AP Calculus BC. Revised: Aug 11, 2009

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1 Curse name: AP Calculus BC Revised: Aug 11, 2009 Curse Descriptin: In AP Calculus BC, students will explre fur main ideas frm calculus: limits, derivatives, indefinite integrals, and definite integrals. Prblems will be apprached using a balance f multiple representatins including: graphically, numerically, analytically/algebraically, and verbally. Wherever practical, cncepts will be applied t analyze real-wrld situatins. Graphing calculatrs and/r cmputers will be used n a regular basis t help slve prblems, experiment, interpret results, and supprt cnclusins. Students will learn t cmmunicate mathematics thrugh the use f a math jurnal and having time each day t wrk cperatively in grups t exchange ideas and appraches and t reflect n hmewrk assignments. A variety f assessments will be used t including paper based tests and quizzes, hmewrk, grup and individual labratry wrk. In rder t prepare students fr the AP exam, tests and quizzes will cntain questins similar t thse that appear n the AP exam in terms f cntent, difficulty, and structure. Primary text(s) and ther majr resurces: Calculus f a single variable Larsn, Hstetler, and Edwards (D.C. Heath and Cmpany, 1994) Page 1 f 20

2 Objectives unit number & title Essential Cncepts (specific skills and knwledge students will have) U1 Prerequisites and Analysis f Graphs U2 Limits and Cntinuity Objective set 1: 1 week Student will explre the fllwing tpics review the real number system review the cartesian plane review graphs f equatins review lines in the plane review functins review trignmetric functins Hw are real numbers and sets f real numbers be represented, classified, and rdered? What algebraic techniques are cmmnly used in calculus? Hw can data and relatinships be rganized and represented graphically? Objective set: 4-5 weeks Students will: Evaluate Limits at a Pint 1 sided limits 2 sided limits Sandwich Therem Evaluate Limits invlving Infinity Asympttic Behavir End Behavir mdels Prperties f Limits (Algebraic Analysis) Visualizing Limits (Graphic Analysis Determine Cntinuity f a functin Cntinuity at a Pint Cntinuus Functins Discntinuus Functins Remvable Discntinuity Jump Discntinuity Infinite Discntinuity Determine and analyze Rates f Change and Tangent Lines Average rate f change Page 2 f 20 Assessment Students must shw prficiency with MLR skills and knwledge in assessments marked Essential in rder t prgress t the next curse level. What is a limit? Hw d yu find a limit with a table, graph, r analytically? When des a limit nt exist? What is the definitin f cntinuity fr a functin n an pen r clsed interval? What is a cntinuus functin? Hw can I determine if a functin is cntinuus? What is meant by rates f change? Hw is the rate f change determined? Hw are tangent lines and a rate f change related? Hmewrk and quizzes will be given n a regular basis thrughut the curse Test #1

3 Objectives unit number & title Essential Cncepts (specific skills and knwledge students will have) U3 The Derivative Objective Set 5-6 Weeks Tangent line t a curve Slpe f a curve (algebraically and graphically) Nrmal line t a curve (algebraically and graphically) Instantaneus rate f change Students will explre the fllwing tpics: Determine Rates f Change Average Speed Instantaneus Speed Use a variety f methds t determine the Derivative f a Functin Definitin f the derivative (difference qutient) Derivative at a Pint Relatinships between the graphs f f and f Graphing a derivative frm data One sided derivatives Determine if a functin is Differentiable Cases where f (x) might fail t exist Lcal linearity Derivatives n the calculatr (Numerical derivatives using NDERIV) Symmetric difference qutient Relatinship between differentiability and cntinuity Intermediate Value Therem fr Derivatives Use Rules fr Differentiatin t determine derivatives f functins Cnstant, Pwer, Sum, Difference, Prduct, Qutient Rules Page 3 f 20 Assessment Students must shw prficiency with MLR skills and knwledge in assessments marked Essential in rder t prgress t the next curse level. What is a nrmal line and hw can I determine the frmula fr a nrmal line? Prject #2 Test #2 Hw d I determine a rate f change? What is the difference between an average rate f change and an instantaneus rate f change? Hw I determine the derivative f a functin? Hw d I determine if a functin is differentiable? Hw d I determine if a functin is cntinuus? What facts can we cnclude abut a cntinuus functin?

