The Ultimate AP Calculus BC Course

Understand 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 detailed 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

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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

Frequently Asked Questions

What college course is AP Calc BC equivalent to?

AP Calculus BC is roughly equivalent to two semesters of college calculus, usually called Calculus I and Calculus II. It covers limits, derivatives and integrals, then adds the Calc II topics: integration by parts, partial fractions, improper integrals, infinite series, and polar and parametric equations. Exactly how much credit a qualifying score earns depends on each college’s policy, so check the school’s AP credit page.

Is AP Calculus BC a college course?

AP Calculus BC is taught in high school, but it is designed to match the content and pace of college-level calculus. Students who earn a qualifying score on the AP exam may receive college credit or placement, depending on the college. Because BC moves fast, a strong grasp of algebra, trigonometry and precalculus functions before starting makes the year much smoother.

What is the AP Calculus BC Course and Exam Description?

The Course and Exam Description (CED) is the College Board’s official framework for AP Calculus BC. It lists the 10 units, the skills students should build, and how the exam is organized. It is the best place to confirm what is tested, and the College Board posts the current version as a free PDF on its AP Central website.

What topics are in the AP Calculus BC course content?

AP Calculus BC includes everything in Calculus AB: limits and continuity, differentiation, applications of derivatives, integration, differential equations and applications of integrals. BC then adds advanced integration techniques, parametric equations, polar coordinates, vector-valued functions, and infinite sequences and series, including Taylor and Maclaurin series. Series is often the unit students find most new, so it is worth starting review early.

Can I take an AP Calculus BC course online?

Yes. Many students study AP Calculus BC online, either through a school-approved online class or with self-paced lessons and worksheets for review. If you want the course listed on your transcript as AP, it must be an authorized AP course, but anyone can register for the AP exam through a school that offers it. Self-study works best with a weekly plan unit by unit.

What is a post-BC AP Calculus course?

A post-BC course is a math class taken after finishing AP Calculus BC, usually in senior year. Common options are multivariable calculus, linear algebra or differential equations, often offered through a local college or dual-enrollment program. There is no AP exam for these courses, so students should ask the college they plan to attend how the credit will transfer.

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Recommended EffortlessMath Books

For full AP Calc BC prep with multiple practice tests, the AP Calculus BC Test Prep Bundle (resource currently unavailable) covers every topic with detailed answer explanations. For a workbook to learn the underlying calc, see the Calculus for Beginners (resource currently unavailable).

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