1 of 28 Calculus AB Syllabus Calculus BC I teach Calculus as the culmination of all of the student's previous work in mathematics. This allows me to help the student organize their previous knowledge. This goal is met by, in addition to the presentation of calculus topics, but also by constant exposure to competition type questions covering the entire range of high school math topics. This syllabus, shows all topics taught in the BC course. Bibliography Larson, Ron, Robert P. Hostetler, and Bruce H. Edwards. Calculus with Analytic Geometry. 7th ed. Boston: Houghton Mifflin, 2002. Hockett, Shirley 0., Barron's How To Prepare For The AP Calculus (7th Ed.) COWculus on the Web: An online homework manage system sponsored by Temple University. Each student has an individual account and does homework problems online. The results are available to the teacher online. Good practice questions. http://www.math.temple.edu/~cow/ WebWork: Another homework management system sponsored by Rochester University. WebWork gives different problems to each student. Students can work together, but must each do their own problems. Student progress is viewable online by the teacher and can be downloaded to the teachers gradebook. http://webwork.rochester.edu/ Teacher Website: http://teachers.dadeschools.net/akoski A collection of programs, demonstrations, worksheets and other materials to enhance the learning of mathematics. Assessments: Chapter Tests and Quizzes from Larson TestBank Midterm Exam: Selected Questions from Released Exam 1985-1988 Final Exam: Released Exam 2003 Online Assignments from COWculus Online Assignments from WebWork Many individual worksheets developed (and borrowed) over the years. Calculator Usage: Each student is expected to be comfortable with calculator usage. Much like graphing calculators, all COWculus and WebWork assignments depend on calculator-like input. Like the TI-89 both systems will display the "pretty" form of the mathematics entered if the student so desires. Since WebWork gives each student a different form of the problem, it is common that problems will use numbers that are not easily done without a calculator. Student get a great deal of calculator practice this way. In addition, I have collected a vast number of calculator active worksheets to be used when I notice a student who needs more assistance. Finally, students work through many calculator active problems from released exams, both in class and for homework. The graphing calculator and Geometer's Sketchpad are essential element in exploring abstract mathematics. With it the student is able to experiment, visualize and conjecture. Students become comfortable with calculator use and are able to perform the basic tasks required by the AP curriculum. (Graphs, Solutions, Derivatives and Integrals)
28
Embed
Calc BC Work...5of 28 Calculus AB Syllabus Applications of Differentiation 3 weeks • Extrema on an interval and the Extreme Value Theorem AB26 • Rolle’s Theorem and the Mean
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
1 of 28 Calculus AB Syllabus
Calculus BCI teach Calculus as the culmination of all of the student's previous work in mathematics. This allows me to helpthe student organize their previous knowledge. This goal is met by, in addition to the presentation of calculustopics, but also by constant exposure to competition type questions covering the entire range of high schoolmath topics. This syllabus, shows all topics taught in the BC course.
BibliographyLarson, Ron, Robert P. Hostetler, and Bruce H. Edwards. Calculus with Analytic Geometry. 7th ed. Boston:Houghton Mifflin, 2002.
Hockett, Shirley 0., Barron's How To Prepare For The AP Calculus (7th Ed.)
COWculus on the Web: An online homework manage system sponsored by Temple University. Each studenthas an individual account and does homework problems online. The results are available to the teacher online.Good practice questions. http://www.math.temple.edu/~cow/
WebWork: Another homework management system sponsored by Rochester University. WebWork givesdifferent problems to each student. Students can work together, but must each do their own problems. Studentprogress is viewable online by the teacher and can be downloaded to the teachers gradebook.http://webwork.rochester.edu/
Teacher Website:http://teachers.dadeschools.net/akoskiA collection of programs, demonstrations, worksheets and other materials to enhance the learning ofmathematics.
