanding+Task Inthistask,youwilldevelopyourunderstandingoftheendbehaviorofrationalfunctionsaswellas discoverthebehaviorofevenandoddfunctions. PartI:Endbehaviorofrationalfunctions
3.9"My"Irrational"and" Imaginary"Friends" A"Solidify"Understanding"Task"
55 3.9MyIrrationaland ImaginaryFriends ASolidifyUnderstandingTask Part1:Irrationalnumbers Findtheperimeterofeachofthefollowingfigures. Expressyouranswerassimplyaspossible. 2013www.flickr.com/photos/lel4nd
1.5 Look Out Below! A Solidify Understanding Task
22 1.5 Look Out Below A Solidify Understanding Task What happens when you drop a ball? It falls to the ground. That question sounds as silly as Why did the chicken cross the road? (To get to the other
1.6 The Tortoise and the Hare A Solidify Understanding Task
29 1.6 The Tortoise and the Hare A Solidify Understanding Task In the children s story of the tortoise and the hare, the hare mocks the tortoise for being slow. The tortoise replies, Slow and steady wins
Quadratic)Functions) 1.6)
Name: Ready,Set,Go Quadratic)Functions) 1.6) Ready 2013www.flickr.com/photos/88394234@N04/8139271342 Topic:Recognizingfunctions Identifywhichofthefollowingrepresentationsarefunctions.IfitisNOTafunctionstate
8.2 Getting Centered A Solidify Understanding Task
8.2 Getting Centered A Solidify Understanding Task Malik s family has decided to put in a new sprinkling system in their yard. Malik has volunteered to lay the system out. Sprinklers are available at the
( a ) m "they"are" If"n"is"a"positive"integer"greater"than"1"and"both"a"and"b"are"positive"real"numbers"then,"
22 3.4RadicalIdeas APracticeUderstadigTask NowthatTiaadTehaikowthat a m = a ) m theyare woderigwhichform,radicalformorexpoetialform,is besttousewheworkigwithumericaladalgebraic expressios. 2013www.flickr.com/photos/zjootsuite
Advanced Mathematics II
1 Advanced Mathematics II Module 7 Circles a Geometric Perspective By The Mathematics Vision Project: Scott Hendrickson, Joleigh Honey, Barbara Kuehl, Travis Lemon, Janet Sutorius www.mathematicsvisionproject.org
6.10!Finding&the&Value&of&a&Relationship! A"Solidify"Understanding"Task!
6.10Finding&the&Value&of&a&Relationship A"Solidify"Understanding"Task Part%I:%What s%your%angle?% AndreaandBonitaaregoingforawalkstraightupthesideofahill.Andreadecidedto stretchbeforeheadingupthehillwhilebonitathoughtthiswouldbeagoodtimetogeta
5.3 It s All In Your Head A Solidify Understanding Task
16 5.3 It s All In Your Head A Solidify Understanding Task In the previous task you were asked to justify some claims by writing paragraphs explaining how various figures were constructed and how those
Secondary Two Mathematics: An Integrated Approach Module 3 - Part One Imaginary Number, Exponents, and Radicals
Secondary Two Mathematics: An Integrated Approach Module 3 - Part One Imaginary Number, Exponents, and Radicals By The Mathematics Vision Project: Scott Hendrickson, Joleigh Honey, Barbara Kuehl, Travis
Secondary Two Mathematics: An Integrated Approach Module 8 Circles and Other Conics
1 Secondary Two Mathematics: An Integrated Approach Module 8 Circles and Other Conics By The Mathematics Vision Project: Scott Hendrickson, Joleigh Honey, Barbara Kuehl, Travis Lemon, Janet Sutorius www.mathematicsvisionproject.org
Up a Little, Down a Little
Up a Little, Down a Little A Solidif Understanding Task One of the most common applications of eponential growth is compound interest. For eample, Mama Bigbucks puts $20,000 in a bank savings account that
Secondary One Mathematics: An Integrated Approach Module 3 Linear and Exponential Functions
1 Secondary One Mathematics: An Integrated Approach Module 3 Linear and Exponential Functions By The Mathematics Vision Project: Scott Hendrickson, Joleigh Honey, Barbara Kuehl, Travis Lemon, Janet Sutorius
Appendix on complex numbers:
Appendix A Appendix on complex numbers: A. complex numbers We begin with a review of several properties of complex numbers, their representation, and some of their basic properties. The use of complex
GUIDED NOTES 5.6 RATIONAL FUNCTIONS
GUIDED NOTES 5.6 RATIONAL FUNCTIONS LEARNING OBJECTIVES In this section, you will: Use arrow notation. Solve applied problems involving rational functions. Find the domains of rational functions. Identify
Cluster Heading Standard MVP. Analyze proportional relationships and use them to solve real- world and mathematical problems.
