AP Calculus BC

Mrs. Moskal                                                                            Piscataway High School

Mathematics Dept.                                                                  732-981-0700  ext. 7052

Room A05                                                                              dmoskal@pway.org

 

Welcome to AP Calculus BC.  This is a 10-credit course.  The primary text for this course is Calculus Graphical, Numerical, Algebraic.  To help you be successful in this course, every day you should bring the following items to class:

  • Your assigned text book.Your book must be covered.
  • A math notebook (spiral or 3-ring with multiple sections), folder/portfolio, pencil, and planner.
  • Completed homework assignment including any assigned worksheets or explorations.
  • A TI-89 graphing calculator.

 

The grading system is based on points.  Chapter tests are 100 points.  Announced and/or unannounced quizzes are 10 – 40 points.  Point values for other items such as class assignments, explorations, homework, presentations, and projects will be announced.   Your grade will be determined by dividing your total points by the total possible earned points.  The final exam is a final project.

 Homework will usually be assigned on a daily basis.  It will be reviewed and discussed during class.    Homework assignments will be posted on the board, posted in Schoology, and sent via “Remind” which then posts on my PHS website.  If you are absent from class, it is your responsibility to make up the work. Check with a classmate for in-class assignments and class notes.  If you are absent for one day, you have one day to make up the work, and so on.

 I am usually available for extra help any Monday, Wednesday, or Thursday after school or before first block begins.  Please let me know if you need extra help.

 I welcome student and parental contact.  By working together we will have a successful year.

 

APCalculus BC  - Syllabus

Course description:

 

This course is comprised of college material, and is conducted in a rigorous manner.It is designed for students who have outstanding skills and interests in the field of mathematics and who want to gain college credit by taking the College Board Advanced Placementexamination in May.

 

Both practical and theoretical approaches are presented at an accelerated pace, thereby requiring a serious commitment on the student’s part.Explorations are used extensively to engage and motivate the student.

 

Topics presented in this course include functions, limits, derivatives, integrals, and numerical approximations.A detailed scope and sequence is shown below.Every area that is referenced in theAPCalculus BC Course Descriptionis studied.In addition, some other topics such asNewton’s Method, Simpson’s rule, integration by trigonometric substitution, volume by cylindrical shells, and work are studied.Some of these topics are studied after the APExam.

 

Students continually work with functions that are represented analytically, graphically, numerically, or described verbally.They must routinely demonstrate the connections between these representations.Emphasis is placed on understanding and using the mathematical modeling process to set up and solve a variety of problems.

 

 


Course Schedule: Scope and Sequence

 

Approximate time frame

Topic

First semester:

 

First marking period

4-6 days

Functions

  • Multiple representations of functions
  • Absolute value and interval notation
  • Domain and range.
  • Categories of functions, including linear, polynomial, rational, power, exponential, logarithmic, and trigonometric
  • Even and odd functions
  • Evaluating functions including composite functions
  • Inverse functions
  • Parametric relations

First marking period

6-9 days

Limits

·Informal concept of limit

·Formal definition of limit

·Language of limits, including notation and one-sided limits

·Properties of Limits

·Finite limits, infinite limits, and asymptotes

·Analyzing limits numerically and graphically; using the Sandwich Theorem and end behavior models

·Informal concept of continuity

·Formal definition of continuity

·Types of discontinuities

·Properties of continuous functions

·The Intermediate Value and Extreme Value Theorems

·Average and instantaneous rates of change

·Tangent and normal to a curve

First marking period

18-21 days

Derivatives

  • Evaluating derivatives using the difference quotient definition, the derivative at a point definition, and the symmetric difference quotient definition
  • The notation of derivatives
  • The relationships between the graph off(x) and f ’(x)
  • Graphing the derivative from data
  • One-sided derivatives
  • Determining where a function is not differentiable
  • Linear functions and local linearity
  • Differentiability and continuity
  • The Intermediate Value Theorem for Derivatives
  • Rules and formulas for computing derivatives including the power rule and chain rule
  • Derivative as a rate of change
  • Higher order derivatives
  • Implicit differentiation

First and second marking periods

14-16 days

Applications of Derivatives

  • Finding extrema
  • The Extreme Value Theorem
  • Increasing and decreasing behavior
  • The Mean Value Theorem
  • Critical values and local extrema
  • The first and second derivative tests
  • Concavity and points of inflection
  • Modeling and Optimization
  • Linearization and Newton’s method
  • Estimating change with differentials
  • Related Rates Problems

Second marking period

13-16 days

The Definite Integral

  • Area under a curve and distance traveled
  • Rectangular Approximation methods
  • Summation notation and partitions
  • Riemann sums
  • Definition of the definite as a limit of Riemann sums
  • Terminology and notation of integration
  • Definite integrals and area
  • Properties of definite integrals
  • Average value of a function
  • Mean Value Theorem for definite integrals
  • Connecting differential and integral calculus
  • The Fundamental Theorem of Calculus – Parts I and II
  • Definition of antiderivative
  • Numerical approximation techniques for integration including trapezoidal and Simpson’s Rule

