AP Calculus BC Course Outline

Understanding 1.0 Credit for Online Education at Forest Trail Academy

This is a full-year course in the calculus of functions of a single variable. It includes all topics covered in Calculus AB plus additional topics. Both courses represent college-level mathematics for which most colleges grant advanced placement and credit. The content of Calculus BC is designed to qualify the student for placement and credit in a course that is one course beyond that granted for Calculus AB.

AP Calculus BC is the study of limits, derivatives, definite and indefinite integrals, polynomial approximations and (infinite) series. Though this is considered a study of single-variable calculus, parametric, polar, and vector functions will be studied. Consistent with AP philosophy, concepts will be expressed and analyzed geometrically, numerically, analytically, and verbally. Calculus BC covers topics that are usually included in the first 3 semesters of college calculus.

Students should be able to work with functions represented in a variety of ways: graphical, numerical, analytical, or verbal. They should understand the connections among these representations.

Students should understand the meaning of the derivative in terms of a rate of change and local linear approximation, and should be able to use derivatives to solve a variety of problems.

Students should understand the meaning of the definite integral both as a limit of Riemann sums and as the net accumulation of change, and should be able to use integrals to solve a variety of problems.

Students should understand the relationship between the derivative and the definite integral as expressed in both parts of the Fundamental Theorem of Calculus.

Students should be able to communicate mathematics and explain solutions to problems both verbally and in written sentences.

Students should be able to model a written description of a physical situation with a function, a differential equation, or an integral.

Students should be able to use technology to help solve problems, experiment, interpret results, and support conclusions.

Students should be able to determine the reasonableness of solutions, including sign, size, relative accuracy, and units of measurement.

Students should develop an appreciation of calculus as a coherent body of knowledge and as a human accomplishment.

Course Outline

Introduction & Review

Welcome to AP Calculus BC
Course Syllabus
MLA Formatting Word 2007
MLA Documentation Updates
Completing Math Assessments
MLA Citation
MLA Incorporating Sources
Calculus On The Web
Review — Stuff You Should Already Know

Limits & Continuity

An Introduction to Limits & Continuity
Quiz: Introduction to Limits & Continuity
Tangent Lines & Rates of Change
Handout: Tangent Lines & Rates of Change Worksheet
Assignment: Tangent Lines & Rates of Change
The Limit
Handout: The Limit Worksheet
Assignment: The Limit
One-Sided Limits
Handout: One-Sided Limits Worksheet
Assignment: One-Sided Limits
Computing Limits
Handout: Computing Limits Worksheet
Assignment: Computing Limits
Infinite Limits
Handout: Infinite Limits Worksheet
Assignment: Infinite Limits
Limits at Infinity
Handout: Limits at Infinity Worksheet
Assignment: Limits at Infinity
Continuity
Handout: Continuity Worksheet
Assignment: Continuity
Additional Resources
Handout: Limits & Continuity Exam
Assignment: Limits & Continuity Exam

Derivatives

Introduction to Derivatives
Definition & Interpretation of the Derivative
Handout: Definition & Interpretation of the Derivative Worksheet
Assignment: Definition & Interpretation of the Derivative
Differentiation Formulas
Handout: Differentiation Formulas Worksheet
Handout: Derivatives Exam 1
Assignment: Differentiation Formulas
Assignment: Derivatives Exam 1
Product & Quotient Rule
Handout: Product & Quotient Rule Worksheet
Assignment: Product & Quotient Rule
Derivatives of Trig Functions
Handout: Derivatives of Trig Functions Worksheet
Assignment: Derivatives of Trig Functions
Derivatives of Exponential & Logarithm Functions
Handout: Derivatives of Exponential & Logarithm Functions Worksheet
Assignment: Derivatives of Exponential & Logarithm Functions
Derivatives of Inverse Trig Functions
Derivatives of Hyperbolic Functions
Handout: Derivatives of Inverse & Hyperbolic Trig Functions Worksheet
Assignment: Derivatives of Inverse & Hyperbolic Trig Functions
The Chain Rule
Handout: The Chain Rule Worksheet
Handout: Derivatives Exam 2
Assignment: The Chain Rule
Assignment: Derivatives Exam 2
Implicit Differentiation
Handout: Implicit Differentiation Worksheet
Assignment: Implicit Differentiation
Related Rates
Handout: Related Rates Worksheet
Assignment: Related Rates
Higher Order Derivatives
Handout: Higher Order Derivatives Worksheet
Assignment: Higher Order Derivatives
Additional Resources
Handout: Derivatives Exam 3
Assignment: Derivatives Exam 3

Application of Derivatives

What’s the Big Deal About Derivatives, Anyway?
Rates of Change
Critical Points & Extrema
Handout: Critical Points & Extrema Worksheet
Assignment: Critical Points & Extrema
The Shape of a Graph
Handout: The Shape of a Graph Worksheet
Assignment: The Shape of a Graph
The Mean Value Theorem
Handout: The Mean Value Theorem Worksheet
Assignment: The Mean Value Theorem
Optimization
Handout: Optimization Worksheet
Assignment: Optimization
L’Hopital’s Rule
Handout: L’Hosptial’s Rule Worksheet
Assignment: L’Hospital’s Rule
Linear Approximations
Handout: Linear Approximations Worksheet
Assignment: Linear Approximations
Differentials
Handout: Differentials Worksheet
Assignment: Differentials
Newton’s Method
Handout: Newton’s Method Worksheet
Assignment: Newton’s Method
Midtest
Assignment: Calculus Midtest
Handout: Motion Part 1 Worksheet
Handout: Motion Part 2 Worksheet
Handout: Motion Part 3 Worksheet
Assignment: Motion Part 1
Assignment: Motion Part 2
Assignment: Motion Part 3