Assessments:Chapter Tests and Quizzes from Larson TestBankMidterm Exam: Selected Questions from Released Exam 1985-1988Final Exam: Released Exam 2003Online Assignments from COWculusOnline Assignments from WebWorkMany individual worksheets developed (and borrowed) over the years.
Calculator Usage:Each student is expected to be comfortable with calculator usage. Much like graphing calculators, allCOWculus and WebWork assignments depend on calculator-like input. Like the TI-89 both systems willdisplay the "pretty" form of the mathematics entered if the student so desires. Since WebWork gives eachstudent a different form of the problem, it is common that problems will use numbers that are not easily donewithout a calculator. Student get a great deal of calculator practice this way.In addition, I have collected a vast number of calculator active worksheets to be used when I notice a studentwho needs more assistance. Finally, students work through many calculator active problems from releasedexams, both in class and for homework.
The graphing calculator and Geometer's Sketchpad are essential element in exploring abstract mathematics.With it the student is able to experiment, visualize and conjecture. Students become comfortable withcalculator use and are able to perform the basic tasks required by the AP curriculum. (Graphs, Solutions,Derivatives and Integrals)
2 of 28 Calculus AB Syllabus
Teaching StrategiesProjects
The course includes several major projects1. Cowculus and/or WebWork
a. Online homework and math practice. Provided in conjunction with Temple University andRochester University. See Homework Correlation at end of this document.
2. 16 Functionsa. Students create detailed and complete analysis of 16 basic functions. They must include ALL
information about roots, discontinuities, limits etc.3. Geometer's Sketchpad
a. A large collection of demonstrations and activities. A favorite is Dynagraphs which has beenmodified to match the 16 Functions activity.
4. Volume Projecta. Students create a volume model based on one of the volume integration techniques.
Accompanying paper work must show complete calculus solution, plus the actual value of theapproximation.
5. Daily Journal A collection of 1) assigned topics, 2) class impressions 3) independent research and 4)free writing. Students get daily practice in putting their mathematics ideas into words. Students will usedaily writing to explore calculus topics to develop an appreciation of calculus as a coherent body ofknowledge and as a human accomplishment.
6. Major Reviewa. Several "8-hour" review sessions are scheduled for struggling students. During these reviews a
complete review on basic topics is completed.7. Full Length practice exams
a. During the 3rd marking period, all students are required to take from 1-6 full length AP practiceexams. Exams are graded and returned during the sessions.
8. Mu Alpha Theta competitions and practicea. Students are encourage to participate in a myriad of mathematics competitions and practices.
Students can choose from as simple as ½ hour in class quizzes (Continental Math League) toweek-long major competitions.
b. Mu Alpha Theta, Florida Math League, Continental Math League , American Scholastic MathAssociation, AMC-8/10/12, AIME, USAMO
BC students, in many cases, are able to work independently. These students work through theCOWculus course as well as assignments from Larson. Progress is monitered through regularexaminations from the Larson Test Bank. In addition, these students spend a great deal of time workingthrough problems from mathematics competitions. Pacing for these students is self-directed.
Rule of Four __________
The "Rule of Four" is a fundamental theme in all my math classes. I have the four domains: symbolic, numeric,analytic, and verbal permanently written on my board. During all presentations and discussions, students areexpected to present their work in as many of the domains as they can. I find that most of my students have littleexperience in transferring thinking from one domain to the other. During most discussions and presentations Iwill pose the question in one format and then ask students to reinterpret in one of the other domains. Extensiveuse of Geometer's sketchpad allows me to create graphic situations. Students are asked to describe that theythink will happen under certain conditions. Three particularly good sketches are Dynagraphs, Color Bars andPolar Plots. In addition, each student has a notebook in which they keep a daily journal. Students use thejournal to discuss their impressions of math and lessons. The course teaches students how to communicatemathematics and explain solutions to problems both verbally and in written sentences.