Quarter 1 Review of 7 th and 8 th grade Standards: Review Total Days 45 REVIEW OF 7 th and 8 th grade standards: Ratios and Proportional Relationships Analyze proportional relationships and use them to
Worksheet 9. Topics: Taylor series; using Taylor polynomials for approximate computations. Polar coordinates.
ATH 57H Spring 0 Worksheet 9 Topics: Taylor series; using Taylor polynomials for approximate computations. Polar coordinates.. Let f(x) = +x. Find f (00) (0) - the 00th derivative of f at point x = 0.
Integrated Math 1 Honors Module 11H Exponents Ready, Set, Go! Homework Solutions
1 Integrated Math 1 Honors Module 11H Exponents Ready, Set, Go! Homework Solutions Adapted from The Mathematics Vision Project: Scott Hendrickson, Joleigh Honey, Barbara Kuehl, Travis Lemon, Janet Sutorius
Lecture 11. Andrei Antonenko. February 26, Last time we studied bases of vector spaces. Today we re going to give some examples of bases.
Lecture 11 Andrei Antonenko February 6, 003 1 Examples of bases Last time we studied bases of vector spaces. Today we re going to give some examples of bases. Example 1.1. Consider the vector space P the
Sequences of height 1 primes in Z[X]
![Sequences of height 1 primes in Z[X]](/thumbs/87/96347391.jpg "Sequences of height 1 primes in Z[X]")
Sequences of height 1 primes in Z[X] Stephen McAdam Department of Mathematics University of Texas Austin TX 78712 mcadam@math.utexas.edu Abstract: For each partition J K of {1, 2,, n} (n 2) with J 2, let
Integrated Math 3 Module 5 Honors Modeling with Functions Ready, Set, Go! Homework Solutions
1 Integrated Math 3 Module 5 Honors Modeling with Functions Ready, Set, Go! Homework Solutions Adapted from The Mathematics Vision Project: Scott Hendrickson, Joleigh Honey, Barbara Kuehl, Travis Lemon,
4.7 We All Scream for Ice Cream
RATIONAL EXPRESSIONS & FUNCTIONS - We All Scream for Ice Cream A Practice Understanding Task The Glacier Bowl is an enormous ice cream treat sold at the neighborhood ice cream parlor. It is so large that
ARE YOU READY 4 CALCULUS
ARE YOU READY 4 CALCULUS TEACHER NAME: STUDENT NAME: PERIOD: 50 Problems - Calculator allowed for some problems SCORE SHEET STUDENT NAME: Problem Answer Problem Answer 1 26 2 27 3 28 4 29 5 30 6 31 7 32
MPEG-4 AVC/H.264 Motion Vector Prediction Rev. 4
MPEG-4 AVC/H.264 Motion Vector Prediction Rev. 4 Author: Shevach Riabtsev (Israel, Haifa), riabtsev@yahoo.com Reviewer: Mohsen Abdoli, MS.c Student of Computer Engineering, Sharif University of Technology,
Unit 0. Unit 1. OUHSD Mathematics Pacing Guide. OUHSD Mathematics Pacing Guide. Total Days 15. Total Days 23. Unit 0: Bridge to Math 1.
OUHSD Mathematics Pacing Guide Quarter 1 Unit 0: Bridge to Unit 0 Total Days 15 Cluster Heading Standard MVP "REVIEW: Use own Resources Solving Equations: One- step; Two- step; Multi- step; Variables on
Z = F(X) Combinational circuit. A combinational circuit can be specified either by a truth table. Truth Table
Lesson Objectives In this lesson, you will learn about What are combinational circuits Design procedure of combinational circuits Examples of combinational circuit design Combinational Circuits Logic circuit
Appendix: Synthetic Division
Appendix: Synthetic Division AP Learning Objectives In this section, we will learn how to: 1. Divide polynomials using synthetic division. Synthetic division is a short form of long division with polynomials.
