
ADVANCED PLACEMENT CALCULUS AB – AP level
Code: M662 Full Year (12) (1 credit)
Prerequisite:Precalculus.
Note: Each student is expected to take the Advanced Placement AB Calculus Examination in May. The fee is determined by the College Board and is the responsibility of the student. In the event that a student does not take the AP Exam, the student’s report card and transcript will reflect only a course in high school Honors Calculus.
This course may be offered as a Distance Learning course between John Jay and Roy C. Ketcham High Schools.
Areas of Study Include:
· Precalculus Review
 Simplify expressions, solve equations, transform functions, and graph functions involving the functions used in calculus.
 Model problems by transforming functions to fit the problem
 Use of a graphing calculator to draw a complete graph in a suitable window
· Develop an Intuitive Understanding of Limits
 Distinguish between determinate and indeterminate limits.
 Evaluate limits graphically, numerically and analytically.
 Recognize the three ways a limit does not exist
 Change variables to evaluate limits
 Define and test for continuity graphically, and analytically
 Find and define horizontal and vertical asymptotes using limits
 Recognize the ways pixels can be deceiving on a graphing calculator
· Derivatives analytically, graphically, and numerically
 Find derivatives using the limit definition of a Fermat quotient
 Evaluate derivatives numerically, and recognize when a numerical derivative may be defined even when the exact derivative does not exist
 Recognize the four ways a derivative does not exist
 Understand the relationship between continuity and differentiability
 Given the graph of a function approximate the rate of change and produce a feasible graph of the derivative of the function
 Determine derivatives using differentiation rules and techniques for polynomial rational, radical, trigonometric, inverse trigonometric, logarithmic, exponential, absolute values, and piecewise define functions and for linear combinations, products, quotients, powers, and compositions of these functions
 Determine derivatives of functions and their inverses using implicit differentiation
· Applications with Derivatives
 Define a function which models the changing process
 Find equations of tangent lines and estimate function evaluations using linearization
 Demonstrate and apply the intermediate value theorem, extreme value theorem, Rolle’s Theorem, and the mean value theorem
 Use the first and second derivative to accurately graph a function identifying extrema, intercepts, inflection points, and regions of increase, decrease, monotonicity, positive concavity, and negative concavity
 Use derivatives to study rates of change at a variety of phenomena including motion.
 Use derivatives to model and solve a variety of optimization problems
 Use derivatives to model a variety of related rates problems
· Integration Analytically, Numerically, and Graphically
 Define a function which models the integrand
 Find integrals using the limit definition of a Riemann Sum
 Evaluate integrals numerically using the left, right, midpoint, and trapezoidal rules, and realize the possible errors
 Given the graph of the function, produce a feasible graph of the antiderivative as the net accumulation of a rate of change
 Evaluate integrals using the Fundamental Theorem of Calculus
 Evaluate definite and indefinite integrals of various functions using integration rules and techniques based on antiderivatives including linearity, change of variable, and by parts
 Use the Variable Limits Theorem to evaluate the derivative of an integral with variable limits
 Approximate definite integrals using the trapezoidal rule with and without a graphing calculator
· Mathematical Modeling
 Antidifferentiation by substitution
 Modeling problems with separable differential equations and analytically solving them
 Investigate differential equations with slope fields
· Applications with Integrals
 Determine area of a region
 Determine volumes of solids of revolution by discs and washers
 Determine volumes of solids of known cross section
 Determine travel distance of a particle
 Demonstrate the mean value theorem for integrals and find the average value of a function
· Optional Post AP Topics
 Calculus of Parametric Functions
 Integration by parts
 L’Hopital’s Rule
 Improper Integrals
 Challenge problems
Assessment: Final project and/or final exam
For the complete AP Curriculum see http://apcentral.collegeboard.com/apc/Controller.jpf
Textbook: Calculus: Graphical, Numerical, Algebraic, 3^{rd} Edition, published by Pearson Education, © 2007