Advanced Placement Calculus BC Syllabus 
                                                          
              Introduction 
                
                    In this course, we cover the topics that are listed in The Calculus BC Course Description 
              posted on the AP Central website. We divide our studies into three major topics: limits, 
              differential calculus, and integral calculus.  An understanding of limits is the basis for future 
              concepts in calculus. Differential calculus enables us to calculate rates of change, to find the 
              slope of a curve, and to calculate velocities and accelerations of moving bodies.  Integral 
              calculus is used to find the area of an irregular region in a plane and volumes of irregular solids. 
               
                    In this course students acquire knowledge of the basic mechanics of limits, derivatives, 
              and antiderivatives.  Additionally students develop an understanding of the theory of calculus. 
              Each student then uses these concepts in applying calculus to multiple applications.  
               
                    Problems are presented and solved in four distinct ways:  analytically, numerically, 
              graphically, and verbally.  Students will use a graphing calculator to examine continuity and 
              local linearity, to create tables of values for functions and relations, to solve an equation, to 
              graph a function in any window, to determine the value of a derivative at a point, and to find the 
              value of a definite integral. Each student is required to have his or her own graphing calculator. 
              A TI-84 or TI-89 is required. An overhead graphing calculator (TI-84 and TI-89) is available for 
              use. 
               
                    The first seven months of the class are devoted to studying the topics covered in a typical 
              college Calculus I and II course.  The next month is review and preparation for the AP exam. 
              Throughout the year, information concerning the administration, scoring, and content of the 
              exam are discussed and examples are provided for student analysis.  After taking the exam in 
              May, students will complete a project and study additional topics in calculus not included in 
              Calculus BC. 
               
                    The school operates on a 4-day rotation of classes. Each class is held for 45-minutes on 
              two of the days, 90-minutes on one of the days, and not at all on the last day of the cycle. This 
              amounts to an average of 45 minutes per day, but gives the flexibility to complete learning 
              activities that require a longer amount of time occasionally.  
               
              Topics Covered: 
               
              Review of Pre-Calculus Math                                       5 days 
                    Students are given a packet of exercises to complete during the summer. This packet is 
              reviewed in class and assessed as a test grade. The topics include: graphing equations, intercepts, 
              intersection points, symmetry, mathematical models, equations of lines, parallel and 
              perpendicular, domain and range, trigonometric functions, line of best fit, transformations of 
              functions, and parametrically defined functions. 
               
              Limits and Their Properties                                       10 days       
                 •  General properties of limits 
                     •   Finding limits algebraically, analytically and graphically 
                     •   Formal definition of limits (ε - δ) including proofs of limits of linear functions 
                     •   Horizontal and vertical asymptotes 
                     •   Continuity 
                     •   Removable, jump, and infinite discontinuities 
                     •   Infinite limits 
                     •   Theorems involving limits: Squeeze Theorem, Extreme Value Theorem, and the 
                         Intermediate Value Theorem. 
                  
                 Differentiation                                                                   15 days 
                     •   Definition of the derivative 
                     •   Relating the graph of a function with the derivative 
                     •   Non-differentiability 
                     •   Basic Differentiation Rules 
                     •   Rates of change 
                     •   The Product Rule ad Quotient Rule 
                     •   Derivatives of trigonometric functions 
                     •   Higher order derivatives 
                     •   The Chain Rule 
                     •   Implicit Differentiation 
                     •   Related Rates 
                     •   Rectilinear motion 
                  
                 Applications of Differentiation                                                   20 days 
                     •   Extrema on a closed interval and critical points 
                     •   Rolle’s Theorem 
                     •   The Mean Value Theorem 
                     •   Increasing and decreasing functions and the First Derivative Test 
                     •   Concavity and the Second Derivative Test 
                     •   Limits at infinity (horizontal asymptotes, slant asymptotes and other end behavior) 
                     •   Summary of curve sketching 
                     •   Optimization problems 
                     •   Newton’s Method 
                     •   Differentials 
                  
