Calculus AB (Pre Requisites: Pre-Calculus)

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1 Curse Descriptin and Overview Calculus AB (Pre Requisites: Pre-Calculus) Overview The gal f the Calculus AB is t intrduce the first part f Calculus t students. The curse will g int depth t Limits and Cntinuity (Chapter 2), Derivatives (Chapter 3 &4), Applicatins f Derivatives (Chapter 5) and The Definite Integral (Chapter 6). By the end f the curse, students will be able t: Represent functins graphically, numerically, analytically and verbally. Graphing Calculatrs will be used t facilitate this prcess. Use graphing calculatrs t graph functins, slve equatins, evaluate derivatives and integrate while slving prblems. Understand the meaning f a derivative as a limit, instantaneus rates and slpe f a tangent line. Understand the meaning f integratin as a limit f Riemann sum and accumulatins f a set f intervals. They will be able t make cnnectins frm derivatives and integratin. Mdel real life examples and slve a variety f prblems by using prper mathematical language. This wuld include functins, derivative, and integratin. Primary Textbk - Bressud, D., Demana, F., Finney, R., Kennedy, D., Waits,B. (2016). Calculus: Graphical, Numerical, Algebraic (Fifth Editin. Pearsn Educatin Inc. Required Technlgy Yu will have access t either TI 89 Silver Editin r TI-Nspire with Current Updates s that yu are able t graph equatins. We will be using the graphing calculatr t verify limits, explratin, analyze and interpret data and explain results frm the table, graph, and equatins. (Nte: Calculatrs are used t supprt answers graphically r numerically after yu have slved the prblem using pencil r paper.) Students will have access t cmputers where they will take ntes frm mymathlab.cm pwerpints as well as cmplete assignments nline. Mymathlab.cm ffers a variety f tls fr student t practice test questins and has extra practice fr all sectins f the bk. Once we cmplete the Chapters 1-8, we will take practice test that have been cmpiled n the website. Mathematical Practices 1. Reasning f definitins and therems Students are expected t understand vcabulary and therems in the curse. They must be able t express all cncepts verbally and in writing. In multiple lessns students will be able t demnstrate cncepts such as MVT and Euler s Methd t their peers. 2. Cnnecting cncepts and Prcess

2 Students will have pprtunities t explain cncepts and prcesses using graphs, tables, verbally, in writing and analytically. They will be able t display cncavities f functins and explain real life situatins f limits. 3. Understanding representatins Students will create, draw and identify different mdels f velcity. Students will take nte f their parents driving frm 2 lcatins and create a graph that shws the entire trip. Anther student will then calculate the distance traveled. Students will understand that explaining cncepts verbally can be displayed graphically, analytically and in a wrd prblem. 4. Fluency Students are required t shw all steps invlving cmputatin. They will be able t prperly explain ideas and results using the crrect terminlgy. Thrughut the curse multiple examples f incrrect slutins will be given and the student must be able t lcate the errr and express why the mistake was made. 5. Cmmunicatin Students are required t use accuntable talk and be able t explain prcedures either verbally r n the bard t their peers. In varius assignments and tests, students will have at least ne prblem that requires students t explain their prcess r identify an errr and explain why the errr was made. Big Idea 1 Limits Learning Objectives prvided by Calculus: Graphical, Numerical, Algebraic 5 th Editin EU 1.1 Cncept f a limit t understand the behavir f a functin LO 1.1A Express limits symblically 2.1, 2.2 LO 1.1A(b) Interpret limits expressed symblically 2.1, 2.2 LO1.1B Estimate limits 2.1,2.2 LO1.1C Determine limits f a functin 2.1,2.2 LO1.1D Deduce and interpret behavir f functins using limits 2.1,2.2 EU1.2 Cntinuity is a key prperty f functins that is defined using limits LO1.2A Analyze functin fr intervals f cntinuity r pints f discntinuity LO1.2B Determine the applicability f imprtant Calculus therems using cntinuity 2.3, 5.1, 5.2, Big Idea Tw: Derivatives EU2.1 The derivative f a functin is defined as a limit f a different qutient and can be determined using a variety f strategies. LO2.1A Identify the derivative f a functin as the limit f a different qutient 3.1 LO2.1B Estimate the derivative 3.1, 3.2

