AP Calculus BC Syllabus Ms. Mulligan Northside College Preparatory High School x26910

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1 Instructor: Phone: E-mil AP Clculus BC Syllus Ms. Mullign Northside College Preprtory High School Introduction In this course, students will not only see the euty of mthemtics, ut they will deepen their understnding of ll previously studied mth courses. In mny wys, Clculus is the glue tht holds together mthemtics. It provides the rtionle for why mth works. Why does n oject thrown into the ir rech mimum nd how do we know when tht mimum occurs? Wht is the reltionship etween position, velocity nd ccelertion? Why cn we dd n infinite list of numers nd still rrive t n nswer other thn infinity? Clculus nswers these nd mny more questions. Students will hve more thn cursory understnding of Clculus nd its pplictions. They will understnd concepts s well s procedures. Students will lern nd pply Clculus concepts y employing the rule of four numericlly, geometriclly, symoliclly, nd verlly so tht they understnd tht how ll four representtions of the mthemtics re relted nd integrted. Students will communicte their understnding in vriety of wys: through mthemticl representtion, through written justifiction nd through verl eplntions to their clssmtes. Course Outline (with pproimte numer of weeks nd ssessments) Continuity nd Limits (one week, one quiz) 1.7 Introduction to continuity 1.8 Limits Review of different kinds of functions to emine nd understnd vrious limits: liner, polynomil, rtionl, eponentil, sinusoidl, logistic Intermedite Vlue Theorem nd Etreme Vlue Theorem Properties of Limits Limit t point, limit t infinity nd negtive infinity, infinite limits (symptotes) sin Use of grphing clcultor to find lim 0 Estimting limits from tles nd grphs Clculting limits using lger Understnding continuity nd limits Key Concept: The Derivtive (3 weeks, one quiz nd one test) 2.1 How do we mesure speed? 2.2 The derivtive t point 2.3 The derivtive function 2.4 Interprettions of the derivtive 2.5 The second derivtive 2.6 Differentiility Differentiility vs. continuity Mtching grphs of functions to grphs of its first nd second derivtives

2 Using tles of numeric vlues of functions to estimte first nd second derivtives Interpreting situtions with respect to their functions nd derivtives (e.g. displcement, velocity nd ccelertion) Repeted prctice of verlly eplining the mening nd interprettion of derivtive in vriety of wys Using nderive on the TI-89 to mke conjectures on the reltionship etween polynomil functions nd their derivtives Eplore different definitions of derivtive: instntneous rte of chnge, slope of tngent line f ( ) f( h) s definition of derivtive, limit of difference quotient + lim h Shortcuts to Differentition (3 weeks, one quiz nd one test) 3.1 Powers nd Polynomils 3.2 The eponentil function 3.3 The product nd quotient rules 3.4 The chin rule 3.5 The trigonometric functions 3.6 The Chin Rule nd Inverse Functions 3.7 Implicit functions 3.9 Liner pproimtion nd the derivtive 3.10 Theorems out differentile functions More eplortion of polynomils functions to formlize power rule Use nderiv on TI-89 to find the constnt of proportionlity of eponentil functions dy derivtives: y =, = lni d Lern how to tke the symolic derivtive on TI-89 nd use it to test conjectures for the derivtive of product (product rule) nd derivtive of composite function (chin rule) Grphing y = sin nd y = nderiv(sin,, ) to mke conjecture out the derivtive of sine nd cosine Emine the slope of circle t vrious points to demonstrte tht the slope is dependent on oth the nd y vlues of point (segue to implicit differentition) Slope of tngent line, locl linerity nd errors of liner pproimtions Men Vlue Theorem, Incresing Function Theorem, Constnt Function Theorem, The Rcetrck Principle Using the Derivtive (3 weeks, one test) 4.1 Using first nd second derivtives 4.2 Fmilies of curves 4.3 Optimiztion 4.5 Optimiztion nd modeling 4.6 Rtes nd relted rtes 4.7 L Hopitl s Rule, growth nd dominnce 4.8 Prmetric equtions Connecting concvity to mim nd minim nd points of inflection Verl nd nlyticl eplntions for mim nd minim Interpreting optimiztion prolems nd creting relevnt nlyticl representtions of situtions (understnding quntity to e optimized, drwing supporting sketches, formul for function to e optimized, finding criticl points, evluting criticl points nd endpoints) h 0