CSCI 1590 Intro to Computational Complexity
CSCI 1590 Intro to Computational Complexity NP-Complete Languages John E. Savage Brown University February 2, 2009 John E. Savage (Brown University) CSCI 1590 Intro to Computational Complexity February
WCPSS Math 2 Unit 5: MVP MODULE 3 Solving Quadratics & Other Equations
SECONDARY MATH TWO An Integrated Approach WCPSS Math 2 Unit 5: MVP MODULE 3 Solving Quadratics & Other Equations The Scott Hendrickson, Joleigh Honey, Barbara Kuehl, Travis Lemon, Janet Sutorius 2017 Original
Quantum Mechanics for Scientists and Engineers. David Miller
Quantum Mechanics for Scientists and Engineers David Miller Wavepackets Wavepackets Group velocity Group velocity Consider two waves at different frequencies 1 and 2 and suppose that the wave velocity
v = 1(1 t) + 1(1 + t). We may consider these components as vectors in R n and R m : w 1. R n, w m
20 Diagonalization Let V and W be vector spaces, with bases S = {e 1,, e n } and T = {f 1,, f m } respectively Since these are bases, there exist constants v i and w such that any vectors v V and w W can
Analysis of Algorithms Generating Functions II. Holger Findling
Generating Functions II Holger Findling Lecture notes are obtained from Fundamentals of Algorithmics, Gilles Brassard and Paul Bratley. Inhomogeneous Recurrences. The solution of a linear recurrence with
Math 113 Homework 5 Solutions (Starred problems) Solutions by Guanyang Wang, with edits by Tom Church.
Math 113 Homework 5 Solutions (Starred problems) Solutions by Guanyang Wang, with edits by Tom Church. Exercise 5.C.1 Suppose T L(V ) is diagonalizable. Prove that V = null T range T. Proof. Let v 1,...,
TAYLOR AND MACLAURIN SERIES
TAYLOR AND MACLAURIN SERIES. Introduction Last time, we were able to represent a certain restricted class of functions as power series. This leads us to the question: can we represent more general functions
Simple Harmonic Motion Concept Questions
Simple Harmonic Motion Concept Questions Question 1 Which of the following functions x(t) has a second derivative which is proportional to the negative of the function d x! " x? dt 1 1. x( t ) = at. x(
8 sin 3 V. For the circuit given, determine the voltage v for all time t. Assume that no energy is stored in the circuit before t = 0.
For the circuit given, determine the voltage v for all time t. Assume that no energy is stored in the circuit before t = 0. Spring 2015, Exam #5, Problem #1 4t Answer: e tut 8 sin 3 V 1 For the circuit
Student Exploration: Diffusion
Name: Date: Student Exploration: Diffusion Vocabulary: absolute zero, controlled experiment, diffusion, dynamic equilibrium, Kelvin scale, kinetic energy Prior Knowledge Question (Do this BEFORE using
Quadratic and Polynomial Inequalities in one variable have look like the example below.
Section 8 4: Polynomial Inequalities in One Variable Quadratic and Polynomial Inequalities in one variable have look like the example below. x 2 5x 6 0 (x 2) (x + 4) > 0 x 2 (x 3) > 0 (x 2) 2 (x + 4) 0
C/CS/Phys C191 Quantum Gates, Universality and Solovay-Kitaev 9/25/07 Fall 2007 Lecture 9
C/CS/Phys C191 Quantum Gates, Universality and Solovay-Kitaev 9/25/07 Fall 2007 Lecture 9 1 Readings Benenti, Casati, and Strini: Quantum Gates Ch. 3.2-3.4 Universality Ch. 3.5-3.6 2 Quantum Gates Continuing
For more information visit
If the integrand is a derivative of a known function, then the corresponding indefinite integral can be directly evaluated. If the integrand is not a derivative of a known function, the integral may be
A field trips costs $800 for the charter bus plus $10 per student for x students. The cost per student is represented by: 10x x
LEARNING STRATEGIES: Activate Prior Knowledge, Shared Reading, Think/Pair/Share, Note Taking, Group Presentation, Interactive Word Wall A field trips costs $800 for the charter bus plus $10 per student
POLYNOMIAL EXPRESSIONS PART 1
POLYNOMIAL EXPRESSIONS PART 1 A polynomial is an expression that is a sum of one or more terms. Each term consists of one or more variables multiplied by a coefficient. Coefficients can be negative, so
Student Exploration: Temperature and Particle Motion
Name: Date: Student Exploration: Temperature and Particle Motion Vocabulary: absolute zero, Kelvin scale, kinetic energy, molecule, temperature, Prior Knowledge Questions (Do these BEFORE using the Gizmo.)
