Mater Academy Charter High School
AP Calculus BC
Syllabus
Introduction
The course will cover all topics associated with Functions and their Graphs, Limits of a Function,
Derivatives and Integrals, in accordance with the AP Calculus BC Course Description.
AP Calculus BC is organized as a full-year course that includes all topics taught in Calculus AB (AP
Calculus BC is an extension of the previous Calculus AB course) plus supplementary topics such as
Parametric, Polar, and Vector Functions, Equations Involving Derivatives, several Integration Techniques
like Integration by Parts, Simple Partial Fractions, Improper Integrals, Polynomial Approximations and
Series among others.
Technology
The AP Calculus BC lessons will enforce the use of graphing calculators as an essential goal of the
course.
C5— The course teaches students how to use graphing calculators to help solve problems, experiment,
interpret results, and support conclusions.
In addition, the following resources may be helpful for those students looking for online tutoring or for
those who are interested in consulting references regarding the AP Exam:
 www.collegeboard.com.
 www.flvs.net (for free online tutoring)
 www.calculus-help.com
 www.skylit.com/calculus/fr.html (to find questions and solutions from previous AP Exams)
 http://math.exeter.edu/rparris/winplot.html (this site allows the students to explore excellent
visual presentations, and it is a great graphing tool)
 http://www.univie.ac.at/future.media/moe/tests/diff1/ablerkennen.html (an interactive way to
learn about derivatives)
Primary Textbook

Stewart, James. “Calculus”. Books/Cole Cengage Learning. 6th Edition. 2008
Page
1
Alternative Textbooks
 Finney, Ross L., Franklin Demana, Bert Waits, and Daniel Kennedy. “Calculus: Graphical,
Numerical, and Algebraic”. Reading, Mass.: Addison-Wesley, 2007.
 “Cracking the AP Calculus AB & BC Exams”, 2010 Edition / Princeton Review.
Magda de la Torre
Mater Academy Charter High School
AP Calculus BC
Syllabus
Course Outline
(Organized based on the College Board’s Course Description)
I.
Functions, Graphs and Limits ( 5 weeks)
C2— The course teaches all topics associated with Functions, Graphs, and Limits; Derivatives; Integrals;
and Polynomial Approximations and Series as delineated in the Calculus BC Topic Outline in the AP
Calculus Course Description.
•
•
•
•
•
•
Derivatives ( 5 weeks)
C2— The course teaches all topics associated with Functions, Graphs, and Limits; Derivatives; Integrals;
and Polynomial Approximations and Series as delineated in the Calculus BC Topic Outline in the AP
Calculus Course Description.
 Concept of the derivative. Derivative presented graphically, numerically and
analytically. Derivative interpreted as an instantaneous rate of change. Derivative defined
as the limit of the difference quotient. Relationship between differentiability and
continuity.
 Derivative at a point. Slope of a curve at a point. Examples are emphasized, including
points at which there are vertical tangents and points at which there are no tangents.
Tangent line to a curve at a point and local linear approximation. Instantaneous rate of
change as the limit of average rate of change. Approximate rate of change from graphs
and tables of values.
 Derivative as a function. Corresponding characteristics of graphs of ƒ and ƒ∙.
Relationship between the increasing and decreasing behavior of ƒ and the sign of ƒ∙. The
Mean Value Theorem and its geometric interpretation. Equations involving derivatives.
Verbal descriptions are translated into equations involving derivatives and vice versa.
Page
2
II.
Analysis of graphs.
(2 weeks)
Limits of functions (3 weeks)
An intuitive understanding of the limiting process. Language of limits, including notation
and one-sided limits. Calculating limits using algebra. Properties of limits. Estimating
limits from graphs or tables of data. Estimating limits numerically and graphically.
Asymptotic and unbounded behavior. Understanding asymptotes in terms of graphical
behavior. Describing asymptotic behavior in terms of limits involving infinity.
Comparing relative magnitudes of functions and their rates of change
Continuity as a property of functions. An intuitive understanding of continuity.
Understanding continuity in terms of limits. Types of discontinuities. Geometric
understanding of graphs of continuous functions. Intermediate Value and Extreme Value
Theorem
Parametric, polar and vector functions.
Magda de la Torre
Mater Academy Charter High School
AP Calculus BC
Syllabus

Second derivatives. Corresponding characteristics of the graphs of ƒ, ƒ∙ and ƒ ∙.
Relationship between the concavity of ƒ and the sign of ƒ ∙. Points of inflection as places
where concavity changes.

