# AP® Calculus BC

## Course Summary

Learn about the mathematical study of change to help you succeed in a career in economics, science, video-game engineering, and more. Gain the skills needed to build future technologies by working with functions and using derivatives to solve a variety of problems. In this course, students engage with a range of lessons, assessments, and more to ensure successful course completion, and learn the application of limits, change, implicit differentiation, integration, advanced sequences and series, and other calculus topics.

This course prepares students for the AP® Calculus BC exam.

## Curriculum

This is a 2-semester course. We do not recommend taking both semesters simultaneously.

##### Unit 0: Review

- Lines
- Functions and Graphs
- Exponential Functions
- Functions and Logarithms
- Trigonometric Functions

##### Unit 1: Limits and Continuity

- Finding Limits Graphically and Numerically
- Evaluating Limits Analytically
- One-sided Limits
- Continuity and Intermediate Value Theorem
- Infinite Limits
- Limits at Infinity

##### Unit 2: Differentiation Definition and Fundamental Properties

- The Derivative and the Tangent Line Problem
- Basic Differentiation Rules
- Equations of a tangent Line
- The Natural Logarithmic Functions: Differentiation
- The Product and Quotient Rules

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

- The Chain Rule
- Implicit Differentiation
- Inverse Functions
- Inverse Trigonometric Functions and Differentiation

##### Unit 4: Contextual Applications of Differentiation

- Velocity and Acceleration
- Related Rates - Introduction
- Related Rates - Advanced Examples
- Indeterminate Forms and L'Hopital's Rule

##### Unit 5: Analytical Applications of Differentiation

- Rolle's Theorem
- The Mean Value Theorem
- Extrema on an Interval
- Extrema on a Closed Interval
- Increasing and Decreasing Functions and the First Derivative Test
- Concavity and the Second Derivative Test
- A Summary of Curve Sketching
- Optimization Problems

##### Unit 6 Part 1: Integration and Accumulation of Change

- Left and Right Side Approximations
- Area - Midpoint and Trapezoidal Approximations
- Riemann Sums and Definite Integrals
- Antidifferentiation and Indefinite Integration
- The Fundamental Theorem of Calculus
- Integration by Substitution
- The Natural Logarithmic Functions: Integration

##### Unit 6 Part 2: More Integration Techniques

- Exponential Functions: Differentiation and Integration
- Bases Other Than e and Applications
- Inverse Trigonometric Functions and Integration
- Trigonometric Integrals
- Trigonometric Substitution
- Integration by Parts
- Partial Fractions
- Improper Integrals

##### Unit 7: Differential Equations

- Slope Fields
- Separation of Variables
- Differential Equations: Growth and Decay

##### Unit 8: Applications of Integration

- Area of a Region Between Two Curves
- Area of a Region Between Two Curves - Advanced Examples
- Volume: Solids with Known Cross Sections
- Volume: The Disk Method
- Volume: The Shell Method
- Arc Length

##### Unit 9: Parametric Equations, Polar Coordinates, and Vector-Valued Functions

- Parametric Equations
- Differentiation of Parametric Equations
- Vector-Valued Functions
- Differentiation and Integration of Vector-Valued Functions
- Velocity and Acceleration Using Parametric and Vector-Valued Functions
- Polar Coordinates and Polar Graphs
- Polar Form of a Derivative
- Areas in Polar Coordinates

##### Unit 10: Infinite Sequences and Series

- Sequences
- Infinite and Geometric Series
- The Integral Test and p-Series
- Comparisons of Series and the Ratio Test
- Alternating Series
- Taylor Polynomials and Approximations
- Power Series
- Taylor and Maclaurin Series
- Representation of Functions by Power Series