Course Overview

The AP Calculus AB course develops a foundational understanding of differential and integral calculus, emphasizing analytical, graphical, numerical, and verbal representations. The curriculum is structured around the College Board's eight-unit framework, focusing on change, limits, function analysis, modeling, justification, and precise mathematical communication.

Skills and Practices

• Implementing Mathematical Processes: Determine expressions and values using appropriate calculus procedures.

• Connecting Representations: Translate and connect analytical, graphical, numerical, and verbal information.

• Justification: Support conclusions with definitions, theorems, properties, and mathematical reasoning.

• Communication and Notation: Use correct notation, terminology, units, and conventions.

Unit 1: Limits and Continuity

Key Topics

• Definition and notation of limits

• Estimating limits from graphs and tables

• Algebraic properties and manipulation of limits

• Squeeze Theorem

• Types of discontinuities and continuity at a point

• Infinite limits, vertical and horizontal asymptotes

• Intermediate Value Theorem

Example:

To evaluate , factor the numerator and simplify:

Unit 2: Differentiation – Definition and Fundamental Properties

Key Topics

• Average and instantaneous rates of change

• Definition of the derivative and notation

• Estimating derivatives from data

• Relationship between differentiability and continuity

• Power Rule, Constant Rule, Sum/Difference Rule, Constant Multiple Rule

• Derivatives of trigonometric, exponential, and logarithmic functions

• Product and Quotient Rules

Key Formula:

The derivative of at is defined as:

Unit 3: Differentiation – Composite, Implicit, and Inverse Functions

Key Topics

• Chain Rule for composite functions

• Implicit differentiation

• Derivatives of inverse and inverse trigonometric functions

• Higher-order derivatives

Example:

If , then by implicit differentiation:

Unit 4: Contextual Applications of Differentiation

Key Topics

• Interpreting derivatives in context (e.g., velocity, acceleration)

• Related rates problems

• Linearization and local linearity

Example:

If a balloon's radius increases at 2 cm/s, find the rate at which volume increases when cm:

At , cm3/s

Unit 5: Analytical Applications of Differentiation

Key Topics

• Mean Value Theorem

• Extreme Value Theorem, critical points, and extrema

• First and Second Derivative Tests

• Concavity and inflection points

• Optimization problems

Key Formula:

Mean Value Theorem: If is continuous on and differentiable on , then there exists in such that:

Unit 6: Integration and Accumulation of Change

Key Topics

• Riemann sums and area approximation

• Definite and indefinite integrals

• Fundamental Theorem of Calculus

• Antiderivatives and integration techniques (substitution, parts, partial fractions)

• Improper integrals

Key Formula:

Fundamental Theorem of Calculus:

, where

Unit 7: Differential Equations

Key Topics

• Modeling with differential equations

• Verifying solutions

• Slope fields and Euler's Method

• Separation of variables

• Exponential models

Example:

For , the general solution is

Unit 8: Applications of Integration

Key Topics

• Average value of a function

• Position, velocity, and acceleration via integrals

• Area between curves

• Volumes by cross-sections, disc, and washer methods

Key Formula:

Area between curves and from to :

Study & Assessment Plan

• Preview each unit and build a concept summary before starting.

• Engage in mixed practice during units, focusing on explanation and application.

• Take timed unit checks and review errors at the end of each unit.

• Begin cumulative review in January and AP-style practice in March.

• Use full-length practice exams and error logs in April leading up to the exam.

Primary Alignment Sources

• College Board AP Calculus AB Course and Exam Description

• Official College Board clarifications and exam resources

Additional info: This syllabus is designed for personalized study planning and aligns with the official AP Calculus AB framework but does not replace the official school syllabus or AP Classroom resources.