Applications of derivatives
 Analysis of curves, including the notions of monotonicity and concavity.
 Analysis of planar curves given in parametric form, polar form and vector form,
including velocity and acceleration.
 Optimization, both absolute (global) and relative (local) extreme.
 Modeling rates of change, including related rates problems.
 Use of implicit differentiation to find the derivative of an inverse function.
 Interpretation of the derivative as a rate of change in diverse applied contexts,
including velocity, speed and acceleration.
 Geometric interpretation of differential equations via slope fields and the
relationship between slope fields and solution curves for differential equations.
 Numerical solution of differential equations using Euler’s method.
 L’Hospital’s Rule, including its use in determining limits and convergence of
improper integrals and series.
Computation of derivatives
 Knowledge of derivatives of basic functions, including power, exponential,
logarithmic, trigonometric and inverse trigonometric functions.
 Derivative rules for sums, products and quotients of functions.
 Chain rule and implicit differentiation.
 Derivatives of parametric, polar and vector functions.

III.
Integrals (10 weeks)
C2— The course teaches all topics associated with Functions, Graphs, and Limits; Derivatives; Integrals;
and Polynomial Approximations and Series as delineated in the Calculus BC Topic Outline in the AP
Calculus Course Description.

Page
3

Interpretations and properties of definite integrals. Summation notation. Definite
integral as a limit of Riemann sums. Basic properties of definite integrals. Linearity
properties of definite integrals.
Applications of integrals. A variety of applications to model physical, biological or
economic situations. Finding the area of a region (including a region bounded by
polar curves), the volume of a solid with known cross sections, the average value of a
function, the distance traveled by a particle along a line, the length of a curve
(including a curve given in parametric form), and accumulated change from a rate of
change.
Magda de la Torre
Mater Academy Charter High School
AP Calculus BC
Syllabus




IV.
Fundamental Theorem of Calculus. Use of the Fundamental Theorem to evaluate
definite integrals. Use of the Fundamental Theorem to represent a particular
antiderivative, and the analytical and graphical analysis of functions so defined.
Techniques of antidifferentiation. Antiderivatives following directly from
derivatives of basic functions. Antiderivatives by substitution of variables (including
change of limits for definite integrals), parts, and simple partial fractions
(nonrepeating linear factors only). Improper integrals (as limits of definite integrals).
Applications of antidifferentiation. Finding specific antiderivatives using initial
conditions, including applications to motion along a line. Solving separable
differential equations and using them in modeling (including the study of the
equation y∙ = ky and exponential growth). Solving logistic differential equations
and using them in modeling.
Numerical approximations to definite integrals. Use of Riemann sums and
trapezoidal sums to approximate definite integrals of functions represented
algebraically, graphically and by tables of values.
Polynomial Approximations and Series (10 weeks)
C2— The course teaches all topics associated with Functions, Graphs, and Limits; Derivatives; Integrals;
and Polynomial Approximations and Series as delineated in the Calculus BC Topic Outline in the AP
Calculus Course Description.