Second semester:

 

Second and third marking periods

20-24 days

Differential Equations and Mathematical Modeling

  • Initial value problems
  • Antiderivatives and slope fields
  • Antiderivatives and indefinite integrals
  • Techniques of antidifferentiation: including substitution, integration by parts, and tabular integration
  • Solving separable differential equations
  • Exponential growth and decay
  • Logistic growth and regression
  • Solving initial value problems by Euler’s method
  • Solving initial value problems visually using slope fields

 

Third marking period

15-18 days

Applications of Definite Integrals

  • Integral as net change
  • Applications of particle motion – net and total distance traveled
  • Finding the area under a curve and between curves
  • Integrating with respect to y
  • Finding the volume of solids, using area of cross sections and solids of revolution; Cavalieri’s principle
  • Surface area for solids of revolution
  • Arc length including parametric functions
  • Applications from science and statistics: i.e. work
  • Evaluating indeterminate forms using L’Hopital’s Rules
  • Growth rates of functions
  • Improper integrals
  • Convergence tests including direct and limit comparison tests
  • Antiderivatives by partial fractions and deduction formulas

Third and fourth making periods

16-20 days

 

 

 

 

 

 

 

 

Infinite Series

  • Concepts and notation for sequences and series
  • Limit of a sequence of partial sums
  • Geometric and power series
  • Convergence and divergence
  • Differentiating and integrating term by term
  • Maclaurin and Taylor series and polynomials
  • Taylor’s theorem with remainder
  • Center, radius, and interval of convergence
  • Tests for convergence or divergence: nth term test, direct comparison test, ratio test, integral test, limit comparison test, alternating series test, and nth-root test
  • Absolute and conditional convergence
  • Harmonic series and p-series

 

Fourth making period

8-10 days

 

 

 

Parametric, Vector, and Polar Functions

  • Second derivative for a parametrized curve
  • Surface area for solids of revolution formed by parametric curves
  • Vectors and vector-valued functions
  • Differentiating and integrating vector functions
  • Polar coordinates and polar graphs
  • Calculus of polar curves, including slope, area, and length of a curve

 

Fourth making period

15-20 days

 

 

Review for APExam

 

 

 

 

 

 

Teaching Strategies:

 

Calculus is where seemingly separate concepts that students have learned in previous years come together.It is where that often asked question of“Why do I need to know this?’ is answered.

Explorations are used extensively to engage and motivate the student.Students frequently work in small, assigned groups to discuss explorations, solve problems, and review homework prior to discussing them as a whole class. The groups are changed a few times each marking period.

 

Students use TI-89 calculators on a daily basis. Each student has a TI-89 calculator. Instruction is given on the appropriate use of the calculator throughout the course.Programs are used to demonstrate RAM methods, Euler and improved Euler’s method, and Simpson’s Rule, to name a few.

 

 

Assessment:

 

Students are frequently assessed.Assessment is done in a variety of ways: quizzes, tests, assigned problems, a summer project, mid-term exam, final exam, and a final project.Assessments are made individually and in groups. Emphasis is placed on the justification of solutions.Students are required to explain and validate their solutions.Most tests are two periods in length and they are divided into two sections.In one section, they are encouraged to use their calculators.In the other section, no calculators are permitted.

 

 

APExam Preparation:

 

Students prepare for the exam throughout the course.Multiple choice questions and open-ended questions from released exams are included in quizzes and tests as appropriate.Some problems are assigned for homework.Calculator use is as prescribed by the College Board.Solutions and the scoring of the open-ended questions are thoroughly reviewed.

 

The three weeks before the AP exam are devoted to additional intense review.Students practice on the most current complete released exam and on other released exams.Students work individually and in groups.Addition resources are used.

 

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References and Materials:

 

Primary Textbook

 

Finney, Ross L., Franklin Demana, Bert Waits, and Daniel Kennedy.CalculusGraphical, Numerical, Algebraic. 4th ed. Pearson/Prentice Hall, 2012.

 

 

 

Additional Resources

 

Anton, Howard.Calculus. 3rd ed./brief ed. New York: John Wiley & Sons, 1988.

Barton, R., Diehl, J.,Advanced Placement Calculus with the TI-89, 1stEdition, Texas Instruments, Inc.,

1999.

Foerster, Paul A.,Calculus: Concepts and Applications, 1sted., Key Curriculum Press 1998.

Larson, Ron, Hostetler, Robert P., and Edwards, Ralph,Calculus of a Single Variable, 7thed., Houghton

Mifflin Company, 2002.

 

 

Last Modified on September 7, 2012