4 Objectives unit number & title Essential Cncepts (specific skills and knwledge students will have) U4 Applicatins f the Derivative Higher rder derivatives Apply cncepts f the derivative t analyze mtin prblems Psitin, velcity, acceleratin, and jerk Particle mtin L HÔpital s Rule Apply cncepts f the derivative t analyze ecnmic prblems Marginal Cst Marginal Revenue Marginal Prfit Determine Derivatives f Trignmetric Functins Use the Chain Rule t derive cmpsitin functins Determine the rate f change using Implicit Differentiatin Differential Methd Y Methd Determine Derivatives f Inverse Trignmetric Functins Determine Derivatives f Expnential and Lgarithmic Functins Hw can we use derivatives t analyze mtin? Hw can we use derivatives t analyze ecnmic prblems? Hw d we determine the derivative f trignmetric functins? What are cmpsite functins and hw d we determine the derivative? What are implicit functins and hw d we determine the derivative? What are inverse trignmetric functins and hw d we determine the derivative? What are expnential functins and lgarithmic functins and hw d we determine the derivative Objective Set 5-6 Weeks Students will: Page 4 f 20 Lcate Extreme Values f a functin Relative Extrema Abslute Extrema Extreme Value Therem Definitin f a Critical Pint Interpret Implicatins f the Derivative Assessment Students must shw prficiency with MLR skills and knwledge in assessments marked Essential in rder t prgress t the next curse level. What are extrema and where d they ccur? What can we interpret abut a Prject #3 Test #3

5 Objectives unit number & title Essential Cncepts (specific skills and knwledge students will have) Rlle s Therem Mean Value Therem Increasing and Decreasing functins Prduce accurate graphs by using the relatinships f f and f with f(x) First derivative test fr relative max/min Secnd Derivative Cncavity Inflectin Pints Test fr relative max/min Analyze Optimizatin Prblems using calculus Estimate functins using Linearizatin mdels Lcal Linearizatin Tangent Line apprximatin Differentials Analyze Related Rates prblems functin frm the derivative? Hw is a functin related t its first and secnd derivatives? What is an ptimizatin prblem and hw can we use derivatives t slve them? Hw can we prduce estimates f the value f a curve using linearizatin mdels? What is a related rate prblem and hw can we use derivatives t slve them? U5 The Definite Integral Objective Set 3-4 Weeks Students will explre:: Page 5 f 20 Apprximate Areas between curves using summatins Riemann sums Left Right Midpint Trapezidal Cmpare and cntrast Definite Integrals t Riemann sums Use Prperties f Definite Integrals Assessment Students must shw prficiency with MLR skills and knwledge in assessments marked Essential in rder t prgress t the next curse level. What methds can be used t apprximate areas underneath curves? Hw can we mdify ur apprximatin methds t imprve accuracy? What are the basic prperties f Prject #4 Test #4

6 Objectives unit number & title Essential Cncepts (specific skills and knwledge students will have) Pwer Rule Mean Value Therem fr Definite Integrals Understand hw The Fundamental Therem f Calculus describes the relatinship between the integral and the derivative Part 1 Part 2 U6 Differential Equatins and Mathematical Mdeling Assessment Students must shw prficiency with MLR skills and knwledge in assessments marked Essential in rder t prgress t the next curse level. definite integrals? Prject #5 Test #5 Hw are derivatives and integrals related? Objective Set 3 Weeks Students will explre the fllwing tpics: Unit 7 Applicatins f Definite Integrals Prduce Slpe Fields fr differential equatins Evaluate Antiderivatives using cmmn frmulas Indefinite Integrals Pwer Frmulas Trignmetric Frmulas Expnential and Lgarithmic frmulas Analyze Lgistic Grwth mdels What is a differential equatin? Hw can we visualize the slutins fr a differential equatin? What are sme cmmn antiderivatives? Prject #6 Test #6 What are lgistic grwth mdels and hw d I analyze them using calculus? Objective Set 3-4 Weeks Students will explre the fllwing tpics: Page 6 f 20 Evaluate and interpret Integrals as net change Calculating distance traveled Cnsumptin ver time Net Change frm data Calculate areas between curves Integrating with respect t x Integrating with respect t y Hw can integrals be used t describe mtin r ther changing variables? Hw can we use integrals t calculate areas between curves?