6.2 Mean value theorem and maximum principles
Hence g = u x iu y is analytic. Since is simply connected, R g(z) dz =forany closed path =) g has an integral function, G = g in. Call G = U + iv. For analytic functions d G = d G. Hence dz dx g = d dz
dt (0) = C 1 + 3C 2 = 3. x(t) = exp(3t).
M445: Dynamics; HW #5. SOLUTIONS so 1. a Characteristic polynomial is Corresponding general solution is λ 2 2λ 3 = λ 12 = 1 ± 1 + 3 = 1 ± 2 = { 1 3 xt = C 1 exp t + C 2 exp3t. is The derivative is dt t
August 2018 ALGEBRA 1
August 0 ALGEBRA 3 0 3 Access to Algebra course :00 Algebra Orientation Course Introduction and Reading Checkpoint 0.0 Expressions.03 Variables.0 3.0 Translate Words into Variable Expressions DAY.0 Translate
Algebra 2 Summer Work Packet
Algebra Summer Work Packet Covering Prerequisite Concepts for Incoming Algebra 1 Students This workbook contains problems designed to ensure the student's readiness for Algebra. The nine topics covered
Week 8 Exponential Functions
Week 8 Exponential Functions Many images below are excerpts from the multimedia textbook. You can find them there and in your textbook in sections 4.1 and 4.2. With the beginning of the new chapter we
Transmission of Nerve Impulses (see Fig , p. 403)
How a nerve impulse works Transmission of Nerve Impulses (see Fig. 12.13, p. 403) 1. At Rest (Polarization) outside of neuron is positively charged compared to inside (sodium ions outside, chloride and
Student Exploration: Air Track
Name: Date: Student Exploration: Air Track Vocabulary: air track, approach velocity, conservation of energy, conservation of momentum, elasticity, kinetic energy, momentum, separation velocity, velocity
arxiv:cs/ v1 [cs.cc] 13 Jun 2006
![arxiv:cs/ v1 [cs.cc] 13 Jun 2006](/thumbs/73/68967537.jpg "arxiv:cs/ v1 [cs.cc] 13 Jun 2006")
Approximability of Bounded Occurrence Max Ones Fredrik Kuivinen Department of Computer and Information Science, Linköpings Universitet, S-581 83 Linköping, Sweden, freku@ida.liu.se ariv:cs/0606057v1 [cs.cc]
Pacing Calendar Mathematics Algebra I AU G U S T
AU G U S T 2 0 1 6 Unit 0: Basic Calculator Applications Students use the graphing calculator as a tool for solving problems. This unit may be completed as needed. Unit 1: Equations and Inequalities Students
Section 4.1 Polynomial Functions and Models. Copyright 2013 Pearson Education, Inc. All rights reserved
Section 4.1 Polynomial Functions and Models Copyright 2013 Pearson Education, Inc. All rights reserved 3 8 ( ) = + (a) f x 3x 4x x (b) ( ) g x 2 x + 3 = x 1 (a) f is a polynomial of degree 8. (b) g is
3.9 Is This The End? A Solidify Understanding Task
3.9 Is This The End? A Solidify Understanding Task In previous mathematics courses, you have compared and analyzed growth rates of polynomial (mostly linear and quadratic) and exponential functions. In
Supplementary Material
ec1 Supplementary Material EC.1. Measures in D(S, µ, Ψ) Are Less Dispersed Lemma EC.1. Given a distributional set D(S, µ, Ψ) with S convex and a random vector ξ such that its distribution F D(S, µ, Ψ),
Mealy & Moore Machines
Mealy & Moore Machines Moore Machine is a finite-state machine whose output values are determined solely by its current state and can be defined as six elements (S, S 0, Σ, Λ, T, G), consisting of the
Polynomial one or more monomials added or subtracted. (i.e. : 5x or 6xy-3 or 6xy - 5x + 3 or
Polynomials Necessary Terms for Success Welcome back! We will now tackle the world of polynomials. Before we get started with performing operations or using polynomials for applications, we will need some
Lecture 7: Positive Semidefinite Matrices
Lecture 7: Positive Semidefinite Matrices Rajat Mittal IIT Kanpur The main aim of this lecture note is to prepare your background for semidefinite programming. We have already seen some linear algebra.
