Description and Objectives

The course AP Calculus (BC) is a one year course which serves to teach the student calculus at the university level while at the same time preparing him for the AP exam given by the College Board in May.  When successfully completed, this course should replace a one-year course (i.e. two semesters) in calculus given at an engineering school (e.g. Math 140 -141 at the University of Maryland).  Here we go into more depth topics that were covered in AP Calculus (AB) and introduce new concepts to complete what is taught in one year at the university level.  Once the material for the BC exam is covered, additional topics in Linear Algebra, Differential Equations, Complex Analysis and Dynamical Systems are covered in small blocks.

The course can be broken down into four (4) basic segments of unequal lengths:  Limit theory, Differential Calculus and Integral Calculus, Series.  The time allocation for these blocks is as follows: Approximately three (3) weeks are spent on limit theory, five (5) weeks on Differential Calculus and eight (8) weeks on Integration and six (6) weeks on series with four (4) weeks remaining for special topics.  Interspersed throughout will be review of and further development of calculus concepts already introduced in Calculus AB as well as student activities using Matlab and Mathematica on machines in the Computer Lab.

Homework is given daily and comprises 20% of the overall grade.  Quizes are given without warning but usually weekly and make-up 20% of the grade.  There are typically six (6) full-period exams per semester (three per quarter) and a two (2)-hour midyear and Final exam.

As we approach the May AP exam date, our classes will increasingly concentrate on review of the material for the exam.  We will have mini-mockAP exams daily in the final weeks before the real thing – seven multiple choice questions in 15 minutes followed by an immediate discussion of the solutions.  Additionally there will be after-school reviews.

Students are required to obtain (whether by purchase or borrow) a graphing calculator.  We will use the TI-84 plus in class for overhead presentations, but many others are suitable.  Please consult with the instructor prior to purchasing a different machine.

Textbooks

Our textbook is Calculus: One Variable 10^thed., Salas, Hille, Etgen.  Additionally we will make use of the Princeton Review Cracking the AP Calculuspreparation book.

Course Requirements

Limit Theory

Objective:  To more formally introduce the student to the concept of a limit of a function.  Here the interplay between domain and range are explored so that the student has not only a conceptual understanding of the symbols, but understands how to construct a proof.

Text:  Chapters 1 & 2

Differential Calculus

Objective:  To help the student understand more fully the concept of a derivative, we re-introduce the derivative:  the instantaneous velocity of a position function.  We discuss right and left derivatives and review all topics from Calculus AB.  Additionally we will make use of Paramteric forms of expressing functions and work in Polar and Rectangular Coordinate systems.

Text:  Chapters 3, 4 , 7,  9

Integral Calculus

Objective:  Integration is re-introduced with more emphasis on the development of the Integral from Riemann Sums.  Next we explore integration techniques: u-substitution, integration by parts, partial fraction decomposition, trigonometric substitution, numerical methods.

Lastly we look at differential equations.  The student reviews separation of variables and writing equations in differential form.  Additionally we will program the TI-84 to solve differential equations numerically using Euler’s method.  We use Matlab on an overhead projector to demonstrate the slope fields and how to generate them to aid in the solution of differential equations.

Text:  Chapters 5, 6, 8 & 9

Polynomial Approximation and Series

Objective:  A series is defined as a sequence of partial sums and convergence is defined as a limit of partial sums.  Students will use technology to explore convergence and divergence.  We will discuss tests for convergence of geometric series, harmonic series.  Additionally we will discuss Taylor series expansions of functions as well as Maclaurin series.

Text:  Chapters 10 and 11

Linear Algebra, Differential Equations, Complex Analysis and Dynamics

Objective:  Students are exposed to matrix solutions to linear systems.  Also we will look at singular and orthogonal matrices and how they apply to vector transformations.  Some attention will be given to different methods of solving inexact ODE’s using integrating factors.  We look at the Cauchy-Riemann equations for complex variables.  Lastly, if time permits we study orbits of quadratic functions and observe some basic Dynamical Systems.

Text:  Handouts from the instructor

Successful Students

Students may ask questions any time.  Office hours are plentiful and include before school every day, lunch hour every day and after school by appointment.  Students are encouraged to make use of the instructor’s email address: mhernandez@heights.edu  or his extension on campus 301.365.0227×216

Summer Assignment

Do practice tests and practice Calculus AB problem sets from your Princeton review prep book all summer long. Stay loose and nimble with the material: there will be an exam on the first day of classes in September (a mini mock AP Calculus AB exam) to see how much you retained over the Summer. It will count as your first exam grade of the year.