3 LO2.1C Calculate derivatives 3.3, 3.5, D Determine high rder derivatives 3.3, 4.2 EU 2.2 A functin s derivative can be used t understand the behavir f the functin LO2.2A Use the derivative t analyze prperties f functins LO2.2B Recgnize the cnnectin between differentiability and cntinuity 3.2 EU2.3 The derivative has multiple interpretatins and applicatins including thse that invlve instantaneus rates f change LO2.3A Interpret the meaning f a derivative within a prblem 2.4, 3.1, 3.4, 5.5 LO2.3B Slve prblems invlving the slpe f a tangent line 2.4, 3.4, 5.5 LO2.3C Slve prblems invlving related rates, ptimizatin, rectilinear mtin 3.4, 5.1, 5.3 LO2.3D Slve prblems invlving related rates f change in applied cntexts 5.5, 5.6 LO2.3E Verify slutins t differential equatins 7.1 LO2.3F Estimate slutins t differential Equatins 7.1 EU2.4 The Mean Value Therem cnnects the behavir f a differentiable functin ver an interval f the derivative f that particular functin at a particular pint in the interval LO2.4 Apply the Mean Value Therem t describe the behavir f functin ver an interval 5.2 Big Idea 3: Integrals and the Fundamental Therem f Calculus EU3.1 Antidifferentiatin is the invers prcess f differentiatin LO3.1A Recgnize the antiderivative f basic functins 6.3 EU3.2 The definite integral f a functin ver an interval is the limit f a Riemann sum ver that interval and can be calculated using a variety f strategies 3.2A(a) Interpret the definite integral as a limit f a Riemann Sum 6.1, A(b) Express the limit f a Riemann sum in integral ntatin B Apprximate a definite integral 6.1, 6.2, C Calculate a definite integral using areas and prperties f definite integrals 6.2,6.3 EU3.3 The Fundamental Therem f Calculus, which has tw distinct frmulatins, cnnects differentiatin and integratin. LO3.3A Analyze functins defined by an interval 6.1-4, 8.1 LO3.3B(a) Calculate antiderivatives 6.3, 6.4, 7.2 LO3.3B(b) Evaluate definite integrals 6.3, 6.4, 7.2 EU3.4 The definite integral f a functin ver an interval is a mathematical tl with many interpretatins and applicatins invlving accumulatin.

4 LO3.4A Interpret the meaning f a definite integral within a prblem 6.1, 6.2, 8.1, 8.5 LO3.4B Apply definite integrals t prblems invlving the average value f a functin 6.3 LO3.4C Apply definite integrals t prblems invlving mtin 6.1, 8.1 LO3.4D Apply definite integrals t prblems invlving area, vlume 8.2, 8.3 LO3.4E Use the definite integral t slve prblems in varius cntexts 6.1, 8.1, 8.5 EU3.5 Antidifferentiatin is an underlying cncept invlved in slving separable differential equatins. Slving separable equatins invlves determining a functin r relatin given its rate f change. LO3.5A Analyze differential equatins t btain general slutins 7.1, 7.4 LO3.5A Interpret, Create, and slve differential equatins frm prblems in cntext. 7.1, 7.4 Grading Students will have a variety f multiple chice and free respnse questins thrughut the curse. They must be able t justify their answers either verbally r written. This wuld allw students t prepare fr the AP test in May. District Grading Plicy states students shall receive: 50% Assignments and Hmewrk (Any chapter assignment, handut, r bell wrk) 30% Tests and Prjects (Pp quizzes, chapter tests, research prjects r applied prjects) 20% Majr Tests (9 weeks test) Semester 1 Big Idea One- Limits - Preparatin fr Calculus (Chapter 1) 2 weeks Newtn and Leibniz calculus cntrversy. Yu have a lt f freedm in this paper. Here are sme tpics yu shuld include. (a) Discuss Newtn s cntributin t calculus. (b) Discuss Leibniz s cntributin t calculus. (c) Discuss hw their view f the calculus differs frm tday s view. (d) Discuss the cntrversy ver the inventin f calculus. (e) What was the pinin at the time? What is the pinin tday? Linear functins Slpe f a line Pint Slpe equatins Parallel and Perpendicular lines Applicatins f Linear Functins Slving system f equatins Functins and graphs Dmain and Range Viewing and interpreting graphs Piecewise functins Abslute functins Cmpsite functins Expnential Functins Expnential grwth and decay Cmpund interest The number e

