1
AP Calculus BC
Syllabus
2015-2016
Course Overview
AP Calculus BC is an enriched mathematics course and curriculum that is designed
to help students in their understanding of the calculus curriculum and to provide
and prepare them for the mathematics needed to be successful in post secondary
education. Students are introduced to this curriculum through a comprehensive
study of all of the objectives outlined in the AP Calculus Course Description.









Goal from the AP Calculus Course Description
Students should be able to work with functions numerically, graphically,
analytically, and verbally
The derivative should be understood as the instantaneous rate of change of a
function and as the local linear approximation of the function
The definite integral should be understood as the limit of a Riemann sum and as
the net accumulation of the rate of change
The relationship between derivatives and the definite integral should be
understood in terms of both parts of the Fundamental Theorem of Calculus
Students should learn to communicate about mathematics verbally and in
writing
Students should be able to model a written description of a physical situation
with a function, a differential equation, or an integral
Students should learn to use technology to analyze problems, experiment, and
verify and interpret results
Students are expected to learn to judge the reasonableness of their solutions
Develop an appreciation of calculus as a coherent body of knowledge and as a
human accomplishment
Primary Textbook:
Larson, Ron, and Bruce Edwards. Calculus of a Single Variable AP Edition Tenth
Edition. Boston: Brooks/Cole, Cengage Learning, 2014
Teaching Strategies
Connections in mathematics are stressed frequently. For instance: not all students
realize at the beginning of the study of derivatives that the definition relates back to
the study of slope in Algebra I. For comprehension of calculus concepts, students
must make the mathematical connections to previous learning in order to have a
true understanding of new calculus concepts and applications. Solutions to
problems are found graphically, numerically, analytically, and verbally in order to
demonstrate knowledge of the calculus curriculum being studied. In addition,
proper vocabulary and symbolism are used in the classroom and expected of the
2
students. To further illustrate these connections students will derive many formulas
such as the definition of derivative and the derivative rules.
During the summer, students completed a three-part packet that reviews many key
components of pre-calculus and necessary algebra skills. For the first four days of
class students will review this in class as well as take a quiz over this review. This is
a great chance to get all students acclimated to the class.
Students are encouraged to ask questions and call out answers immediately during
lecture. Consequently, problems are cleared up quickly and no classmates are left
behind and in a quandary due to a lack of understanding. The instructor will meet
with struggling students before or after school to give students additional assistance
as needed.
Students practice on multiple choice and free response questions from old AP exams
on a weekly basis. These problems will be worked on individually, whole class, or
small groups. During the free response component students are expected to
explain/justify their solution in well-written sentences and through the use of oral
communication to their classmates.
Student Activities
Students review parent functions, domain, and range early in the year when they
review their summer work assignment. Students approach their study of calculus
with a multi-representational view (graphically, numerically, analytically, and
verbally).
Daily, students are working in and adding to their notebooks. Cooperative learning
groups will be used frequently in the classroom on assignments. In addition,
students will show answers on the board to demonstrate calculus solutions to their
classmates. The importance of explaining their work with calculus orally and in
complete sentences will be stressed at this time.
Throughout each section of each unit the instructor will implement and discuss AP
testing strategies as well as discussing AP questions. This will illustrate the current
content the students are on as well as the material they have already covered.
During the instructor’s lecture and modeling of example problems and content
students are encouraged to ask questions and to participate in the class discussion
of the day’s lesson.
Graphing Calculators and Technology
Graphing calculators will be used on a daily basis to reinforce the calculus concepts
and to help interpret results. The instructor will use the TI-84 Plus. All students will
3
have either the TI-84 Plus, TI-89, or the TI-Nspire. The calculator will be used in a
variety of ways, including but not limited to the following:
 Conducting explorations
 Graphing functions within arbitrary windows
 Estimating the coordinates of a point of intersection of two functions
 Estimating the value of a limit from a table
 Determining the asymptotic behavior of a function
 Finding zeros of functions
 Analyzing and interpreting results
 Verifying results found analytically
 Justifying and explaining results of graphs and equations
 Performing numerical differentiation and integration
 Investigate the slope of a the line tangent to a function at a point using the
draw tangent feature
Along with the TI-84 plus calculator, all students will have a MacBook Air. Through
the adoption of the Larson series listed above, we also have many online resources
that students will be able to use. The sites we will use are:
 Cengage Learning (Coursemate for Calculus)
 Webassign
 www.larsoncalculus.com
 http://tube.geogebra.org
These sites have many features including:
 Interactive quizzes and examples
 Videos attached to examples
 eBook
 Electronic media that demonstrates calculus concepts graphically and
numerically. Calculus in Motion will also be used to illustrate certain
concepts graphically and numerically.
 Quizzes asking students to explain their work in sentences
 AP practice problems attached to every unit and content learned
Course Planner
Review of Precalculus (4 days)
Students will have a chance to review their summer work material in groups and
will begin presenting these problems on the board. At the end of the review, there
will be an assessment covering the big ideas discussed in precalculus.
Chapter 1: Limits and Their Properties (8 days)
1.1: A Preview of Calculus
1.2: Finding Limits Graphically and Numerically
1.3: Evaluating Limits Analytically
1.4: Continuity and One-Sided Limits
4
1.5: Infinite Limits
Chapter 2: Differentiation (15 days)
2.1: The Derivative and the Tangent Line Problem
2.2: Basic Differentiation Rules and Rates of Change
2.3: Product and Quotient Rules and Higher-Order Derivatives
2.4: The Chain Rule
2.5: Implicit Differentiation
2.6: Related Rates
*
Students will find derivatives algebraically using the many differentiation rules
discussed in this unit (i.e. power, product, quotient, chain) and will find the
equation of a tangent line at a point, and when the slope of a tangent line is 0.