3 of 28 Calculus AB Syllabus
AP ReviewsI generally am able to finish all new material several weeks before the AP Exam. I use this time for review andpractice. During this time, students work on the sample questions in the AP Calculus Course Description and onmultiple-choice and free-response questions from AP Released Exams as well as problems from the Barron'sreview book. Some of these are assigned for homework, while others are given as a quiz or test.
Topic OutlineI have included a sample of typical assignments and correlated them to the topics in the Course Outline. I usedthe following codes. There is a more detailed description of some assignments at the end of the outline.
Partial Correlation to Class Assignments:Code Description
AB# or BC# COWculus online assignment
WW# WebWork online assignment
Prj# Class Project
Jrnl Journal Writing Assignment
GSP# Geometer's Sketchpad Demonstration
Lar# Larson page # exercises/discussion
Calc# Graphing Calculator exercise
WS# Work Sheet
FRQ AP Free Response Question
Prg# Teacher written Programs
Demo# Demonstrations, Models, hands on activities
Barrons Various Multiple Choice Sets to coverpractice and applications
Analysis of Graphs, Functions and Precalculus Review (summer or asneeded)
Students in BC should already be comfortable with these topics. Students work independently throughChapter 1 of the Barron's Calculus Book. Students are given a set of 16 functions to graph andcompletely describe. (See Graph Scoring guidelines at the end of this syllabus) At this time allfunctional transformations are discussed related to the graphs students are creating. GeometerSketchpad is use extensively to demonstrate the concepts of transformations. Students will be able towork with functions represented in a variety of ways: graphical, numerical, analytical
Barron's Chapter 1 16 Graphs ColorBar Program Dynagraphs ( ), ( ), , , , , , ,f ax af x f x a f x a f a f a f x f x etc WS4
I cover these topics in Calculus AB and Alebra 2. Many students are able to go directly to the BC curriculum.These students work independently using COWculus and assignments from Larson. Students who are able towork at an advanced pace,
4 of 28 Calculus AB Syllabus
Limits and Their Properties (2 Weeks)• An introduction to limits, including an intuitive understanding of the limitprocess
CAB11
• Using graphs and tables of data to determine limits Lar#48 Proj #1• Properties of limits GSP#1 Proj #1• Algebraic techniques for evaluating limits L#69 AB11• Comparing relative magnitudes of functions and their rates of change GSP#2• Continuity and one-sided limits AB12 AB13
• Geometric understanding of the graphs of continuous functions Calc#1• Intermediate Value Theorem Lar#75• Infinite limits AB13• Using limits to find the asymptotes of a function AB13Barron's Chapter 2 Multiple Choice Questions
Differentiation 2 Weeks• Zooming-in activity and local linearity Calc#2• Understanding of the derivative: graphically, numerically, and analytically WS1• Approximating rates of change from graphs and tables of data FRQ#9• The derivative as: the limit of the average rate of change, an instantaneousrate of change, limit of the difference quotient, and the slope of a curve at a point
variousFRQ
• The meaning of the derivative—translating verbal descriptions into equationsand vice versa
FRQ#7b
• The relationship between differentiability and continuity Lar#101• Functions that have a vertical tangent at a point 2
5y x• Functions that have a point on which there is no tangent y x• Differentiation rules for basic functions, including power functions andtrigonometric functions
AB21AB22
• Rules of differentiation for sums, differences, products, and quotients AB21• The chain rule Lars#133• Implicit differentiation Lars#142 AB22• Related rates Lars#149
BarronsAB17
Barron's Chapter 3 Multiple Choice Questions
5 of 28 Calculus AB Syllabus
Applications of Differentiation 3 weeks• Extrema on an interval and the Extreme Value Theorem AB26• Rolle’s Theorem and the Mean Value Theorem, and their geometricconsequences
AB24
