Course 4: Preparation for Calculus Unit 1: Families of Functions

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1 Course 4: Preparatio for Calculus Uit 1: Families of Fuctios Review ad exted properties of basic fuctio families ad their uses i mathematical modelig Develop strategies for fidig rules of fuctios whose graphs are related by traslatio, reflectio, stretchig, ad compressig to those of basic family members Apply ehaced fuctio skills to build models of more complex relatioships betwee variables Develop the defiitios ad importat properties of arithmetic operatios o fuctios Develop the defiitio, properties, ad uses of fuctio compositio Lesso 1 Fuctio Models Revisited Review properties ad applicatios of liear, expoetial, power, ad iverse variatio fuctios Develop a taxoomy of fuctio rule, graph, ad umerical patters Lesso 2 Customizig Models I: Traslatio ad Reflectio Discover the coectios betwee rules of fuctios whose graphs are related by vertical traslatio, reflectio across the x-axis, ad horizotal traslatio Discover ways that maximum ad miimum poits, zeroes, ad y-itercepts of two fuctios are related if their graphs are related by vertical traslatio, reflectio across the x-axis, ad horizotal traslatio Develop strategies for usig the coectios betwee graph trasformatios ad fuctio rules to develop models for relatioships that are based o the core fuctio families Lesso 3 Customizig Models II: Stretchig ad Compressig Discover the coectios betwee rules for fuctios that are related by horizotal ad/or vertical stretchig ad compressig Develop strategies for adjustig basic sie ad cosie fuctios to vary amplitude ad period Discover relatioships amog maximum ad miimum poits, zeroes, ad y-itercepts of fuctios whose graphs are related by vertical ad/or horizotal stretchig ad compressig Use ideas of vertical ad horizotal stretchig/compressig of graphs to costruct models of periodic variatio Lesso 4 Combiig Fuctios Develop the defiitios ad importat properties of arithmetic operatios o fuctios Discover meaigful ways to combie fuctios by arithmetic operatios Develop uderstadig of fuctio compositio ad skill i costructig rules for composite fuctios from rules of the compoet fuctios Core-Plus Mathematics 2d editio CPMP 25

2 Uit 2: Vectors ad Motio Describe ad use the cocept of vector i mathematical, scietific, ad everyday situatios Represet vectors geometrically ad operate o geometric vectors Describe, represet, ad use vector compoets ad operatios sythetically ad aalytically Ivestigate ad justify geeral properties of vectors ad vector operatios Provide vector proofs of properties of triagles ad parallelograms Use vector cocepts to parametrically represet liear, projectile, circular, ad elliptical motios i a plae Aalyze motios usig parametric models Lesso 1 Liear Motio Represet vectors as directed lie segmets with both directio ad magitude Describe ad illustrate the scalar multiple ad the opposite of a vector Describe ad illustrate the compoets of a vector Add two vectors geometrically ad by usig compoets Model liear motios ad forces with vectors Lesso 2 Vectors ad Parametric Equatios Describe ad illustrate the relatioship betwee the coordiates of the termial poit of a positio vector ad its compoets Explore ad prove properties of scalar products ad additio of vectors Describe ad illustrate the dot product of two vectors, use it to compute the cosie of the agle formed by the vectors ad to test whether or ot those vectors are perpedicular Use vectors to prove properties of triagles ad quadrilaterals Write parametric equatios that model liear motio Simulate liear motio usig techology Lesso 3 Simulatig Noliear Motio Write parametric equatios to simulate projectile, circular, ad elliptical motio Idetify importat forces affectig motio Use both radia ad degree measuremets to describe agular velocity Describe the locatio of a poit o a rotatig circle i terms of its iitial locatio ad the agular velocity Core-Plus Mathematics 2d editio CPMP 26

3 Uit 3: Algebraic Fuctios ad Equatios Review ad exted studet skill i work with polyomials ad polyomial fuctios to model graph patters ad coditios of quatitative relatioships Develop basic properties of polyomial fuctios like zeroes, local max/mi poits, ed behavior, ad represetatio i stadard, factored, ad ested multiplicatio forms Develop uderstadig ad skill i divisio of polyomials ad the divisio algorithm Develop cocepts of complex umbers ad their represetatio i a + bi form ad as poits/vectors o the coordiate grid Develop defiitios ad skill i operatios o complex umbers Review ad exted studet uderstadig ad skill i work with ratioal expressios ad ratioal fuctios to model geeralized iverse variatio relatioships Ehace studet skill i aalyzig ratioal fuctios ad their graphs to idetify domai, asymptotes, zeroes, ad local max/mi poits Exted studet skill i combiig ratioal expressios by additio, subtractio, multiplicatio, ad divisio ad i solvig equatios ivolvig ratioal expressios Develop skill i aalyzig expressios ad solvig equatios that ivolve radicals Reflect o geeralizable strategies for solvig algebraic problems by aalysis of the forms of symbolic expressios ad equatios as well as by routie symbol maipulatio Lesso 1 Polyomial Fuctio Models ad Operatios Fit polyomial fuctio models to data ad graph patters usig problem coditios, statistical regressio, ad the method of udetermied coefficiets Exted the relatioship betwee stadard, factored, ad ested multiplicatio forms of polyomials Develop polyomial divisio ad the divisio algorithm p(x) = (x k)q(x) + r(x) Solve polyomial equatios ad iequalities Lesso 2 Complex Numbers Develop uderstadig of the eed for complex umbers to solve quadratic equatios ad the defiitio of the ew umbers i the form a + bi, with a ad b real umbers ad i = 1 Use defiitios of additio, subtractio, multiplicatio, ad divisio of complex umbers to establish algebraic properties of complex umber operatios Develop geometric represetatio of complex umbers, icludig absolute value for magitude, ad the coectio betwee complex umber operatios ad basic geometric trasformatios Core-Plus Mathematics 2d editio CPMP 27

