The Ultimate AP Calculus AB Course

Calculus, the mathematical study of continuous change, splits into two main branches: differential calculus, concerning rates of change and slopes of curves; and integral calculus, which deals with accumulation of quantities and areas under or between curves.

AP Calculus AB, one of the Advanced Placement courses for high school students, covers fundamental concepts of calculus. It includes limits, derivatives, integrals, and the Fundamental Theorem of Calculus. Approximately equivalent to a first-semester college calculus course, it focuses mainly on differential and integral calculus with applications, providing a solid foundation in these core areas.

The AP Calculus AB test also emphasizes understanding conceptual frameworks, solving real-world problems, and using graphical, numerical, and analytical representations to solve challenges. It thoroughly prepares students for further studies in mathematics, science, and engineering.

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AP Calculus AB Complete Course

Limit and Continuity

• Limit Introduction
• Neighborhood
• Estimating Limits from Tables
• Functions with Undefined Limits
• One Sided Limits
• Limit at Infinity
• Continuity at a Point
• Continuity over an Interval
• Removing Discontinuity
• Direct Substitution
• Limit Laws
• Limit Laws Combinations
• The Squeeze Theorem
• Indeterminate and Undefined
• Infinity Cases
• Trigonometric Limits
• Rationalizing Trigonometric Functions
• Algebraic Manipulation
• Redefining Function’s Value
• Rationalizing Infinite Limits
• The Intermediate Value Theorem
• Asymptotes

Derivative Basics

• Derivative Introduction
• Average and Instantaneous Rates of Change
• The Derivative of a Function
• Derivative of Trigonometric Functions
• Power Rule
• Product Rule
• Quotient Rule
• Chain Rule
• Derivative of Radicals
• Derivative of Logarithms and Exponential Functions
• Differentiability
• Differentiating Inverse Functions
• Derivative of Reciprocal Trigonometric Functions

Applications of Derivatives

• Optimization Problems
• L’Hôpital
• Implicit Relations
• Implicit Differentiation
• Related Rates
• Higher Order Derivatives
• Extreme Value Theorem
• Second Derivatives: Minimum vs. Maximum
• Curve Sketching Using Derivatives
• Mean Value Theorem
• Newton’s Method

Integrals

• What is Integral?
• Applications of Integrals
• Exponential Growth and Decay
• The Anti-Derivative
• Sigma Notation (Summation Notation)
• Riemann Sums
• Rules of Integration
• Power Rule
• Fundamental Theorem of Calculus
• Trigonometric Integrals
• Substitution Rule
• Integration by Parts
• Integral of Radicals
• Exponential and Logarithmic Integrals
• Improper Integrals

Applications of Integrals

• Average Value of a Function
• Arc Length
• Accumulation Problems
• Finding Area Between Curves
• Introduction to Solids of Revolution
• Volume with Cross-section Method
• Disc Method
• Washer Method
• Shell Method
• Surface Area
• Modeling Particle Motion
• Partial Fraction Expansion

Differential Equations

• Introduction and Applications
• Classification of Differential Equations
• First-Order Ordinary Differential Equations
• Linear Differential Equations
• Separable Differential Equations
• Slope Fields
• Euler’s Method for Numerical Solutions
• Modeling Change
• Simple Growth and Decay
• Population Models