• Increasing and decreasing functions and the First Derivative Test Lar#181• Concavity and its relationship to the first and second derivatives AB27• Second Derivative Test Lar#189• Limits at infinity AB13• A summary of curve sketching—using geometric and analytic information aswell as calculus to predict the behavior of a function
AB24
• Relating the graphs of ,f, ,f’, and f''. WS1• Optimization including both relative and absolute extrema AB26• Tangent line to a curve and linear approximations AB16• Application problems including position, velocity, acceleration, andrectilinear motion
AB25AB26Barrons
AB17
Barron's Chapter 4 Multiple Choice Questions
Integration 4 Weeks• Antiderivatives and indefinite integration, including antiderivatives followingdirectly from derivatives of basic functions
AB30
• Basic properties of the definite integral Lar#271• Area under a curve FRQ#1a FRQ#8• Meaning of the definite integral Lar #269• Definite integral as a limit of Riemann sums Demo#1 AB32• Riemann sums, including left, right, and midpoint sums AB32• Trapezoidal sums Lar#301 AB32• Use of Riemann sums and trapezoidal sums to approximate definite integralsof functions that are represented analytically, graphically, and by tables of data
FRQ3a
• Use of the First Fundamental Theorem to evaluate definite integrals AB32• Use of substitution of variables to evaluate definite integrals Lar#297• Integration by substitution Lar#297• Discovery lesson on the Second Fundamental Theorem of Calculus Prog1• The Second Fundamental Theorem of Calculus and functions defined byintegrals
WS#2
• The Mean Value Theorem for Integrals and the average value of a function FRQ#3Select Multiple Choice Questions on Barron's Chapter 5,6,7, and 8
6 of 28 Calculus AB Syllabus
Logarithmic, Exponential, and Other Transcendental Functions 1weeks—
• The natural logarithmic function and differentiation AB35 AB36• The natural logarithmic function and integration AB37 Lar#347• Inverse functions Lar#335• Exponential functions: differentiation and integration — FRQ#8• Bases other than e and applications FRQ#8 Lar#357• Solving separable differential equations Lar#366
Journal:WS#5
FRQ#6
• Applications of differential equations in modeling, including exponential growth FRQ#11• Use of slope fields to interpret a differential equation geometrically WS#3• Drawing slope fields and solution curves for differential equations WS#3 FRQ#6• Euler’s method as a numerical solution of a differential equation Larson
AppendixA
FRQ#12
Barron's Multiple Choice Questions on Differential Equations Chapter 9
Logarithmic, Exponential, and Other Transcendental Functions 1week• Inverse trig functions and differentiation Lar#386• Inverse trig functions and integration Lar#393
Applications of Integration 3 weeks• The integral as an accumulator of rates of change AB34 FRQ#1• Area of a region between two curves FRQ#8a• Volume of a solid with known cross sections FRQ#10c• Volume of solids of revolution FRQ#8b• Applications of integration in problems involving a particle moving along a line,including the use of the definite integral with an initial condition and using the definiteintegral to find the distance traveled by a particle along a line
Barrons6
Barron's Chapter 6 Multiple Choice Questions
Integration Techniques, L’Hopital’s Rule, and Improper Integrals 1week• Review of basic integration rules• Integration by parts Lar#494• Trigonometric integrals Lar#512• Integration by partial fractions Barron's 5 BC#11• Solving logistic differential equations and using them in modeling Lar#523-58 Lar#409-12-15• L’Hopital’s Rule and its use in determining limits BC#12• Improper integrals and their convergence and divergence, includingthe use of L’Hopital’s Rule
Lar#547 AB#40
7 of 28 Calculus AB Syllabus
Infinite Series (6 weeks)• Convergence and divergence of sequences BC#01• Definition of a series as a sequence of partial sums BC#02• Convergence of a series defined in terms of the limit of the sequence ofpartial sumsof a series
BC#02
• Introduction to convergence and divergence of a series by using technologyon two examples to gain an intuitive understanding of the meaning ofconvergence