4 Lesso 3 Ratioal Fuctio Models ad Operatios Write expressios for rules of ratioal fuctios that model patters i experimetal data, geometric curves, ad problem coditios Idetify asymptotes (horizotal, vertical, ad oblique) for graphs of ratioal fuctios Review ad exted skills i maipulatig ratioal expressios ito useful equivalet forms Solve ratioal equatios ad iequalities Lesso 4 Algebraic Strategies Write algebraic expressios for relatioships that ivolve radicals Solve equatios ad iequalities ivolvig radicals Review strategies for maipulatig algebraic expressios ad equatios ito equivalet forms ad the justificatios for those maeuvers Advace strategic thikig usig symbol sese to aalyze problem situatios ad their algebraic models both as a ehacemet of ad method for avoidig algebraic calculatio Core-Plus Mathematics 2d editio CPMP 28

5 Uit 4: Trigoometric Fuctios ad Equatios Kow ad be able to use the defiitios of the six trigoometric fuctios of a agle i stadard positio Derive ad use the fudametal trigoometric idetities Prove trigoometric idetities Solve trigoometric equatios Represet complex umbers geometrically Iterpret multiplicatio ad divisio of complex umbers geometrically Use De Moivre s Theorem to fid powers ad roots of complex umbers Lesso 1 Reasoig with Trigoometric Fuctios Kow ad be able to use the defiitios of the six trigoometric fuctios Describe the graph ad period of each trigoometric fuctio Derive ad use the fudametal trigoometric idetities Develop strategies for provig trigoometric idetities Derive ad use the opposite-agle, cofuctio, sum, differece, ad double-agle idetities for sie ad cosie Lesso 2 Solvig Trigoometric Equatios Solve liear ad quadratic trigoometric equatios Solve equatios of the form af(bx + c) = d, where f is a trigoometric fuctio Express the geeral solutios of a trigoometric equatio i forms such as x = a + 2π or x = a + 360, for ay iteger Use idetities to trasform trigoometric equatios ito more easily solved forms Lesso 3 The Geometry of Complex Numbers Express a complex umber i both stadard ad trigoometric forms Use complex umber multiplicatio ad divisio to size trasform, rotate, or rotate ad size trasform the poit or vector associated with a complex umber Uderstad ad use De Moivre s Theorem Determie all the th roots of a complex umber ad represet them geometrically Core-Plus Mathematics 2d editio CPMP 29

6 Uit 5: Expoetial Fuctios, Logarithms, ad Data Modelig Uderstad e as the limit of ( 1 + ) as Use e rt as a approximatio for ( 1 + r) t 1 Use fuctios of the form y = Ae rt to solve expoetial growth ad decay problems Show how ay expoetial fuctio ca be expressed i equivalet form usig base e ad how ay logarithm ca be expressed i equivalet form usig base 10 or base e Use properties of expoets ad logarithms to write algebraic expressios i equivalet forms ad solve equatios ivolvig logs ad expoets Use residual plots to evaluate the goodess of fit of liear regressio equatios Use logarithmic trasformatios of data to fid liearized data patters Use liear regressio equatios ad back trasformatio (solvig for y) to determie power ad expoetial fuctios that represet data patters Lesso 1 Expoets ad Natural Logarithms Uderstad e as the limit of ( 1 + ) as Use e r as a approximatio for ( 1 + r) ad e rt as a approximatio for ( 1 + r) t Use fuctios of the form y = Ae rt to solve expoetial growth ad decay problems Show how ay expoetial fuctio ca be expressed i equivalet form usig base e ad how ay logarithm ca be expressed i equivalet form usig base 10 or base e Use properties of expoets ad logarithms to write algebraic expressios i equivalet forms ad solve equatios ivolvig logs ad expoets Lesso 2 Liearizatio ad Data Modelig 1 Use residual plots to assess the goodess of fit for liear models of data patters Use logarithmic trasformatios of data to fid liearized data patters Use liear regressio equatios ad back trasformatio (solvig for y) to determie expoetial ad power fuctios that model data patters Core-Plus Mathematics 2d editio CPMP 30

