AP Calculus BC Syllabus Course Overview Textbook Anton, Bivens, and Davis. Calculus: Early Transcendentals, Combined version with Wiley PLUS. 9 th edition. Hoboken, NJ: John Wiley & Sons, Inc. 2009. Course Plan Below is the scope and sequence of our AP Calculus AB course. Unit 0: Functions (12 Days) A. Functions and the analysis of graphical information B. Properties of functions C. Graphing functions on calculators and computers; computer algebra systems D. Transformations of basic functions 1. Shifts 2. Compressing and stretching 3. Reflections 4. Operations with functions 5. Symmetry of functions 6. Odd and even functions E. Lines 1. Slope and equations 2. Uniform rectilinear motion F. Families of functions 1. Polynomial and rational functions 2. Trigonometric functions 3. Absolute Value 4. Parametric equations and graphs G. Inverse functions 1. Inverse Trigonometric functions 2. Exponential functions 3. Logarithmic functions Unit 1: Limits and Continuity (9 Days) A. Limits an intuitive approach 1. Graphic approach 2. Numerical approach 3. Analytical approach B. Computing limits 1. Algebraic approach 2. lim f ( x), lim f ( x) xa xa 3. Vertical asymptotes 4. Piece-wise functions C. Computing limits; end behavior 1. lim f ( x), lim f ( x) x x, lim f ( x) xa 2. Horizontal asymptotes 3. Radical functions 4. Divergent, convergent and oscillating 5. Diverging at varying rates
D. Continuity 1. Definition in terms of limits 2. Intermediate Value Theorem 3. Zeros of a function on the graphing calculator E. Limits and continuity of trigonometric, exponential, and inverse functions Unit 2: The Derivative (8 Days) A. Slopes and rates of change 1. Numerical, graphic, and algebraic approach 2. Secant versus tangent lines 3. Average versus instantaneous rates of change B. The Derivative 1. Definition as a limit of the difference quotient 2. Instantaneous rate of change as a limit of the difference quotient 3. Equation of a tangent line to a curve at a specific point 4. Various forms of the derivative formula 5. x Notation 6. Continuity, limits and differentiability C. Techniques of differentiation 1. Algebraic operations 2. Product and quotient rule 3. Power rule 4. Derivatives with rational exponents 5. Horizontal tangent lines 6. Higher order derivatives D. Derivatives of trigonometric functions E. The Chain rule Unit 3: Topics in Differentiation (10 Days) A. Implicit differentiation 1. Solve for dy dx 2. Slope of tangent lines given an initial value 3. Second order derivatives 4. Chain rule B. Inverse Functions 1. Continuity and differentiability of inverse functions 2. Implicit differentiation 3. Inverse trigonometric functions C. Derivatives of logarithmic and exponential functions 1. Slopes of tangent lines 2. Chain rule 3. Irrational powers D. Related Rates 1. Differential equations 2. Verbal descriptions of the differential expression E. Local linear approximation 1. Differentials and notation F. L Hopital s rule: indeterminate forms
Unit 4: The Derivative in Graphing and Applications (15 Days) A. Analysis of Functions 1. Increasing, decreasing and concavity a. Critical numbers b. Inflection points 2. Relative extrema a. First derivative test b. Second derivative test c. Stationary points d. Critical values 3. Constructing complex graphs from function characteristics a. Applying tools of calculus b. Vertical tangent lines and cusps B. Absolute maxima and minima 1. Critical points and endpoints 2. Extreme Value Theorem 3. Finite closed intervals 4. Infinite intervals 5. Open intervals C. Optimization problems Optimization Project Minimum Firebreaks in a Forest D. Newton s method 1. Graphic and tabular approach 2. Calculators and recursive formulas E. Rolle s Theorem F. Mean Value Theorem 1. Graphic and algebraic interpretation 2. Application to rectilinear motion G. Rectilinear Motion 1. Position, speed, velocity and acceleration 2. Speeding up and slowing down 3. Total distance traveled 4. Free fall model Rectilinear Motion Project (see student activities below) Unit 5: Integration (14 Days) A. Area under a curve 1. Rectangular method 2. General Sigma notation 3. Simple geometric formulas B. Numerical integration 1. Midpoint approximation 2. Trapezoidal approximation 3. Simpson s rule C. Area as a limit of a summation 1. Sigma notation and properties 2. Rectangular method a. left endpoints b. right endpoints c. midpoints 3. Graphic and tabular approach 4. Negative Areas 5. Definite Integral as a limit of a summation - analytical approach Area Under a Curve and Riemann Sums Project (See Student Activities below)
