# AP Calculus AB

Course Description

The AP Calculus AB course provides an introduction to differential and integral calculus. Topics include limits, derivatives, related rates, the Mean-Value Theorem, Max-Min problems, the integral, the Fundamental Theorem of Integral Calculus, areas, volumes, and average values.

Students enrolled in AP Calculus AB will study the same topics as regular Calculus but will have the additional challenge of the AP exam. Because of the timing of the AP exam, AP Calculus AB students will proceed at an accelerated pace. Throughout the course, students will practice AP type problems and questions.

*This course is offered for dual credit through UMSL.

Curricula for Advanced Placement (AP) courses are created by the American College Board, which offers high level coursework and exams to high school students.  Colleges and universities may grant placement and course credit to students who obtain high scores on examinations.  Curriculum for each subject area is created by a panel of experts and college-level educators in that field of study.  An overview of the AP Calculus AB course can be found HERE.  The Course & Exam Description (CED) can be found HERE.

Essential Questions

• How can I evaluate limits, the definition of continuity, and the Intermediate Value Theorem?
• How can I find derivatives, equations of tangent lines and test for differentiability?
• How can I find derivatives of specific types of functions?
• How can I apply derivative techniques to analyze graphs of functions and select applications?
• How can I apply derivative techniques to select applications to deepen my understanding of how derivatives can be used further in other problem-solving processes?
• How can I find antiderivatives?
• How can I use definite integration as an application to determine area?
• How can I use definite integration as an application to determine area between curves, volume, average values of functions and accumulating amounts?

Enduring Understandings

• Students will determine expressions and values using mathematical operations, procedures and rules.
• Students will translate mathematical information from a single representation or across multiple representations in order to develop processes to problem solve.
• Students will recognize mathematical reasoning requires justification of both process and solution.
• Students will use correct notation, language and mathematical convention to classify concepts and communicate results or solutions.

Course-Level Scope & Sequence (Units &/or Skills)

Unit 1: Limits

• Evaluating limits using graphs or tables
• Limits at Infinity
• Formal Definition of a Limit
• Evaluating Limits Algebraically
• Justifying Limits that do not exist
• Limits of Trigonometric Functions
• Continuity
• Intermediate Value Theorem
• Tangent/Velocity

Unit 2: Derivatives

• Definition of a Derivative
• Derivative Rules (Power, Product, Quotient)
• Derivatives of Trig Functions
• Chain Rule
• Differentiability
• Higher Order Derivatives

Unit 3: Inverse Functions

• Derivatives of Inverse Functions
• Exponential Functions & Their Derivatives
• Derivatives of Logarithmic Functions
• Derivatives of Trigonometric Inverse Functions

Unit 4: Applications of Derivatives (part 1)

• Maximum & Minimum Function Values
• Mean Value Theorem
• First Derivative Test (finding Relative Extrema)
• Using the First Derivative to determine where a function increases or decreases
• Second Derivative Test (finding Relative Extrema)
• Using the Second Derivative to determine concavity of a function and inflection points
• Graphs of Functions & Their Derivatives

Unit 5: Applications of Derivatives (part 2)

• Rectilinear Motion
• Implicit Differentiation
• Related Rates
• Linear Approximations
• L’Hopital’s Rule

Unit 6: Antiderivatives

• Antiderivatives
• Antiderivatives by Substitution
• Differential Equations
• Slope Fields

Unit 7: The Definite Integral

• Sigma Notation
• Approximating Area (Riemann Sums including Trapezoidal Rule)
• Exact Area using Limit of Riemann Sums
• The Definite Integral
• The Fundamental Theorems of Calculus

Unit 8: Area & Volume

• Area Between Curves
• Volumes by Slicing
• Volumes using Disk/Washer
• Average Value of a Function
• Definite Integral as an Accumulator

Course Resources & Materials: AP Classroom, Calc (Math) Medic, Desmos

Date Last Revised/Approved: 2011