3 Using the grphing clcultor to support nlysis of mim nd minim (concvity) Written nd orl eplntions of relesed open-response AP prolems on derivtives Grphing in prmetric mode on TI-89 First nd second derivtives of prmetric equtions (to e revisited lter for vectors nd polr equtions) Length of curve in prmetric equtions Key Concept: The Definite Integrl (2 weeks) 5.1 How do we mesure distnce trveled? 5.2 The definite integrl 5.3 The Fundmentl Theorem nd interprettions 5.4 Theorems out definite integrls Understnding re under velocity curve s displcement (constructing tle of vlues from function of ccumulting, signed re nd grphing tht tle of vlues) Reimnn sums: left-hnd rectngles, right-hnd rectngles, trpezoids, midpoint rectngles (introduction) Using the TI-89 to drw rectngles nd find sums of rectngles Fundmentl theorem of clculus: Averge vlue of function: 1 f ' () tdt= f ( ) f ( ) f ( d ) Properties of definite integrls, comprison of definite integrls Signed re: re ove -is is positive nd re elow -is is negtive Constructing ntiderivtives (2 weeks, one test including chpters 5 nd 6) 6.1 Antiderivtives grphiclly nd numericlly 6.2 Constructing ntiderivtives nlyticlly 6.3 Differentil equtions (introduction) 6.4 Second fundmentl theorem of clculus 6.5 The equtions of motion Mtching grphs of f '( ) nd f ''( ) with f ( ) Second fundmentl theorem: F( ) = f( t) dt Antiderivtives of common functions: 1/, e +, sin, cos, reverse power rule Fundmentl theorem of clculus: Averge vlue of function: f ' () tdt= f( ) f( ) 1 f ( d ) (How is this different thn verge rte of chnge?) Properties of definite integrls, comprison of definite integrls Signed re: re ove -is is positive nd re elow -is is negtive

4 Using chin rule for integrls when one of the ounds is function (e.g.: sin tdt ) Using the TI-89 to tke the nlyticl ntiderivtive Integrtion (3 weeks, one quiz) 7.1 Integrtion y sustitution 7.2 Integrtion y prts 7.4 Algeric identities nd trigonometric sustitutions 7.5 Approimting definite integrls 7.7 Improper integrls 7.8 Comprison of improper integrls More on Reimnn sums: under- or over-estimtes with concvity Prtil frctions (denomintors hve non-repeted liner fctors) Integrls with discontinuities in integrnd Integrls with infinite ounds Using the TI-89 to evlute improper integrls nd to test conjectures on students clculted nswer (e.g.: 2 2 d 1 =?) Using the definite integrl (3 weeks, one test on chpters 7 nd 8) 8.1 Ares nd volumes 8.2 Applictions to geometry 8.3 Are nd rc length in polr coordintes 8.4 Density nd center of mss 8.5 Applictions to physics Connecting re nd volume to the integrl Doing the gel chllenge: clculting the volume of gel, given certin ssumptions, then eting the gel Using the TI-89 to find res etween curves Using the TI-89 to evlute integrls for volumes of solids perpendiculr to the es nd solids from rottion Using the TI-89 to evlute integrls of polr functions Work s the integrl of force Severl relesed open-response AP questions on pplictions of the integrl Development of re of polr curves through re of sections of circle Arc length s review from prmetric equtions Using the clcultor to support understnding of the zeros of polr equtions Sequences nd series (3 weeks, one quiz) 9.1 Sequences 9.2 Geometric series 9.3 Convergence of series 9.4 Tests for convergence 9.5 Power series nd intervls of convergence Distinguish etween sequence nd series Sum of infinite nd finite geometric series 2 0

