The Ultimate AP Calculus BC Course

Unlock the secrets of Calculus BC with comprehensive explanations, in-depth examples, and extensive practice problems designed to master advanced concepts and ace your exams!

Dive deep into the world of Calculus BC with this expertly crafted textbook, tailored to guide students through the complex terrain of advanced calculus. Whether you’re preparing for challenging exams or seeking to solidify your understanding of differential equations, infinite series, and integration techniques, this book has it all. Detailed explanations are paired with in-depth examples to illustrate core concepts clearly. Each chapter offers a variety of problems, from basic drills to more intricate questions that stimulate critical thinking and analytical skills. The layout is student-friendly, featuring clear headings and key points highlighted for quick reference. Additionally, the book includes tips for solving typical exam questions and pitfalls to avoid, making it an essential resource for excelling in Calculus BC.

AP Calculus AB Complete Course

Limit and Continuity

• Limit Introduction
• Neighborhood
• Estimating Limits from Tables
• Functions with Undefined Limits (from table)
• Functions with Undefined Limits (from graphs)
• 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
• Properties of Definite Integrals
• 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
• Integrating Using Tables
• Long Division Method to Simplify Integrals

Applications of Integrals

• Average Value of a Function
• Arc Length
• Accumulation Problems
• Finding Area Between Curves
• Area Between Curves Expressed as x and y
• 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 of DEs
• Classification of Differential Equations
• First-Order Ordinary Differential Equations
• Linear Differential Equations
• Separable Differential Equations
• Slope Fields
• Numerical Methods Introduction
• Euler’s Method for Numerical Solutions
• Modeling Change
• Simple Growth and Decay
• Population Models
• Logistic Models

Analytic Geometry

• Ellipses, Parabolas, and Hyperbolas
• Polar Coordinates
• Converting Between Polar and Rectangular Coordinates
• Graphing Polar Equations
• Derivative of Polar Coordinates
• Applications of Polar Coordinates
• Parametric Equations and Differentiating them
• Vector-valued Functions
• Basic Operations on Vectors
• Derivative of Vector-valued Functions
• Integrating Vector Valued Functions
• Velocity, Speed, and Acceleration Along a Curve
• The Total Change in Length
• Finding the Area Between Polar Curves

Sequences and Series

• Introduction to Sequences and Series
• Types of Sequences and Series
• Arithmetic Sequences
• Geometric Sequences
• Sigma Notation (Summation Notation)
• Arithmetic Series
• Geometric Series
• Alternating Series
• p-Series
• The n-th Term Test
• The Integral Test
• The Comparison Test
• Limit Comparison Test
• The Ratio Test
• Alternating Series Test
• Harmonic Series
• Power Series
• Absolute and Conditional Convergence
• Taylor Series
• Maclaurin Series
• Lagrange Error Bound