7 Uit 6: Surfaces ad Cross Sectios Represet three-dimesioal objects ad surfaces with cotour lies or horizotal ad vertical cross sectios Iterpret ad describe three-dimesioal surfaces or objects represeted with cotour diagrams Use the three-dimesioal coordiate system to locate poits ad represet data, objects, ad surfaces i space Idetify ad sketch graphs of coic sectios represeted algebraically ad write equatios matchig graphs of coics Use iformatio revealed by the form of a equatio of a three-dimesioal surface to visualize, characterize, ad sketch the surface Idetify ad sketch surfaces of revolutio ad cylidrical surfaces Lesso 1 Represetig Three-Dimesioal Objects Draw a cotour diagram from appropriate data Describe ad plot the locatio of a poit i three dimesios usig (x, y, z) coordiates Make a topographic profile that correspods to a vertical cross sectio o a map displayig cotour lies Idetify cross sectios of three-dimesioal surfaces or objects Describe a surface give its cotour diagram Sketch a three-dimesioal object give a set of horizotal ad vertical cross sectios Idetify ad sketch graphs of coic sectios give a equatio i the form ax 2 + by 2 + cx + dy + e = 0 Write equatios that match graphs of coic sectios Lesso 2 Equatios for Surfaces Fid the legth of a segmet that jois two poits i space Fid the coordiates of the midpoit of a lie segmet i space Describe the relatioship betwee the coordiates of two poits that are symmetric with respect to a coordiate plae or axis Determie the itercepts, the traces, the symmetry, ad the coordiate-plae-parallel cross sectios of a surface defied by a equatio Recogize a plae as the graph of a equatio Ax + By + Cz = D Sketch surfaces give their equatios Recogize ad sketch surfaces of revolutio ad cylidrical surfaces whe give appropriate equatios Core-Plus Mathematics 2d editio CPMP 31

8 Uit 7: Cocepts of Calculus Develop the cocept of istataeous rate of chage i a cotiuous variable ad strategies for estimatig those rates of chage Defie the derivative of a fuctio at a poit i its domai Coect the derivative of a fuctio to local approximatio of slope of its graph Develop derivative formulas for liear ad quadratic fuctios Develop the coectio betwee area uder a rate fuctio graph ad accumulatio of chage Defie the defiite itegral of a fuctio ad its applicatio to problems Lesso 1 Itroductio to the Derivative Develop the cocept of istataeous rate of chage i a cotiuous variable ad strategies for estimatig those rates of chage Defie the derivative of a fuctio at a poit i its domai Coect the derivative of a fuctio to local approximatio of slope of its graph Develop derivatives formulas for liear ad quadratic fuctios Lesso 2 Itroductio to the Defiite Itegral Develop uderstadig of the coectio betwee cumulative chage ad area bouded by a rate of chage graph Develop begiig uderstadig of the ways that areas (ad thus defiite itegrals) ca be approximated by Riema sums ad the effects of refiig the approximatio by lettig x 0 Core-Plus Mathematics 2d editio CPMP 32

9 Uit 8: Coutig Methods ad Iductio Develop the skill of careful coutig i a variety of cotexts Uderstad ad apply a variety of coutig techiques, icludig the Multiplicatio Priciple of Coutig, the Additio Priciple of Coutig, coutig trees, ad systematic lists Uderstad ad apply the issues of order ad repetitio whe coutig the umber of possible choices from a collectio Solve coutig problems ivolvig combiatios ad permutatios Uderstad ad apply the Biomial Theorem Uderstad ad apply coectios amog combiatios, the Biomial Theorem, ad Pascal s triagle Apply coutig methods to probability situatios i which all outcomes are equally likely Exted uderstadig of ad apply the Geeral Multiplicatio Rule for Probability Develop the skill of combiatorial reasoig, icludig use i proofs Uderstad ad carry out proofs by mathematical iductio Uderstad ad carry out proofs usig idirect reasoig ad the Least Number Priciple Lesso 1 Coutig Strategies Develop the skill of systematic coutig by thikig carefully about the umber of possibilities i a variety of cotexts Uderstad ad apply basic coutig strategies, such as makig tree diagrams (coutig trees), makig systematic lists, ad usig the Multiplicatio Priciple of Coutig Uderstad ad apply the issues of order ad repetitio whe coutig the umber of possible choices from a collectio Solve coutig problems ivolvig combiatios ad permutatios Lesso 2 Coutig Throughout Mathematics Apply coutig methods to probability situatios i which all outcomes are equally likely Exted uderstadig of ad apply the Geeral Multiplicatio Rule for Probability Uderstad ad apply the Biomial Theorem Uderstad ad apply coectios amog combiatios, the Biomial Theorem, ad Pascal s triagle Develop the skill of combiatorial reasoig, icludig its use i proofs Lesso 3 Proof by Mathematical Iductio Uderstad the Priciple of Mathematical Iductio Develop skill i provig statemets by mathematical iductio Uderstad the Least Number Priciple ad proofs usig this priciple alog with idirect reasoig Core-Plus Mathematics 2d editio CPMP 33