Page
4

Concept of series. A series is defined as a sequence of partial sums, and convergence
is defined in terms of the limit of the sequence of partial sums. Technology can be
used to explore convergence and divergence.
Series of constants
 Motivating examples, including decimal expansion.
 Geometric series with applications.
 The harmonic series.
 Alternating series with error bound.
 Terms of series as areas of rectangles and their relationship to
improper integrals, including the integral test and its use in testing
the convergence of p-series.
 The ratio test for convergence and divergence.
 Comparing series to test for convergence or divergence.
Taylor series
 Taylor polynomial approximation with graphical demonstration of
convergence (for example, viewing graphs of various Taylor
polynomials of the sine function approximating the sine curve).
 Maclaurin series and the general Taylor series centered at x = a.
Magda de la Torre
Mater Academy Charter High School
AP Calculus BC
Syllabus





V.
Maclaurin series for the functions ex , sin x, cos x, and 1/1-x
Formal manipulation of Taylor series and shortcuts to computing
Taylor series, including substitution, differentiation,
antidifferentiation and the formation of new series from known
series.
Functions defined by power series.
Radius and interval of convergence of power series.
Lagrange error bound for Taylor polynomials.
Review for AP Exam and Final Exam
The course leaves about 2 weeks to teach the most difficult topics. The remaining time will be used to
review the main topics before the AP exam.
Teacher Strategies





Students are expected to follow the syllabus systematically, completing all assignments with
adequate time to be prepared for the AP exam.
Students will receive materials and course topics on the first day of class.
The teacher will coach the progress of the students and will work on every point on the course
planner to achieve the goals of doing well on the AP exam.
Each topic will be presented in different ways: numerically, geometrically, symbolically, and
verbally in order to teach students the connection among these representations.
The teacher and the students are going to spend two (2) weeks at the beginning of the school year
to review a variety of Pre-Calculus topics, basically functions and their graphs, in order to
familiarize students with the basic functions and be able to represent functions in a variety of
ways (graphically, numerically, analytically, and verbally). Students should identify the
connection among these representations.
C3— The course provides students with the opportunity to work with functions represented in a variety
of ways - graphically, numerically, analytically, and verbally- and emphasizes the connections among
these representations.

Page
5

Students will be instructed to use a graphing calculator to help them in solving problems from the
real world. Students who are taking calculus BC have already been trained to use the graphing
calculator. In any case, the teacher will spend additional time (by means of after school tutoring)
during the first weeks of the school year, training students in the use of the TI-83 calculator, and
addressing concerns about its usage. Students will be provided with a graphing calculator every
class.
Students will use the graphing calculators to:
 Investigate limits of functions
 Explore continuity of a function
 Confirm and discover characteristics of graphs of functions and their derivatives
(e.g. ,Extreme Values, Inflection Points, and Concavity)
Magda de la Torre
Mater Academy Charter High School
AP Calculus BC
Syllabus








Determine the asymptotic behavior of a function.
Perform numerical integration
Show Riemann sums
Compute partial sums
Use Euler’s methods
Show a slope field
Draw a solution curve on a slope field
To sketch implicitly defined functions
C5— The course teaches students how to use graphing calculators to help solve problems, experiment,
interpret results, and support conclusions.

Students will justify responses and support their conclusions as a typical practice in class. To
develop student’s communication skills, the course includes a diversity of teaching strategies to
encourage students to expand their vocabulary and explanation skills.
C4— The course teaches students how to communicate mathematics and explain solutions to problems
both verbally and in written sentences.

Students will be evaluated by different methods. From middle of October throughout the rest of
the school year, students will work on questions from AP released exams at the beginning of
every class, as “the question of the day”). Questions will be graded.
Every week students will have a quiz containing 5-10 multiple-choice questions, and a test after the
conclusion of every topic. Each student will work in building an AP Calculus BC Portfolio. As a final
point, students will be required to take both, a Mid-Term and a Final Exams in AP Exam format.