Calc#3
• Geometric series and applications Lar#569• The nth-Term Test for Divergence BC#03 Lar#571• The Integral Test and its relationship to improper integrals and areas ofrectangles
GSP#4BC#03-4
Lar#580
• Use of the Integral Test to introduce the test for p-series BC#03-4 Lar#577• Comparisons of series BC#05-6 Lar#587• Alternating series and the Alternating Series Remainder Lar#595• The Ratio and Root Tests BC#03-4 WS#6 Lar#603• Taylor polynomials and approximations: introduction using the graphingcalculator
Lar#613
• Power series and radius and interval of convergence BC#07 Lar#623• Taylor and Maclaurin series for a given function BC#08 Lar#638• Maclaurin series for sin x, cos x, xe and ln x
Manipulation of series, including substitution, addition of series,multiplication of seriesby a constant and/or a variable, differentiation of series, integration ofseries, andforming a new series from a known series Taylor’s Theorem with theLagrange Form of the Remainder (Lagrange Error Bound)
LegrangeLar#611-612
Lar#630
Barron's Multiple Choice Questions: Chapter 10
Plane Curves, Parametric Equations, and Polar Curves (2 weeks)• Plane curves and parametric equations BC#13 Lar#672• Parametric equations and calculus Lar#681• Parametric equations and vectors: motion along a curve, position, velocity,acceleration, speed, distance traveled
BC#14
• Analysis of curves given in parametric and vector form BC#13• Polar coordinates and polar graphs Lar#685• Analysis of curves given in polar form Lar#691• Area of a region bounded by polar curves Lar#701Barron's Multiple Choice Questions: Chapter 8
8 of 28 Calculus AB Syllabus
Details for Sample Exercises/Assessments from Topic Outline
Geometer Sketchpad Sketches1. Teacher created sketch which shows dynamically the interaction between the delta neighborhood and theepsilon neighborhood. Examples for the 16 basic graphs demonstrate all types of discontinuities.
2. Dynagraphs. Dynagraphs dynamically demonstrate the relation between domain and range with two parallelreal lines. Dynaquiz- is a version where students must identify the function from unlabelled Dynagraphs.
3. Color Bars. Using the Dynagraph program as a base, a second line is colored to match the image of eachpoint based on student selected function. Students pick transformation and try to describe that they think willhappen.
4. Integral Test: Students had trouble understanding diagram on page 577 of Larson. Teacher created sketch tolet students dynamically move the rectangles. Students see that the translated rectangles now fit under thecurve.
Calculator Exercises1. Epsilon/Delta Discovery: Students must adjust the window of the x-variable and the y-variable to help
understand the epsilon-delta definition of limit.
2. Local Linearity: Students will zoom in on graphs of functions selected from the Basic 16 functions. Inparticular, students will compare the graphs of 2y x and y x .
3. Graphing partial sums to see convergence: Sample problem Larson pg 568 Firgure 8.5 also problems fromsection 8.2
Demonstrations
1. 2-dimensional and 3-dimensional models to illustrate the summation rules.
1
1
2
n
i
n ni
and
2
1
1 2 1
6
n
i
n n ni
.
Worksheets1. Given either f, f' or f", students will draw the graphs of the other two.
2. A collection of 2nd Fundamental Theorem Problems: ie. x
af t dt f x and
222
x
xf t dt f x f x
3. Slope Field Handout from AP Central by Nancy Stephenson4. Teacher made Transformation Worksheet. Students are given a sample function and 9 transformation todraw. i.e. ( ), ( ), , , , , , ,f ax af x f x a f x a f a f a f x f x etc
5. Solving Separable Differential Equations: Antidifferentiation and Domain Are Both Needed by DavidLomen (worksheet downloaded from AP Central)6. Summary of Tests for Series: See page 602 of Larson. Also teacher made worksheet that compares
Programs1. Color Bars – Program deforms a single copy of the real lines based on a transformation function. Studentspick transformation and try to describe that they think will happen.