7 Objectives unit number & title Essential Cncepts (specific skills and knwledge students will have) Calculate areas between intersecting curves Integrating with respect t x Integrating with respect t y Calculate Vlume f slids Crss sectins Disc Methd Shell Methd Objective Set 2-3 Weeks Assessment Students must shw prficiency with MLR skills and knwledge in assessments marked Essential in rder t prgress t the next curse level. Unit 8 Plane Curves, Parametric Equatins, and Plar Crdinates Graph Plane Curves and Parametric Equatins Analyze Parametric Equatins Slpe and Tangent Line Analyze Plar Crdinates and Plar Graphs Slpe and Tangent Lines What are plane curves and parametric equatins? Hw can they be represented graphically? Hw d we determine the rates f change fr plane curves and parametric equatins? Prject #8 Test #8 Objective Set 3 Weeks Students will: Use series ntatin t express and evaluate finite and infinite Series Determine cnvergence using a variety f tests Integral Test p-series Cmparisns f Series Alternating Series Use Rati and Rt Tests t determine divergence Understand Taylr Plynmials and Apprximatins Represent varius functins using Pwer Series Generate Taylr and McLaurin Series Page 7 f 20 Prject #7 Determine distance f a trip using speed and time data. Test #7 Students will: Unit 9 Infinite Series Hw can we use integrals t calculate vlumes f slids? What are series and hw are they expressed? Hw d we determine if a series cnverges? Hw d we determine if a series diverges? What is a Taylr Plynmial? What is a Pwer series? Prject #9 Test #9

8 unit number & title Objectives Essential Cncepts (specific skills and knwledge students will have) Assessment Students must shw prficiency with MLR skills and knwledge in assessments marked Essential in rder t prgress t the next curse level. Hw are Taylr and McLaurin series similar? Hw are they Different? Assessments: All assessments include a variety f multiple chice and free respnse questins mdeling the AP Calculus BC test. A variety f slutin methds will be used depending n each questin, including analytical, numerical, graphical, and verbal respnses. Students are explicitly taught analytical, graphical, and numerical respnses, as well as interpretatin necessary t understand and verbalize their results based n situatinal cntext. Belw is a cpy f the sheet we use t analyze data numerically, graphically, and analytically. *Nte: due t technical difficulties, the curriculum samples belw demnstrating calculatr use and written respnses had t be scanned in, and nt all elements came acrss as the students wuld receive them. Page 8 f 20

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12 Calculus Prject - Optimize ur Z Using Functins (Revised) Intrductin: Yu have been hired by the lcal cmmunity t advise them n hw t best design their new z. They have limited materials t use and they want t ensure that they get the mst ut f the materials. Yu will need t examine their requirements, determine the best way t meet the requirements, and make a scale mdel r map f yur prpsed design. Objectives: 1. Yu will be able t mdel real situatins with functins. 2. Yu will be able t represent functins algebraically, numerically (data table), and graphically. 3. Yu will be able t use prprtins t prduce a map t scale. Z Requirements: 1. Grilla a. We have feet f fence t use fr the grilla cage. Page 12 f 20

13 b. We want the grilla t live in a rectangular space with the maximum area pssible t swing arund. c. 2. Sheep and Pigs a. We have feet f fence t use fr sheep and pigs. b. We want the sheep and pigs t live in tw adjacent rectangular pens. c. One side f the sheep and pigs area will be brdered by a river fr drinking. d. We want t give the sheep and pigs the maximum area pssible. e. The pig area shuld be the same size as the sheep area. Page 13 f 20