Students should make conclusions and justify their responses as a part of the class routine.
Assignments and assessments are conceived to persuade students to express their ideas in
carefully written sentences to support conclusions when soling problems. Presentations and
discussions are been including to improve the student’s oral communication skills.
Major Assessments and Assignments
Page
6
Students will be evaluated by using different methods. Different assessments and assignments will be
used in class in order to expand the student’s understanding of the concepts of calculus and their
applications, and to develop the student’s skill in recognizing a variety of techniques for solving calculus
problems. In addition, these assignments emphasize a multi-representational approach to calculus and the
use of diverse manners to express concepts, problems, and results, including graphically, numerically,
analytically, and verbally, and the correlation among these representations. Power Point presentations,
banners, and posters could be built by students to show their results; as well portfolios should be created
to organize information in written manner.
Students will work in groups in solving real-world problems. Consequently, they should participate in
discussions of notorious calculus topics and explain their solutions and conclusions to the rest of the class
orally.
Magda de la Torre
Mater Academy Charter High School
AP Calculus BC
Syllabus
The following activities represent some examples of the most important assignments of the course:
Activity #1: Discussion about how derivatives affect the shape of a graph. (Working in groups)
 Points to stress:
1.
2.
3.
4.
The use of the first derivative to determine whether a function is increasing or decreasing.
The first and second derivative test for local maxima and minima
The use of the second derivative to determine concavity and points of inflections.
The geometric description of concavity, and the relationship between concavity and the
behavior of the first derivative.
All students will answer the following question by using both, a graphical display and by written
sentences to justify their response: Why is it true that if f is concave upward, then all tangent lines to f lie
bellow the curve? Students will discuss in group in order to formulate the team conclusion and then, each
team will show their conclusions to the rest of the class.
In addition, each group will receive a graph of f’ in order to sketch the function f from the graph of its
first derivative. After all the teams have had the chance to reconstruct a function from the derivative, they
will show and explain their answers to rest of the class. Students also will work in sketch the graph of the
second derivative of f (f”).
This activity enriches the students’ communication skills and teaches them how to work in groups in
order to prepare for the discussion.
Activity # 2 “Graphing with Calculus and Calculators” (Working in groups)
 Points to stress:
1. The interaction between graphical methods and the analytical aspects of calculus.
2. The use of graphing calculators for estimation of local extrema and inflection points, contrasted
with the use of calculus for precise computation of such points.
3. The need for special care when using graphing technology.
4. The use of graphing calculators as a tool to explore families of curves.
Working in groups, students will examine the functions f(x) = (X+1)5 sin (X-3) from x=-3 to x=0 (At first
glance, the maximum appears to be at x= - 1). Some maxima or minima may be very small compared to
the scale of the draw graphs. Calculus helps us determine an appropriate viewing scale. One of the focal
points of this activity will be the proper use of the graphing calculator and the correct way to enter the
information to graph a function and look for its interesting points.
Working in groups students will investigate about application of differentiation in a variety of problems,
and they will show their conclusion to rest of the class. Each group should make a written report and use
manipulative tools to facilitate student’s comprehension. The following list represent some of the
problems will be used for this activity:
Page
7
Activity #3 “Application of Differentiation” (Working in groups).
Magda de la Torre
Mater Academy Charter High School
AP Calculus BC
Syllabus