9 of 28 Calculus AB Syllabus
Free Response QuestionsCalculus AB 2004 Form B
1. Question 1: a) Area under a curve b) volume between two curves c)2. Question 2 a) calculator eval y' b) cal eval y'' c) integral: accumulation of y' = y+C d)absolute maximum3. Question 3: a)Reimann b) MVT c) accel d) average value4. Question 4 graphic of f' a) find inflection b) abs max and min c) tangent line to f5. Question 5 slope field b) interpret data c) solve differential equation6. Question 6 a) area under curve b) area of triangle c) area of region between two graphs. Maximimze area.
Calculus AB 20047. Question 1: a) Accumulation b) interpret derivative verbally c) average value d) average rate of change
Calculus AB 20028. Question 1: a) Area between two curves (logs e^x) b) Volume between two curves c)abs max and min
Calculus AB 19989. Question 3: Graph of velocity a) find acceleration is positive b) average acc of car c) approx acc d) Reimansum explain the meaning of the integal.
Calculus AB 199610. Question 2: a) Area under curve b) divide region in half c) volume by cross-sections
Calculus BC 199111. Question 6: Rate of Rumor a) max rumor speed b) solve differential equations c)
Calculus BC 200
10 of 28 Calculus AB Syllabus
Graph Scoring Guidelines
1. Axes labeled.2. Equal scales.3. Graph labeled with function name.4. Graph is neat.5. Graph is accurate.6. Reference line (y=x) is drawn.7. Good interval. (Shows important function values)8. Good interval. (Shows all symmetries)9. Good interval. (Shows periodic behavior)10. Good interval. (Shows discontinuities including assymptotes)11. Table of values (Shows interesting examples of above)12. Mapping13. Colorbars (see 7)14. Colorbars (see 8)15. Colorbars (see 9)16. Colorbars (see 10)17. Domain is stated
a. Identify and Discuss all discontinuities18. Range is stated.19. Roots and intercepts are listed.20. Symmetries are stated
a. X-axisb. Y-axisc. Y = Xd. Origin
21. Odd-Even Analysis22. Function Analysis
a. Well-Definedb. One-to-Onec. Ontod. Periodic
23. Identify and state all special limit valuesa. x b. x c. x ad. x ae. x a
11 of 28 Calculus AB Syllabus
Cowculus- AB Online homework Assignments 2468 Problems
Complete COWculus Online Assignment ListAB 00 First Assignment
Precalculus Book
o Functions
Linear Functions
Equation of a line: problems: 1-20
Normal lines: problems: 1-27
Parallel lines: problems: 1-28
Slope of a line: problems: 1-20
AB 02 Sequences
Precalculus Book
o Numbers
Sequences
Arithmetic and geometric sequences: problems: 1-16
Linearly recursive sequences: problems: 1-31
AB 03 Linear Equations
Precalculus Book
o Equations
Linear Equations
Fractional linear equations: problems: 1-16
Linear equations with parameters: problems: 1-8
Solving linear equations: problems: 1-10
AB 04 Plotting
Precalculus Book
o Plotting, Graphs
Lines
Intersecting two lines: problems: 1-13
Lines, point-slope: problems: 2,4,6,8,10,12,14,16
12 of 28 Calculus AB Syllabus
Sketching lines: problems: 1-10
Three collinear points: problems: 2,4,6,8,10,12,14,16,18,20,22
Plotting
Locating Points on Curves: problems: 2,4,6,8,10
Plotting points on a line: problems: 2,4,6,8,10,12
Points and midpoints in the plane: problems: 2,4,6,8,10,12,14,16
Points and midpoints on a line: problems: 2,4,6,8,10,12,14
AB 05 Polynomials
Precalculus Book
o Polynomials
Factoring, Roots
Factoring quadratic polynomials: problems: 2-24
Finding rational roots: problems: 1-19
Polynomial Algebra
Linear combinations: problems: 2,4,6,8,10,12,14,16,18,20