Syllabus  

Lesson 1 – AP Calculus AB, BC

Limits and Continuity

•How limits help us 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

Lesson 2 – AP Calculus AB, BC

Differentiation – Definition and Fundamental Properties

•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

Lesson 3 – AP Calculus AB, BC

Differentiation – Composite, Implicit, Inverse Functions

•The chain rule for differentiating composite functions

•Implicit differentiation

•Differentiation of general and particular inverse functions

•Determining higher-order derivatives of functions

 

Lesson 4 – AP Calculus AB, BC

Contextual Applications of Differentiation

•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

 

Lesson 5 – AP Calculus AB, BC

Review and practice of contents in Lesson 1- Lesson 4

Exam skills will also be discussed

 

Lesson 6 – AP Calculus AB, BC

Analytical Applications of Differentiation

•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

 

Lesson 7 – AP Calculus AB, BC

Integration and Accumulation of Change

•Using definite integrals to determine accumulated change over an interval

•Approximating integrals using Riemann Sums

•Accumulation functions, the Fundamental Theorem of Calculus, and definite integrals

•Antiderivatives and indefinite integrals

•Properties of integrals and integration techniques

Lesson 8 – AP Calculus AB, BC

Differential Equation

•Interpreting verbal descriptions of change as separable differential equations

•Sketching slope fields and families of solution curves

•Solving separable differential equations to find general and particular solutions

•Deriving and applying a model for exponential growth and decay

 

Lesson 9 – AP Calculus AB, BC

Application of Integration

•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

 

Lesson 10 – AP Calculus AB, BC

Comprehensive Review of all contents in AP calculus AB

Exam skills will also be discussed

 

Lesson 11 – AP Calculus BC

Parametric Equations, Polar Coordinates, Vector-Valued Functions

•Finding derivatives of parametric functions and vector-valued functions

•Calculating the accumulation of change in length over an in serval 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

 

Lesson 12 – AP Calculus BC

Infinite Sequences and Series

•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 Tylor series or a Maclaurin series on an appropriate interval