“The Shape of a Can”. Students will investigate the most economical shape for a can. They must
interpret this mean that the volume V of a cylinder is given and we need to find the height h and
radius r that minimize the cost of the metal to make the can.
“Creating a Pyramid” . Given a sphere with radius r, find the height h of a pyramid of minimum
volume whose base is a square and whose base and triangular faces are all tangent to the sphere.
Students will use the formula V= 1/3 Ah, where A is the area of the base.
“Snowball”. Assume that a snowball melts so that its volume decreases at a rate proportional to
its surface area. If it takes three hours for the snowball to decrease the half its original volume,
how much longer will it take for the snowball to melt completely?
“Speed of a Bullet Train” A high-speed bullet train accelerates and decelerates at the rate of 4ft/s2.
Its maximum cruising speed is 90 mi/h. What is the maximum distance the train travel if it
accelerates from rest until it reaches its cruising speed and then runs at that speed for 15 minutes?
Suppose the train starts from rest and must come to a complete stop in 15 minutes. What is the
maximum distance it can travel under these conditions?
Each team will create a portfolio including a summary of the main concepts and vocabulary involving in
the solution of the problems. Also there are going to “make a picture” to help the team in explain methods
and concepts behind their answers. The activity will be a contribution to expand the student’s
organizational skills and their abilities in communicating mathematics and explain solutions to problems
verbally and in written sentences.
Activity # 4 “Newton, Leibniz, and the Invention of Calculus” (Individual Project and Class
Discussion)
Student will read about the contribution of these men in one or more of the given references and write a
report on one of the following three topics. Students could include biographical details but basically the
report should show a description of their methods and notations.
1. The Role of Newton in the Development of Calculus
2. The Role of Leibniz in the Development of Calculus
3. The controversy between the Followers of Newton and Leibniz over Priority in the
Invention of Calculus.
After students complete their individual reports, the class will be organized in three different teams. Each
team will be in charge of a topic in order to formulate ideas to support the presentation and discussion in
class.
Activity #5 “The Origins of L’Hospital’s Rule”( Writing Project)
Students will write a report on the historical and mathematical origins of L’Hospital’s Rule, including
biographical details of two mathematicians involving in this mathematical discovery and a description of
the statement of this rule.
Page
8
Activity #6 “Complementary Coffee Cups” . (Application of Integrals/ The use of a graphing calculator
will be required)
Students will work in the following individual project:
Suppose you have a choice of two coffee cups, one that bends outward and one inward, and you
notice that they have the same height and the shapes fit together snugly. You wonder which cup
Magda de la Torre
Mater Academy Charter High School
AP Calculus BC
Syllabus
holds more coffee. Of course you could fill one with water and pour it into the other one but,
being a calculus student, you decide on a more mathematical approach. Ignoring the handles, you
observe that both cups are surfaces of revolution, so you can think of the coffee as a volume of
revolution.
a. Suppose the cups have height h, cup A is formed by rotating the curve x=f(y)
about the y-axis, and cup B is formed by rotating the same curve about the line
X=K. Find the value of K such that the two cups hold the same amount of coffee.
b. Based in your own measurements and observations, suggest a value for h and an
equation for x= f(y) and calculate the amount of coffee that each cup holds.
Activity #6 “Investigating about the Fundamental Theorem of Calculus” (Reading Activity and
Discussion)



“Functions defined By Integrals” By Ray Cannon
“Exploring the FTC from numerical and graphical points of view” By Mark Howell
“Using the Fundamental Theorem of Calculus in a variety of AP Questions” By Larry Riddle
These articles will help students to understand and assess the Fundamental Theorems of Calculus (FTC).
These materials give excellent examples of concepts and focus on some special calculus topic and its
connection to the FTC. This activity enriches the students’ communication skills and teaches them how to
work in groups in order to prepare for the discussion.
Activity #7 “A Rose is a Rose is a Rose”( Group Activity Exploration/ Using a graphing calculator to
experiment)
This activity introduces students to the polar curves described by mathematicians as rose curves.
Students will graph r =2 sin (nθ) for various positive integer values of n until they can state a rule for how
n determines the number of petals in the rose. Also students will find the total area enclosed by the petals
of r =2 sin (2θ), and they will repeat the process for different values of n. Students will make a conjecture
about the area of the rose generated by r =2 sin (nθ) for an arbitrary positive integer n. At the end of the
activity students will be able to write the general rule for the area of the rose generated by r =2 sin (nθ) for
any positive integer n.
Through of this activity students could interpret the effect of varying a parameter on a family of functions
by using the graphing calculator.
C5— The course teaches students how to use graphing calculators to help solve problems, experiment,
interpret results, and support conclusions.
Additionally students will receive sets of questions in AP format for each topic to help them prepare for
the AP exam. They will be required to take a Mid-Term in the AP Exam format. As a part of the students’
Page
9
preparation to accomplish the class, they should complete selected questions from the section “Chapter
Review” which appear at the end of each chapter in their textbook.
Magda de la Torre