AP Calculus BC

Topics may include:

• How limits help us to handle change at an instant
• Definition and properties of limits in various representations
• Definitions of continuity of a function at a point and over a domain
• Asymptotes and limits at infinity
• Reasoning using the Squeeze theorem and the Intermediate Value Theorem

Topics may include:

• Defining the derivative of a function at a point and as a function
• Connecting differentiability and continuity
• Determining derivatives for elementary functions
• Applying differentiation rules

Topics may include:

• The chain rule for differentiating composite functions
• Implicit differentiation
• Differentiation of general and particular inverse functions
• Determining higher-order derivatives of functions

Topics may include:

• Identifying relevant mathematical information in verbal representations of real-world problems involving rates of change
• Applying understandings of differentiation to problems involving motion
• Generalizing understandings of motion problems to other situations involving rates of change
• Solving related rates problems
• Local linearity and approximation
• L’Hospital’s rule

Topics may include:

• Mean Value Theorem and Extreme Value Theorem
• Derivatives and properties of functions
• How to use the first derivative test, second derivative test, and candidates test
• Sketching graphs of functions and their derivatives
• How to solve optimization problems
• Behaviors of Implicit relations

Topics may include:

• Using definite integrals to determine accumulated change over an interval
• Approximating integrals with Riemann Sums
• Accumulation functions, the Fundamental Theorem of Calculus, and definite integrals
• Antiderivatives and indefinite integrals
• Properties of integrals and integration techniques, extended
• Determining improper integrals

Topics may include:

• Interpreting verbal descriptions of change as separable differential equations
• Sketching slope fields and families of solution curves
• Using Euler’s method to approximate values on a particular solution curve
• Solving separable differential equations to find general and particular solutions
• Deriving and applying exponential and logistic models

Topics may include:

• Determining the average value of a function using definite integrals
• Modeling particle motion
• Solving accumulation problems
• Finding the area between curves
• Determining volume with cross-sections, the disc method, and the washer method
• Determining the length of a planar curve using a definite integral

Topics may include:

• Finding derivatives of parametric functions and vector-valued functions
• Calculating the accumulation of change in length over an interval using a definite integral
• Determining the position of a particle moving in a plane
• Calculating velocity, speed, and acceleration of a particle moving along a curve
• Finding derivatives of functions written in polar coordinates
• Finding the area of regions bounded by polar curves

Topics may include:

• Applying limits to understand convergence of infinite series
• Types of series: Geometric, harmonic, and p-series
• A test for divergence and several tests for convergence
• Approximating sums of convergent infinite series and associated error bounds
• Determining the radius and interval of convergence for a series
• Representing a function as a Taylor series or a Maclaurin series on an appropriate interval