D. Antiderivatives 1. Antiderivative curves 2. Initial value problems 3. Constant of integration E. Antiderivatives by substitution F. Definite Integral 1. Graphic approach simple geometric formulas 2. Properties of the Definite Integral 3. Intermediate Value Theorem 4. Continuity and integration G. The Fundamental Theorem of Calculus b 1. Part 1 f ( xdx ) Fb ( ) Fa ( ) a 2. Graphic and algebraic connection 3. Theorem applied to accumulation of change problems x d 4. Part 2 f () tdt f( x) dx a 1. Chain rule Multi-layering approach to Fundamental Theorem of Calculus (See Teaching Style below) H. Rectilinear Motion 1. Relationship between position, velocity, speed and acceleration function 2. Change in position over given interval of time interpreted as the definite integral of the velocity versus time function over that same interval. 3. Displacement versus total distance traveled 4. Free fall model 5. Average value problems I. Evaluating definite integrals by substitution 1. Change in limits J. Logarithmic functions from an integral point of view Unit 6: Application of Integrals (13 Days) A. Area between two curves 1. Respect to x-axis 2. Respect to y-axis B. Volumes by slicing 1. Solids of revolution; disk method 2. Rotating the area between two curves; washer method 3. Rotation about a vertical or horizontal line; other than x- or y-axis 4. Complex volumes; defined base region with geometric cross-sections. 5. All solids with respect to x-axis and y-axis. C. Volumes by cylindrical shells 1. Respect to x-axis 2. Respect to y-axis D. Length of a plane curve E. Area of a surface of revolution WINPLOTS Computer Lab F. Work 1. Spring 2. Pump G. Fluid force on a horizontally and vertically submerged object
Unit 7: Principles of Integration (10 Days) A. Integration by parts 1. Indefinite integrals 2. Definite integrals B. Trigonometric integrals 1. Substitution C. Partial fractions D. Improper integrals 1. Integrals over an open infinite interval 2. Integrals over an open finite interval 3. Vertical asymptotes Unit 8: Mathematical Modeling with Differential Equations (7 Days) A. Separable differential equations 1. Initial condition problems B. Slope Fields 1. 2 variable functions 2. Differential equations a. Curves of differential equations b. Initial condition problems 3. Euler s Method Calculator Based Lab Differential Equations and Slope Fields on the TI-86 C. Modeling with first order differential equations 1. Exponential growth and decay models 2. Initial value problems 3. Double-life and half-life models 4. Newton s law of cooling 5. Learning curve Unit 9: Infinite Series (17 Days) A. Sequences 1. Monotone B. Convergence of series 1. Sequence of partial sums 2. Geometric 3. Absolute 4. Conditional C. Harmonic Series D. Tests 1. Term divergence 2. Integral 3. P-Series 4. Comparison 5. Limit comparison 6. Ratio a. Absolute convergence 7. Root 8. Alternating series E. Power Series F. Taylor and MaClaurin Series 1. Radius and interval of convergence G. Computing error 1. Lagrange error bound 2. Alternating series method
Unit 10: Analytic Geometry in Calculus (8 Days) A. Polar 1. Coordinates 2. Graphs 3. Tangent lines 4. Arc Length 5. Area B. Parametric 1. Tangent line 2. Arc length 3. Surface Area Unit 11: Three Dimensional Space; Vectors (10 Days) A. Rectangular coordinates in 3 space 1. Spheres B. Vectors 1. Dot product 2. Projections 3. Cross products 4. Equations of lines C. Parametric equations of lines