5 Parametric Equatins Circles Ellipses Inverse functins and Lgarithms One t ne functins Finding inverses Lgarithmic functins Prperties f Lgarithms Trignmetric functins Radian Measure Graphs f trignmetric functins Peridicity Transfrmatin f Trignmetric graphs Inverse f trignmetric graphs. - Cncept f Limits and Cntinuity (Chapter 2) 4 weeks Rates f Change and limits Average and Instantaneus speed Definitin f Limit One and tw sided limits Limits invlving infinite Find limits as x Infinite limits as x a End Behavirs Limits as x ± Using a calculatr, students will create an equatin and find the limit at x= 4.9, x=4.99, x=5.01, x=5.001 frm +/-. Then they will fllw the same steps t a ratinal equatin with the denminatr f (x-5) and (x-5) 2 t determine the pint f discntinuity. The students will be able t identify whether the pint is nn-remvable r remvable. Cntinuity Cntinuity at a pint Intermediate Value Therem fr Cntinuus Functins Rates f change, Tangent lines and Sensitivity Average Rate f Change Tangent t a curve Slpe f a curve Big Idea Tw- Derivatives - Derivatives (Chapter 3) 4 weeks Derivative f a functin Definitin f derivative Ntatin f derivatives Relatins between f and f Graphing the derivative frm data Differentiability Hw f (a) may nt exist Numerical derivatives n a calculatr Differentiability implies cntinuity Intermediate Value Therem fr Derivatives

6 Rules f Differentiatin Psitive integer pwers, multiples, sums and differences Negative integer Pwers f x Secnd and high rder derivatives Velcity and ther rates f change Instantaneus rates f change Mtin alng a line Derivatives in ecnmics **Rcket Launch Prject. Students will launch Estee Rckets t create a relevant equatin, find instantaneus velcity at a given time, and acceleratin. Derivatives f Trignmetric Functins Derivative f trignmetric functins. - Mre Derivatives (Chapter 4) 4 weeks Chain rule Derivative f cmpsite functin Pwer Rule Outside Inside Rule Implicit Differentiatin Implicitly defined functins Ratinal pwers f differentiable functins Derivative f higher rders Derivatives f inverse trignmetric functins Derivatives f inverse functins Derivatives f arctan, arcsin, and arccs Derivatives f expnential and lgarithmic functins Derivative f e x Derivative f a x Derivative f lg Derivative f ln Pwer f Arbitrary Real Pwers. - Applicatin f Derivatives (Chapter 5) 4 weeks Extreme Values f functins Finding Glbal Extremes values Finding lcal extreme values Mean Value Therem Physical Interpretatin Increasing and decreasing functins Cnnecting f and f with the graph f f First derivative test f lcal extrema Cncavity Pint f inflectin Secnd derivative test fr lcal extrema Mdeling and Optimizatin A strategy fr ptimizatin Examples f Business and Industry Examples f ecnmics Mdeling discrete phenmena with Differentiable functins.