That is, when 𝑓 ′ (𝑥) = 0.
*
Students will see problems presented to them verbally when they solve
application problems involving related rates. In this case, students will have to
take what they read and derive a mathematical model from that to solve the
problem they are given. Examples of related rate problems to be discovered
include finding the rate of change of the radius of a balloon when it is being
inflated at a specific rate, and finding the rate of change of a ladder up against a
building when it is pulled away from the building at a specific rate.
Chapter 3: Applications of Differentiation (18 days)
3.1: Extrema on an Interval
3.2: Rolle’s Theorem and the Mean Value Theorem
3.3: Increasing and Decreasing Functions and the First Derivative Test
3.4: Concavity and the Second Derivative Test
3.5: Limits at Infinity
3.6: A Summary of Curve Sketching
3.7: Optimization Problems
3.9: Differentials/Linear Approximations
*
Students will work with functions represented numerically and graphically
during this unit. Students will be given either a table or graph of the first or
second derivative and will be asked to determine key features of the function
such as: relative extrema, increasing/decreasing behavior, and concavity.
Students will justify their answers using complete sentences and the information
from the table and/or graph given to them.
*
Students will see problems presented to them verbally when they solve
application problems involving optimization. In this case, students will have to
derive a formula and will have to determine what quantity is the one to be
optimized. Students will work with many examples in collaborative groups and
5
will use calculus to explain in complete sentence or orally why their answer
either maximizes or minimizes the situation they are given.
Chapter 4: Integration (18 days)
4.1: Antiderivatives and Indefinite Integration
4.2: Area
4.3: Riemann Sums and Definite Integrals
4.4: The Fundamental Theorem of Calculus
4.5: Integration by Substitution
4.6: Numerical Integration
Chapter 5: Logarithmic, Exponential and Other Transcendental Functions (16
days)
5.1: The Natural Logarithmic Function: Differentiation
5.2: The Natural Logarithmic Function: Integration
5.3: Inverse Functions
5.4: Exponential Functions: Differentiation and Integration
5.5: Bases Other Than e and Applications
5.6: Inverse Trigonometric Functions: Differentiation
5.7: Inverse Trigonometric Functions: Integration
*Students will be given a function and inverse function numerically in a table and
will be asked to reach conclusions about the function and the derivative of the
inverse using the intermediate value theorem, the mean value theorem, and
properties of inverse functions. This problem will tie into many topics and will be
represented in the form of a free response questions. Students must use the calculus
to explain their answers in complete sentences.
Chapter 6: Differential Equations (12 days)
6.1: Slope Fields
6.2: Differential Equations: Growth and Decay
6.3: Separation of Variables
*Students will work with functions analytically by solving separable differential
equations. Students will be given a differential equation and an initial condition and
will be asked to find the particular solution that satisfies this initial condition given.
During this time, students will be given many previous AP tests to practice these
types of questions dealing with slope fields and separable differential equations.
Students will solve these one-on-one and in collaborative groups and will present
the problems to their classmates.
6
Chapter 7: Applications of Integration (12 days)
7.1: Area of a Region between Two Curves
7.2: Volume: The Disk Method and Washer Method
7.3: Volume: The Shell Method
Chapter 8: Integration Techniques, L’Hopital’s Rule, and Improper Integrals
(20 days)
8.1: Basic Integration Rules
8.2: Integration By Parts
8.3: Trigonometric Integrals
8.4: Trigonometric Substitution
8.5: Partial Fractions
8.6: Integration by Tables
8.7: Indeterminate Forms and L’Hopital’s Rule
8.8: Improper Integrals
Chapter 9: Infinite Series (28 days)
9.1: Sequences
9.2: Series and Convergence
9.3: The Integral Test and p-series
9.4: Comparisons of Series
9.5: Alternating Series
9.6: The Ratio and Root Tests
9.7: Taylor Polynomials and Approximations
9.8: Power Series
9.9: Representation of Functions by Power Series
9.10: Taylor and Maclaurin Series
Chapter 10: Conics, Parametric Equations, and Polar Coordinates (20 days)
10.1: Conics and Calculus
10.2: Plane Curves and Parametric Equations
10.3: Parametric Equations and Calculus
10.4: Polar Coordinates and Polar Graphs
10.5: Area and Arc Length in Polar Coordinates
10.6: Polar Equations of Conics
Review: (10 days)
A. Multiple-choice practice (Items from past exams are used, as well as items from
review books that I have accumulated)
1. Test-taking strategies are emphasized
2. Individual and group practice are both used
B. Free-response practice (Released items from the AP Central website are used
liberally; solutions to these problems will include the use of written
sentences.)
7
1. Rubrics are reviewed so students see the need for complete answers
2. Students collaborate to formulate team responses
3. Individually written responses are crafted. Attention to full explanations is
emphasized
Student Evaluation
In the course planner, the number of days listed includes assessment days. Students
will be assessed through many measures including homework assignments, quizzes,
tests, midterms, finals, AP free-response questions, AP multiple choice questions. All
students will take multiple full-length practice exams prior to sitting for the AP
Calculus AB Exam.
Homework assignments, quizzes, and tests will include both free response and
multiple choice type AP questions. During the free response questions students are
expected to explain and/or justify their solutions to problems in well-written
sentences. On all free response questions, solutions alone will not be given full
credit. Answers must be accompanied by the appropriate work.
A high level of expectation is maintained at all times. Students must keep up with
their studies and are expected to come to class everyday with the rules and formulas
known so that the class can run more smoothly.