MEMORIAL UNIVERSITY OF NEWFOUNDLAND
MEMORIAL UNIVERSITY OF NEWFOUNDLAND DEPARTMENT OF MATHEMATICS AND STATISTICS Section 5. Math 090 Fall 009 SOLUTIONS. a) Using long division of polynomials, we have x + x x x + ) x 4 4x + x + 0x x 4 6x
Anand Raghunathan MSEE 348
ECE 595Z: Digital VLSI Design Automation, Spring 2012 2012 Anand Raghunathan 1 ECE 595Z Digital Systems Design Automation Module 2 (Lectures 3-5) : Advanced Boolean Algebra Lecture 5 Anand Raghunathan
Linear Algebra and Matrix Theory
Linear Algebra and Matrix Theory Part 3 - Linear Transformations 1 References (1) S Friedberg, A Insel and L Spence, Linear Algebra, Prentice-Hall (2) M Golubitsky and M Dellnitz, Linear Algebra and Differential
4.5 Rational functions.
4.5 Rational functions. We have studied graphs of polynomials and we understand the graphical significance of the zeros of the polynomial and their multiplicities. Now we are ready to etend these eplorations
Alg1 Texas TEKS/STAAR/EOC (Second Semester)
Alg 1_sem2; Texas-TEKS/STAAR/EOC page 1 Alg1 Texas TEKS/STAAR/EOC (Second Semester) Legend: Example 3[R]-A.5(B) 3, The reporting category [R], Either Readiness or Supporting A.5, The TEKS (B) Expectation
Algebra 2 (3) Apex - Semester 1 Part A
Algebra 2 (3) Apex - Semester 1 Part A Name Algebra I Literacy Advantage Unit 1 : The Need to Read Lesson 1.1 : Reading and Vocabulary Activity 1.1.1 : Study - Active Reading Learn skills and strategies
Find two positive factors of 24 whose sum is 10. Make an organized list.
9.5 Study Guide For use with pages 582 589 GOAL Factor trinomials of the form x 2 1 bx 1 c. EXAMPLE 1 Factor when b and c are positive Factor x 2 1 10x 1 24. Find two positive factors of 24 whose sum is
Equations and Solutions
Section 2.1 Solving Equations: The Addition Principle 1 Equations and Solutions ESSENTIALS An equation is a number sentence that says that the expressions on either side of the equals sign, =, represent
Algebra I, 2nd 4.5 weeks
07-08 The following practice standards will be used throughout 4.5 weeks:. Make sense of problems and persevere in solving them.. Reason abstractly and quantitatively. 3. Construct viable arguments and
Partial Fractions. Combining fractions over a common denominator is a familiar operation from algebra: 2 x 3 + 3
Partial Fractions Combining fractions over a common denominator is a familiar operation from algebra: x 3 + 3 x + x + 3x 7 () x 3 3x + x 3 From the standpoint of integration, the left side of Equation
Unit 2. Quadratic Functions and Modeling. 24 Jordan School District
Unit Quadratic Functions and Modeling 4 Unit Cluster (F.F.4, F.F.5, F.F.6) Unit Cluster (F.F.7, F.F.9) Interpret functions that arise in applications in terms of a contet Analyzing functions using different
Conservation of Linear Momentum
Conservation of Linear Momentum Objective In this series of experiments, the conservation of linear momentum and kinetic energy will be tested for different types of collisions. Equipment List Air track,
Total Time = 90 Minutes, Total Marks = 100. Total /10 /25 /20 /10 /15 /20
University of Waterloo Department of Electrical & Computer Engineering E&CE 223 Digital Circuits and Systems Midterm Examination Instructor: M. Sachdev October 30th, 2006 Total Time = 90 Minutes, Total
There are two things that are particularly nice about the first basis
Orthogonality and the Gram-Schmidt Process In Chapter 4, we spent a great deal of time studying the problem of finding a basis for a vector space We know that a basis for a vector space can potentially
7.7H The Arithmetic of Vectors A Solidify Understanding Task
7.7H The Arithmetic of Vectors A Solidify Understanding Task The following diagram shows a triangle that has been translated to a new location, and then translated again. Arrows have been used to indicate
Numerical Methods in Physics and Astrophysics
Kostas Kokkotas 2 November 6, 2007 2 kostas.kokkotas@uni-tuebingen.de http://www.tat.physik.uni-tuebingen.de/kokkotas Kostas Kokkotas 3 Error Analysis Definition : Suppose that x is an approximation to
Shortest reconfiguration paths in the solution space of Boolean formulas
Shortest reconfiguration paths in the solution space of Boolean formulas Amer Mouawad Department of Informatics University of Bergen Joint work with Naomi Nishimura, Vinayak Pathak, and Venkatesh Raman
SIGNALS AND SYSTEMS. Unit IV. Analysis of DT signals
SIGNALS AND SYSTEMS Unit IV Analysis of DT signals Contents: 4.1 Discrete Time Fourier Transform 4.2 Discrete Fourier Transform 4.3 Z Transform 4.4 Properties of Z Transform 4.5 Relationship between Z
Note on Chebyshev Regression
1 Introduction Note on Chebyshev Regression Makoto Nakajima, UIUC January 006 The family of Chebyshev polynomials is by far the most popular choice for the base functions for weighted residuals method.