Integrals

Working Backwards
Indefinite Integrals
Handout: Indefinite Integrals Worksheet
Assignment: Indefinite Integrals
Computing Indefinite Integrals
Handout: Computing Indefinite Integrals Worksheet
Assignment: Computing Indefinite Integrals
Substitution Rule for Indefinite Integrals
Handout: Substitution Rule Worksheet
Handout: Integrals Exam 1
Assignment: Substitution Rule
Assignment: Integrals Exam 1
Area Problem
Handout: Area Problem Worksheet
Assignment: Area Problem
Definite Integrals
Handout: Definite Integrals Worksheet
Assignment: Definite Integrals
Computing Definite Integrals
Handout: Computing Definite Integrals Worksheet
Assignment: Computing Definite Integrals
Substitution Rule for Definite Integrals
Handout: Substitution Rule for Definite Integrals Worksheet
Assignment: Substitution Rule for Definite Integrals
Integration by Parts
Handout: Integration by Parts Worksheet
Assignment: Integration by Parts
Partial Fractions
Handout: Partial Fractions Worksheet
Assignment: Partial Fractions
Improper Integrals
Handout: Improper Integrals Worksheet
Assignment: Improper Integrals
Handout: Motion Part 4 Worksheet
Handout: Motion Part 5 Worksheet
Assignment: Motion Part 4
Assignment: Motion Part 5

Applications of Integrals

What’s the Big Deal About Integrals, Anyway?
Area Between Curves
Handout: Area Between Curves Worksheet
Assignment: Area Between Curves
Volumes of Solids of Revolution (Part 1)
Handout: Method of Rings Worksheet
Assignment: Method of Rings
Volumes of Solids of Revolution (Part 2)
Handout: Method of Cylinders Worksheet
Assignment: Method of Cylinders
Volume
Work
Handout: Work Worksheet
Assignment: Work
Handout: Applications of Integrals Exam
Assignment: Applications of Integrals Exam

Sequences & Series

Sequences & Series
Additional Resources
Sequences
Handout: Sequences Worksheet
Assignment: Exploration: The Limit of a Sequence
Assignment: Sequences
Series: The Basics
Handout: Series Worksheet
Assignment: Series
Series: Convergence & Divergence
Special Series
Integral Test
Essay: Integral Test
Comparison Test
Handout: Comparison Test Worksheet
Assignment: Comparison Test
Alternating Series Test
Handout: Alternating Series Test Worksheet
Assignment: Alternating Series Test
Absolute Convergence
Handout: Absolute Convergence Worksheet
Assignment: Absolute Convergence
Ratio & Root Tests
Handout: Ratio & Root Test Worksheet
Assignment: Ratio & Root Test
Strategy for Series
Estimating the Value of a Series
Power Series
Handout: Power Series Worksheet
Taylor Series
Handout: Taylor Series Worksheet
Assignment: Power Series
Assignment: Taylor Series
Post-Test
Assignment: Calculus Post-test

Review & Test Preparation

Exam Format & Scoring
Exam Strategies & Tips
Practice, Practice, Practice
Handout: Calculus BC Section I Part A
Handout: Calculus BC Section 1 Part B
Handout: Calculus BC Section II Part A
Handout: Calculus BC Section II Part B
Assignment: AP Calculus BC Practice – Section I Part A
Assignment: AP Calculus BC Practice – Section I Part B
Assignment: AP Calculus BC Practice – Section II Part A
Assignment: AP Calculus BC Practice – Section II Part B

Course Survey

Handout: Course Survey
Assignment: Course Survey

Understanding the AP Calculus BC Exam Format

The AP Calculus BC exam is structured to assess students' understanding of calculus concepts through a combination of multiple-choice and free-response questions. Familiarity with the exam format is crucial for effective preparation, as it helps students manage their time and approach each section strategically.

The exam consists of two sections: Section I, which includes 45 multiple-choice questions, and Section II, featuring 6 free-response questions. Students are given a total of three hours to complete the exam, with specific time limits for each section. Understanding these components allows students to focus their studies on the types of questions they will encounter and the skills they need to demonstrate.

Essential Resources for AP Calculus BC Students

To excel in AP Calculus BC, students should leverage a variety of resources that enhance their understanding and problem-solving skills. These resources include textbooks, online tutorials, and practice exams that cover the full range of topics in the course outline.

For instance, the College Board provides a wealth of materials, including past exam questions and scoring guidelines. Additionally, online platforms like Khan Academy and various educational YouTube channels offer video explanations and interactive exercises that can help clarify complex concepts and improve retention.

Tips for Success in AP Calculus BC

Success in AP Calculus BC requires a combination of strong study habits, effective time management, and a solid grasp of calculus concepts. Students should create a study plan that allocates time for reviewing material, completing assignments, and practicing past exam questions.

Moreover, forming study groups can be beneficial, as discussing problems with peers often leads to deeper understanding. Utilizing office hours with instructors for additional help and clarification on challenging topics can also enhance a student's grasp of the material and boost confidence going into the exam.

Common Challenges in AP Calculus BC and How to Overcome Them

Many students face challenges in AP Calculus BC, particularly with the abstract nature of calculus concepts such as limits, derivatives, and integrals. A common struggle is applying these concepts to real-world problems, which can feel overwhelming without proper guidance.

To overcome these challenges, students should practice regularly and seek help when needed. Utilizing additional resources, such as tutoring or online forums, can provide support. Engaging with calculus applications in physics or engineering contexts can also help solidify understanding by showing the practical implications of calculus concepts.