# MTH520: AP Calculus BC

This list is representative of the materials provided or used in this course. Keep in mind that the actual materials used may vary, depending on the school in which you are enrolled, and whether you are taking the course as Independent Study.

**For a complete list of the materials to be used in this course by your enrolled student, please visit MyInfo**. All lists are subject to change at any time.

Scope & Sequence : Scope & Sequence documents describe *what* is covered in a course (the scope) and also the *order* in which topics are covered (the sequence). These documents list instructional objectives and skills to be mastered. K12 Scope & Sequence documents for each course include:

## Course Overview

This course is the equivalent of an introductory college-level calculus course. In this course, students study functions, limits, derivatives, integrals, and infinite series. Calculus helps scientists, engineers, and financial analysts understand the complex relationships behind real-world phenomena. Students learn to evaluate the soundness of proposed solutions and apply mathematical reasoning to real-world models. Students also learn to understand change geometrically and visually (by studying graphs of curves), analytically (by studying and working with mathematical formulas), numerically (by seeing patterns in sets of numbers), and verbally. Students prepare for the AP Exam and further studies in science, engineering, and mathematics.

back to top## Prerequisites

Success in MTH204: Honors Geometry, MTH304: Honors Algebra II, MTH403: Pre-Calculus/Trigonometry (or equivalents), and teacher/school counselor recommendation.

back to top## Course Outline

### SEMESTER ONE

#### Unit 1: The Basics

Students prepare to study calculus by reviewing some basic pre-calculus concepts from algebra and trigonometry. They learn what calculus is, why it was invented, and what it is used for.

- Pre-Calculus Review
- Introduction to Calculus
- Using a Graphing Calculator
- Combining Functions
- Composite and Inverse Functions
- Graphical Symmetry
- Patterns in Graphs

#### Unit 2: Limits and Continuity

This unit addresses Topic I: Functions, Graphs, and Limits of the College Board's Calculus BC topic outline. Students learn two important concepts that underlie all of calculus: limits and continuity. Limits help students understand differentiation (the slope of a curve) and integration (the area inside a curved shape). Continuity is an important property of functions.

- Finding Limits Analytically
- Asymptotes as Limits
- Relative Magnitudes for Limits
- When Limits Do and Don't Exist
- Continuity
- Intermediate and Extreme Value Theorems

#### Unit 3: The Derivative

This unit addresses Topic II: Derivatives of the College Board's Calculus BC topic outline. Students learn how to calculate a derivative, the slope of a curve at a specific point. They learn techniques for finding derivatives of algebraic functions (such as y = x2) and trigonometric functions (such as y = sin x). Students also interpret the derivative as a rate of change and move fluidly between multiple representations including graphs, tables, and equations.

- Slope and Change
- Derivative at a Point
- The Derivative
- The Power Rule
- Sums, Differences, Products, and Quotients
- Graphs of Functions and Derivatives
- Continuity and Differentiability
- Rolles and Mean Value Theorems
- Higher-Order Derivatives
- Concavity
- Chain Rule
- Implicit Differentiation

#### Unit 4: Rates of Change

This unit focuses on Second Derivatives and Applications of Derivatives within Topic II: Derivatives of the College Board's Calculus BC topic outline. Students learn how to use calculus to model and analyze changing aspects of our world. In addition to the AB topics in this unit, BC students analyze polar and vector-valued functions.

- Extrema
- Optimization
- Tangent and Normal Lines
- Tangents to Polar Curves
- Tangent Line Approximation
- Rates and Derivatives
- Related Rates
- Rectilinear Motion
- Motion with Vector Functions

#### Unit 5: The Integral, Part 1

This unit focuses on Topic III: Integrals in the College Board's Calculus BC topic outline. Students learn numerical approximations to definite integrals, interpretations and properties of definite integrals, the Fundamental Theorem of Calculus, and techniques of anti-differentiation. They learn how to find areas of curved shapes.

- Riemann Sums
- Area Approximations
- The Definite Integral
- Properties of Integrals
- Graphing Calculator Integration
- Applications of Accumulated Change
- Antiderivatives
- Composite Functions

### SEMESTER TWO

#### Unit 1: The Integral, Part 2

This unit focuses on Topic III: Integrals in the College Board's Calculus BC topic outline. Students learn the Fundamental Theorem of Calculus, and techniques of anti-differentiation. They learn how to find areas of curved shapes.

- The Fundamental Theorems of Calculus
- Definite Integrals of Composite Functions
- Analyzing Functions and Integrals

#### Unit 2: Applications of the Integral

This unit focuses on Topic III: Integrals in the College Board's Calculus BC topic outline. Students learn to use integrals and antiderivatives to solve problems. In addition to the AB topics, BC students learn to calculate arc length for a smooth curve.

- Area Between Curves
- More Areas and Averages
- Volumes of Revolution
- Cross Sections
- Arc Length
- More Rectilinear Motion
- Other Applications of the Definite Integral

#### Unit 3: Inverse and Transcendental Functions

This unit focuses on Topic II: Derivatives and Topic III: Integrals in the College Board's Calculus BC topic outline. Students learn to calculate and use derivatives, antiderivatives, and integrals of exponential functions (such as y = 3x where the input variable is an exponent), logarithmic functions (the inverses of exponential functions), and trigonometric functions (such as y = secant x). In addition to the AB topics, BC students learn how to use L'Hopital's Rule and the methods of partial fractions and integration by parts. Also, students learn how to find improper integrals, and derivatives and integrals of parametric functions.

- Derivatives of Inverses
- Inverse Trigonometric Functions
- Logarithmic and Exponential Review
- Transcendentals and 1/x
- Derivatives of Logarithms and Exponentials
- L'Hopital's Rule
- Analysis of Transcendental Curves
- Integrating Transcendental Functions
- Partial Fractions
- Integration by Parts
- Improper Integrals
- Applications of Transcendental Integrals
- Derivatives of Parametric Functions
- Integrating Parametric and Polar Functions

#### Unit 4: Separable Differential Equations and Slope Fields

This unit focuses on Topic II: Derivatives of the College Board's Calculus BC topic outline, specifically, on Equations Involving Derivatives. Students investigate differential equations and solve the equations using a technique called "separating the variables." In addition to the topics covered in AB, BC students also learn to use Eulers method to estimate the solution of differential equations and use logistic equations to model growth.

- Slope Fields
- Differential Equations as Models
- Euler's Method
- Exponential Growth and Decay
- Logistic Growth
- More Applications of Differential Equations

#### Unit 5: Sequences and Series

This unit focuses on Topic IV: Polynomial Approximations and Series of the College Board's Calculus BC topic outline, specifically, on Series of Constants and Taylor Series.

- Sequences
- Series
- Convergence Tests
- Radius of Convergence
- Functions Defined by Power Series
- Taylor and Maclaurin Series
- Taylor's Theorem and Lagrange Error

#### Unit 6: AP Exam Review and Final Exam

Students review what they have learned and become more familiar with AP-type questions in preparation for the AP Exam. Students are also provided with access to previously released AP Exams for practice.

- Exam Strategies
- Review of Topics
- Practice Exams

#### Unit 7: Calculus Project

Teachers may choose to assign a final project.

- Project Days

### K12 Scope & Sequence documents for each course include:

- Course Overview (as seen above)
- Course Outline
- Lesson Time and Scheduling