                 Integration                                                                       15 days 
                     •   Definition of antiderivative 
                     •   Basic integration rules 
                     •   Initial conditions and particular solutions 
                     •   Area approximation using rectangles 
                     •   Riemann Sums and definite integrals 
                     •   The Fundamental Theorem of Calculus 
                     •   Integration by Substitution (both definite and indefinite integrals) 
                     •   Numerical Integration 
                     •   Integration by Partial Fractions 
                     •   Integration by Parts 
                     •    
                  
                 Logarithmic, Exponential, and Other Transcendental Functions                      10 days 
                     •   The Natural Logarithmic Function: Differentiation 
                     •   The Natural Logarithmic Function: Integration 
                     •   Inverse Functions 
                     •   Exponential Functions: Differentiation and Integration 
                     •   Bases other than e 
                     •   Applications of logarithmic and exponential functions 
                     •   Inverse Trig functions: Differentiation 
                     •   Inverse Trig Functions: Integration 
                     •   Hyperbolic Functions 
                  
                 Differential Equations                                                            8 days 
                     •   Slope Fields 
                     •   Euler’s Method 
                     •   Growth and Decay Models 
                     •   Separation of Variables 
                     •   Logistic Growth 
                     •   First-Order Linear Differential Equations 
                  
                 Applications of Integration                                                       15 days 
                     •   Area of region between two curves 
                     •   Volume: The Disk and Washer Methods 
                     •   Volume: The Shell Method 
                     •   Volumes of solids with known cross sections 
                     •   Arc Length and Surfaces of Revolution 
                     •   Work 
                     •   Moments, Centers of Mass, and Centroids 
                     •   Fluid Pressure and Fluid Force  
                  
                 Integration Techniques, L’Hôpital’s Rule, and Improper Integrals                  15 days 
                     •   Review of basic integration rules 
                     •   Integration by parts 
                     •   Trigonometric Integrals 
                     •   Integration using trigonometric substitution 
                     •   Integration using partial fractions 
                     •   Integration by tables and other integration techniques 
                     •   Applying L’Hôpital’s rule to indeterminate forms of limits 
                     •   Improper integrals 
                  
                 Infinite Series                                                                   20 days 
                     •   Sequences  
                     •   Series and Convergence 
                     •   The Integral test and p-series 
                     •   Comparisons of series 
                     •   Alternating series 
                     •   The Ratio and Root tests 
                     •   Taylor Polynomials and approximations 
                     •   Power series 
                     •   Representation of functions by Power Series 
                     •   Taylor and Maclaurin Series 
                  
                 Parametric Equations and Polar Coordinates                                        8 days 
                     •   Defining a function parametrically and eliminating the parameter 
                     •   Find the first and second derivative (with respect to x) of a parametrically defined 
                         function 
                     •   Finding arc length and the area of a surface of revolution using paramterics 
                     •   Polar equations and graphs 
                     •   Tangent lines to polar equations, area of polar regions and arc length of polar graphs 
                  
                 Vectors                                                                           10 days 
                     •   Component form of vectors 
                     •   Basic vector operations 
                     •   Space coordinates and vectors in space 
                     •   Vector valued functions 
                     •   Differentiation and integration of vector valued functions 
                     •   Velocity and acceleration vectors 
                     •   Tangent and normal vectors 
                     •   Arc length of plane and space curves 
                  
                          
                 Review for AP Exam: 
                 During the course of the school year, students work individually, in pairs or in small groups to 
                 complete many released free-response questions as they pertain to the topics that have already 
                 been taught. They are required to show a “complete” solution to each of the problems and 
                 evaluate their responses in comparison to the solutions provided and example student solutions, 
                 where applicable. On occasion, students are required to present problems to the rest of the class 
                 and use precise mathematical language in their presentations. It is important that students 
                 understand all of the components that must be included when completing the free-response 
                 questions. As the AP Exam date draws closer, the extended time block periods in our schedule 
                 are used to take simulated parts of the AP Exam and the shorter periods are used to review 
                 important concepts and common mistakes that are made.