14 3. Lins, Tigers, and Bears a. We have feet f fence t use fr lins, tigers, and bears. b. The animals will live in ne rectangular area divided int three equal parts. c. We want the lins, tigers, and bears t live in the maximum area pssible. Page 14 f 20

15 4. Shark Aquarium a. We want a rectangular aquarium with the left, right, frnt, back, and bttm made f clear glass s peple can see the sharks. b. The tp is pen (n glass used fr tp). c. The length shuld be times the width. d. The tank shuld hld a vlume f e. We want t minimize the amunt f glass we use because the material is very expensive. Page 15 f 20

16 Grading Rubric Element 4. Mastered 3. Prficient Labeled picture f descriptin Yu always d these with n errrs: Yu d mst f these with few (1-2) minr errrs: 2. Develping 1. Emerging Yu d mst f these Yu few r nne f these with sme (3 t 4) errrs: r have majr errrs: 1. draw a clear picture f 1. draw a clear picture f 1. draw a clear picture f each applicatin clearly 1. draw a clear picture f each applicatin clearly each applicatin clearly labeled with all knwn each applicatin clearly labeled with all knwn labeled with all knwn and unknwn variables labeled with all knwn and unknwn variables and unknwn variables and unknwn variables Use Alegbra and Calculus t create the apprpriate functins Page 16 f 20 Yu always d these with Yu d mst f these n errrs: with few (1-2) minr errrs 1. Write key equatins/functins that 1. Write key shw hw each f the equatins/functins that variables are related t shw hw each f the ne anther. (ie. V=lwh) variables are related t ne anther. (ie. V=lwh) 2. arrange these equatins t write an 2. arrange these apprpriate functin(s) equatins t write an that can be used t apprpriate functin(s) describe each prblem that can be used t (i.e. Surface Area describe each prblem functin (i.e. Surface Area Yu d mst f these with sme (3-6) errrs 1. Write key equatins/functins that shw hw each f the variables are related t ne anther. (ie. V=lwh) 2. arrange these equatins t write an apprpriate functin(s) that can be used t describe each prblem (i.e. Surface Area functin Yu d few r nne f these r have majr errrs: 1. Write key equatins/functins that shw hw each f the variables are related t ne anther. (ie. V=lwh) 2. arrange these equatins t write an apprpriate functin(s) that can be used t describe each prblem (i.e. Surface Area

17 Element 4. Mastered 3. Prficient functin Numerical Behavir f the functin Yu always d these with Yu d mst f these n errrs: with few (1 r 2) errrs Graphical Behavir f Functin Yu always d these with Yu d mst f these n errrs: with few errrs(1 r 2): 1. Emerging functin Yu d mst f these Yu d few r nne f with sme (3 t 6) errrs these r have majr errrs: 1. prduce a clear 1. prduce a clear 1. prduce a clear table(s) f the values that table(s) f the values that table(s) f the values that 1. prduce a clear shw the behavir f shw the behavir f shw the behavir f table(s) f the values that yur functin. yur functin. yur functin. shw the behavir f yur functin. 2. use labels, variables, 2. use labels, variables, 2. use labels, variables, and units n yur and units n yur and units n yur 2. use labels, variables, table(s) s it's very easy table(s) s it's very easy table(s) s it's very easy and units n yur t knw what the data t knw what the data t knw what the data table(s) s it's very easy means. means. means. t knw what the data means. 3. Highlight any data n 3. Highlight any data n 3. Highlight any data n the table(s) imprtant t the table(s) imprtant t the table(s) imprtant t 3. Highlight any data n yur slutin and clearly yur slutin and clearly yur slutin and clearly the table(s) imprtant t explain why it helps yu explain why it helps yu explain why it helps yu yur slutin and clearly slve the prblem slve the prblem slve the prblem explain why it helps yu slve the prblem 1. prduce clear graph(s) that shw the behavir f yur functin(s). Page 17 f Develping 1. prduce clear graph(s) that shw the behavir f yur Yu d mst f these Yu d few r nne f with sme errrs (3 t 6): these r have majr errrs: 1. prduce clear graph(s) that shw the 1. prduce clear behavir f yur graph(s) that shw the behavir f yur