7 ***The calculus f rainbws. Fr this wrk yu may wish t cnsult the article Smewhere within the rainbw by Steven Janke and The Calculus f Rainbws by Rachel Hall and Nigel Higsn. (a) State Fermat s principle f ptics. (b) State Snell s law and indicate hw yu wuld use calculus t prve Snell s law (yu dn t have t prve it). (c) Explain the behavir f sunlight in a raindrp. (d) Suppse the sun is in the sky making an angle θ with the hrizn. Yu stand with yur back t the sun. At what angle shuld yu gaze t find the rainbw? (e) Specifically, what if the sun is setting (θ = 0)? What if the θ = 45? (f) Discuss (but dn t prve) why the rainbw appears as a band f clrs in the sky. (g) Discuss (but dn t prve) why we frequently see a secnd, lighter rainbw abve the main rainbw. Linearizatin, sensitivity, and differentials Linear apprximatin Differentials Abslute, relative, percent change Newtn s Methd Related Rates Related Rate Equatins Slutin Strategy Simulating Related Mtin ***Lengthening Shadw: the stry f related rates. This prject is based n the jurnal article Lengthening Shadw: the stry f related rates, by Austin, Barry, and Berman (Math Mag., 2000). (a) Summarize the article discussing the histry f related rates prblems. (b) Discuss the purpse f intrducing related rates prblems t a calculus curse. (c) Discuss the current mvement away frm related rates prblems. (d) Yu may want t interview the mathematics prfessrs here t determine their pinin f related rates prblems. (e) Find cpies f ld textbks and give examples f the first related rates prblems. (f) Slve these prblems. Semester 2 Big Idea Three: Integrals and the Fundamental Therem f Calculus - The Definite Integral (Chapter 6) 4 weeks Estimating with finite sum Accumulatin Prblems as Area Rectangular Apprximatin Methd Vlume f sphere Cardiac Output Definite Integrals Riemann Sums Terminlgy and Ntatin f integratin Definite integral and area Integrals n a Calculatr Discntinuus Integrable Functins Student will Riemann Sums f a functin, visualize Riemann sums as a series f rectangles and evaluate sums in definite integrals. Student will find left and right sums algebraically in subdivisins as it ges t infinity. Definite integrals and Antiderivatives Prperties f Definite Integrals Average Value f a functin Mean Value Therem fr Definite Integrals Cnnectin Differential and Integral Calculus

8 *** Students create an equatin f their chice and calculate the definite integral frm x= -1 t x=7 using methds frm the sectin. Students will present their wrk n TAKK.cm s shw cmputatin. Students will present their finding and thse bserving will use a different methd t see if they can cme up with a similar answer. Fundamental Therem f Calculus Antiderivative Part x Graphing the functin f(t)dt a Evaluatin Part Area Cnnectin Analyzing Antiderivatives graphically Trapezidal Rule Trapezidal Apprximatins Errr Analysis Other Algrithm - Differential Equatins and Mathematical Mdeling (Chapter 7) 4 weeks Slpe fields and Euler s Methd Slpe fields Euler s Methd Antidifferentiatin by substitutin Indefinite Integrals Leibniz Ntatin and Antideriviatives Substitutin in Indefinite/ Definite Integrals Antidifferentiatin by parts Prduct rule in integral frm Slving fr the unknwn integral Tabular integratin Inverse trignmetric and lgarithmic functins Expnential grwth and decay Laws f Expnential Change Cntinusly cmpund interest Radiactivity Newtn s Law f Cling Lgistic Grwth Ppulatin Grwth Partial fractins Lgistic Grwth Mdels - Applicatins f Definite Integrals (Chapter 8) 4 weeks Accumulatin and net change Linear mtin revisited Cnsumptin ver time Cming and ging Net change frm data Density Areas in a plane Area between curves Area enclsed by intersecting curves Bundaries with changing functins

9 Integrating in Respect fr y Vlumes Vlume as an integral Square crss sectins Circular crss sectins Cylindrical Shells Other Crss Sectins Length f curves Sine waves Length f Smth Curve Vertical tangents, crners and cusps Applicatins frm science and statistics Wrk revisited Fluid Frce and Fluid pressure Nrmal Prbabilities Review/ Test Preparatin (3-4 weeks) Multiple Chice Practice test questins t strengthen test taking strategies Individual and grup practice clarify understanding f test questin Free Respnse Practice writing sentences t explain slutins t prblems Individual and grup practice t frmulate respnses Full explanatin f written respnses will be discussed t make sure students meet requirements frm rubrics. After The Exam (3-4 weeks) Students will lk at cllege math requirements and expectatins fr placement n cllege entrance exam Preparatin fr Calculus BC. (Advance Integratin)