Digital Circuits ECS 371
Digital Circuits ECS 371 Dr. Prapun Suksompong prapun@siit.tu.ac.th Lecture 18 Office Hours: BKD 3601-7 Monday 9:00-10:30, 1:30-3:30 Tuesday 10:30-11:30 1 Announcement Reading Assignment: Chapter 7: 7-1,
3.4 Pascal s Pride. A Solidify Understanding Task
3.4 Pascal s Pride A Solidify Understanding Task Multiplying polynomials can require a bit of skill in the algebra department, but since polynomials are structured like numbers, multiplication works very
Interpolating Accuracy without underlying f (x)
Example: Tabulated Data The following table x 1.0 1.3 1.6 1.9 2.2 f (x) 0.7651977 0.6200860 0.4554022 0.2818186 0.1103623 lists values of a function f at various points. The approximations to f (1.5) obtained
4 Unit Math Homework for Year 12
Yimin Math Centre 4 Unit Math Homework for Year 12 Student Name: Grade: Date: Score: Table of contents 3 Topic 3 Polynomials Part 2 1 3.2 Factorisation of polynomials and fundamental theorem of algebra...........
Algebra I Assessment. Eligible Texas Essential Knowledge and Skills
Algebra I Assessment Eligible Texas Essential Knowledge and Skills STAAR Algebra I Assessment Mathematical Process Standards These student expectations will not be listed under a separate reporting category.
Lab 9 CONSERVATION OF LINEAR MOMENTUM
Lab 9 CONSERVATION OF LINEAR MOMENTUM In this experiment, you will try to verify the Law of Conservation of Linear Momentum using a collision between two cars on the air track. Here are the important concepts
Writing Equations 98 Slope Intercept Form...98 Standard Form Word Problems..107 Using Slope and a Point 109 Given Two Points..
Table of Contents Algebra Readiness Test... 4 Basic Algebra... 11 Integers and Absolute Value... 11 Adding Integers... 13 Subtracting Integers... 15 Multiplying and Dividing Integers 16 The Distributive
MULTIPLYING POLYNOMIALS
GSE Algebra II: Unit #2 Operations with Polynomials Name: MULTIPLYING POLYNOMIALS WARM UP: ANALYZING A VISUAL MODEL FOR POLYNOMIAL MULTIPLICATION The volume of a rectangular prism is the product of the
KINETIC AND POTENTIAL ENERGY
UNIT 1 - ENERGY SECTION 1 - Background Information Energy can be in one of two states: potential or kinetic. Energy can be transferred from potential to kinetic and between objects. Potential energy is
Algebra 1 FINAL EXAM REVIEW Spring Semester Material (by chapter)
Name: Per.: Date: Algebra 1 FINAL EXAM REVIEW Spring Semester Material (by chapter) Your Algebra 1 Final will be on at. You will need to bring your tetbook and number 2 pencils with you to the final eam.
SECONDARY MATH THREE. An Integrated Approach. MODULE 2 Logarithmic Functions
SECONDARY MATH THREE An Integrated Approach MODULE 2 Logarithmic Functions The Scott Hendrickson, Joleigh Honey, Barbara Kuehl, Travis Lemon, Janet Sutorius 2018 Original work 2013 in partnership with
AFM Midterm Review I Fall Determine if the relation is a function. 1,6, 2. Determine the domain of the function. . x x
AFM Midterm Review I Fall 06. Determine if the relation is a function.,6,,, 5,. Determine the domain of the function 7 h ( ). 4. Sketch the graph of f 4. Sketch the graph of f 5. Sketch the graph of f
THE HEIGHT OF THE MIXED SPARSE RESULTANT
THE HEIGHT OF THE MIXED SPARSE RESULTANT Abstract. We present an upper bound for the height of the mixed sparse resultant, defined as the logarithm of the maximum modulus of its coefficients. We obtain