5 Telescoping series Comprison Test, Integrl Test, Limit Comprison Test, Rtio Test, nth term test Using the TI-89 to discover the intervl of convergence for geometric series grph vrious functions with incresing numer of terms to conjecture on the intervl of convergence Using re of rectngles to understnd terms of series Approimting functions using series (3 weeks, one quiz, one test including chpters 9 nd 10) 10.1 Tylor polynomils 10.2 Tylor series 10.3 Finding nd using Tylor series 10.4 The error in Tylor polynomil pproimtions Strting with liner pproimtions t point, then qudrtic pproimtions, then generliztion for higher degree pproimtions (Tylor polynomils) Using the TI-89 to clrify intervls of convergence (grphing incresing numers of terms to find the intervl) Revisiting intervls of convergence vi rtio test Deriving Tylor nd Mclurin series y tking derivtives nd y sustitution Lgrnge form of the reminder nd using the clcultor to confirm errors 1 Memorize Mclurin series for e, sin, cos, 1 Rdius nd intervl of convergence of power series Differentil equtions (3 weeks, one quiz) 11.1 Wht is differentil eqution? 11.2 Slope fields 11.3 Euler s method 11.4 Seprtion of vriles 11.5 Growth nd decy 11.6 Applictions nd modeling 11.7 Models of popultion growth Unit highlights: Drwing slope fields y hnd, then generting them with the clcultor Drwing solutions y hnd, then grphing solutions in slope field in clcultor Creting pproprite models for differentil equtions for sitution involving rtes Emining the slope field ehvior of logistic models on the grphing clcultor to scertin the equilirium (nd stility of it), then relting tht equilirium to the solution of the differentil eqution Vectors (1 week, one quiz) 13.1 Displcement vectors 13.2 Vectors in generl 17.1 Prmeterized curves 17.2 Motion, velocity, nd ccelertion Unit highlights: Wht is vector? How to epress vector y its components? Review of some prmetric ides: position, velocity, speed, ccelertion, rc length Distinguish etween speed nd velocity

6 Sclr products Review (2 weeks, one test) Prctice tests in commercilly ville review ooks ssigned nd discussed ech clss More prctice with relesed AP open-response questions Review of scoring guidelines of open response questions Comprehensive finl em ( mock AP ) to e tken outside of clss time (3.5 hours) under the sme conditions s rel AP em (time limits, sections, clcultors usge, etc.) After AP em (6 weeks) Students will complete n independent or group project for n ssessment grde Project detils will e discussed fter the AP em Communiction nd Prticiption Students will regulrly prctice communicting mthemticl ides effectively. Whether in their groups or in presenttions to the entire clss, ll students re epected to clerly eplin their work nd justify their nswers. This sme kind of communiction is epected on open-response questions where they must show their work nd justify their nswers with words. From the prolems tht re ssigned in previous night s homework, students decide which need to e presented prolems tht they were not successful on. Other students volunteer to present those prolems. If no student is le to present some prolems, then the entire clss will work with the techer to solve it in n interctive fshion. All students re required to prticipte in whole-clss discussions dily. Whether investigting the grphs of derivtives or determining the pproprite intervl of convergence, clss discussion is n integrl prt of lerning nd deepening understnding. Clss discussion my employ Conceptests, series of PowerPoints tht help students understnd the underlying concepts in Clculus. Some clss discussions my rely hevily on grphing clcultors, which students will either purchse on their own or rent from the school. Fcility with the TI-89 will e enhnced through clss discussions nd reinforced through homework prolems. Some clss discussions will use the Check Your Understnding sttements t the end of ech chpter. Students will red nd either confirm or refute ech of the sttements. Students re epected to e in clss every dy. Clss prticiption grdes re determined y ttendnce nd frequency of student interction in groups: discussing nd deting prolem solutions, shring nd listening to others ides, nd consistent enggement with the mteril nd prolems. All students re epected to regulrly present nd defend ides, s well s question nd suggest ides to the presenter. Prticiption ccounts for 10% of grde. Assessments As noted in the course outline, students will hve ssessments t vrious times throughout the yer. Assessments ccount for 60% of grde. Severl of the ssessments will hve clcultor nd nonclcultor sections, similr to the formt for the AP em. Most ssessments will e cumultive. Homework inders will e checked while students tke the ssessments. Homework grde relies on completeness nd ccurcy of homework prolems nd ccounts for 30% of grde. Required mterils:

7 TI-89 grphing clcultor 1.5 inch three-ring inder with t lest 300 sheets of loose-lef pper nd 100 sheets of grid pper All notes nd homework nd hndouts must e kept in chronologicl order, with ssignments nd dtes clerly leled Pencils Required tet: Hughes-Hllet, Deorh, et l. Clculus: Single nd Multivrile, 4 th ed. New York: John Wiley nd Sons (Techer will use ll supporting techer mterils tht ccompny Hughes-Hllet tet: AP techers guide, Conceptests, Test Bnk, Answers to prolems sets) Supplementl tets: Finney, Ross L., Frnklin D Demn, Bert K. Wits, nd Dniel Kennedy. Clculus: Grphicl, Numericl, Algeric AP Edition, 3 rd ed. Boston: Person Prentice Hll Brodwin, Judith, George Lenchner, nd Mrtin Rudolph. Solutions: AP Clculus Prolems Prt II AB nd BC Bellmore NY: Mthemticl Olympids for Elementry nd Middle Schools. Vrious relesed ems nd other commercilly ville review ooks