18 Element 4. Mastered 2. Use labels, variables, and units n yur graph(s) s it's very easy t knw what the curve means. 3. Highlight any pint(s) n the curve that is imprtant t yur slutin and clearly explain why it helps yu slve the prblem. Slve the Prblem Page 18 f Prficient 2. Develping 1. Emerging functin(s). functin(s). functin(s). 2. Use labels, variables, and units n yur graph(s) s it's very easy t knw what the curve means. 2. Use labels, variables, and units n yur graph(s) s it's very easy t knw what the curve means. 2. Use labels, variables, and units n yur graph(s) s it's very easy t knw what the curve means. 3. Highlight any pint(s) n the curve that is imprtant t yur slutin and clearly explain why it helps yu slve the prblem. 3. Highlight any pint(s) n the curve that is imprtant t yur slutin and clearly explain why it helps yu slve the prblem. 3. Highlight any pint(s) n the curve that is imprtant t yur slutin and clearly explain why it helps yu slve the prblem. Yu always d these with Yu d mst f these n errrs: with few (1 r 2) errrs. Yu d mst f these Yu d few r nne f with sme (3 t 6) errrs: these r have majr errrs: 1. Clearly interpret all 1. Clearly interpret all 1. Clearly interpret all the infrmatin frm yur the infrmatin frm yur the infrmatin frm yur 1. Clearly interpret all graphs, tables, and graphs, tables, and graphs, tables, and the infrmatin frm yur algebra t slve the algebra t slve the algebra t slve the graphs, tables, and prblem. prblem. prblem. algebra t slve the prblem. 2. The slutin culd be 2. The slutin culd be 2. The slutin culd be ne number r a set f ne number r a set f ne number r a set f 2. The slutin culd be numbers s labels and numbers s labels and numbers s labels and ne number r a set f units are used t clarify. units are used t clarify. units are used t clarify. numbers s labels and units are used t clarify. 3. If pssible, a picture 3. If pssible, a picture 3. If pssible, a picture

19 Element 4. Mastered is used t clarify yur slutin and shw that yu understand hw the numbers slve the riginal prblem. 3. Prficient is used t clarify yur slutin and shw that yu understand hw the numbers slve the riginal prblem. Check t see that yur slutin makes sense. 2. Develping is used t clarify yur slutin and shw that yu understand hw the numbers slve the riginal prblem. 1. Emerging 3. If pssible, a picture is used t clarify yur slutin and shw that yu understand hw the numbers slve the riginal prblem. Scale Map Yu always d these with Yu d mst f these n errrs: with few (1 r 2) minr errrs: 1. Make a creative, accurately scaled map 1. Make a creative, that represents yur accurately scaled map prpsal fr the that represents yur cnfiguratin f the z. prpsal fr the cnfiguratin f the z. 2. Shw a legend that indicates the scale f 2. Shw a legend that actual lengths t map indicates the scale f lengths. actual lengths t map lengths. 3. Label all required areas and clearly 3. Label all required indicate any key areas and clearly measurements. indicate any key measurements. On time Yur prject is cmpleted Yur prject is cmpleted Yur prject is cmpleted Yur prject is cmpleted Page 19 f 20 Yu d mst f these Yu d few r nne f with sme (3 t 6) errrs: these r have majr errrs: 1. Make a creative, accurately scaled map 1. Make a creative, that represents yur accurately scaled map prpsal fr the that represents yur cnfiguratin f the z. prpsal fr the cnfiguratin f the z. 2. Shw a legend that indicates the scale f 2. Shw a legend that actual lengths t map indicates the scale f lengths. actual lengths t map lengths. 3. Label all required areas and clearly 3. Label all required indicate any key areas and clearly measurements. indicate any key measurements.

20 Element 4. Mastered and handed in befre r at the beginning f class n the due date. Page 20 f Prficient and handed in within 2 days past the due date 2. Develping and handed in within 4 days past the due date 1. Emerging and handed in within 6 days past the due date