AP Calculus BC
Course Description:
The two fundamental problems of Calculus include: 1) finding the slope of the tangent to a curve,
determined by the derivative, and 2) finding the area of a region under a curve, determined by the
integral. The focus of our study is to learn strategies for differentiation and integration of various
functions (algebraic, trigonometric, polar, piecewise, and transcendental). Differentiation and
integration are applied to problem solving across a wide spectrum of topics including physics,
engineering, economics and architecture, for example. In the process our previous study of algebra,
geometry, trigonometry, and pre-calculus is integrated into a meaningful whole as each of these fields
of mathematics significantly contributes to the study of Calculus. Numerical, graphical, analytical and
verbal approaches are each stressed as meaningful strategies for problem solving throughout the
course.
Students taking this course usually elect to take the Advanced Placement Exam for Calculus BC. The
pace of the course responds to the need to cover all material in the AP Calculus BC topic outline by
the date of the exam. We supplement the textbook with a variety of professional AP Calculus test
preparation materials as well as explorations and activities that emphasize a “discovery” model for
learning.
The goals for our students in AP Calculus are to prepare them for success in college level Calculus
courses and to prepare them for the AP Exam. The requirements for success at the collegiate level
vary widely depending on the university or college that our students will attend. Many colleges
require that students perform all homework and exams without the aid of a calculator for the first two
semesters of Calculus, while other colleges require the use of technology in their assessments.
Some colleges stress a more computational approach to Calculus, while others emphasize a more
conceptual understanding. To meet the varying needs of all of our students, we require that students
solve problems both by hand and with the use of technology. Students will use graphing calculators
as well as Geometer’s Sketchpad to explore concepts in Calculus, perform computations, determine
derivative and integral values for problem solving, and confirm the reasonableness of solutions. We
also encourage memorization of all differentiation and integration formulas, even though some
colleges allow the use of a formula sheet on exams.
To be sure that we are meeting the needs of our Calculus students we also go to great lengths to
learn about the variety of ways that Calculus is being taught at colleges and universities all over the
country. Our students come back to visit with many sample problems and exams to share. We also
visit a wide variety of university web sites that have open access to their Calculus materials in an
attempt to stay current with the expectations of professors around the country.
Textbook:
Calculus AP Edition by Ron Larson and Bruce H. Edwards, Brooks/Cole, Cengage Learning, 2010
Learning Strategies:
1. Class notes are posted by chapter and section at this teacher’s web site. Students are
responsible for printing out the notes before class each day. Access my web site at
http://www.rtmsd.org/74417169134649/site/default.asp.
2. The TI-84 plus graphing calculator is used on a regular basis by all students. Students are taught
to develop their Calculus skills by hand as well as support their understanding of Calculus through the
use of technology. Specific skills that are required using the graphing calculator include:
a. Use the y= screen to graph a variety of functions including algebraic, trigonometric,
exponential, logarithmic, and composite functions.
b. Determine the zeros, asymptotes, and maximum and minimum values of a function.
c. Use the Table feature to view coordinate pairs of a function and evaluate a function at any
given x value.
d. Use the Window feature to appropriately view a function, taking into account the range and
domain of the function.
e. Determine points of intersection of two or more functions.
f. Determine the derivative of a function at a given x-value.*
g. Determine the value of the integral within a given range of x-values.*
h. Determine the asymptotes of a function from the graph and table features.
*taught during the AP Calculus BC course.
3. The Geometer’s Sketchpad is used for presentation and exploration. In particular, the Calculus in
Motion sketches produced by Audrey Weeks are used extensively to explore the definition of the
derivative, rates of change, integration, and volumes of revolution. The Calculus in Motion sketches
also include animated solutions to all of the recent AP exam free response items. Students use these
sketches to gain a deeper understanding of the solutions to such problems.
4. Students work in small groups for problem solving on a regular basis.
5. As the reformed Calculus movement would emphasize, a great deal of discussion takes place
regarding the conceptual nature of problem solving as opposed to a more computational approach to
simply evaluating integrals and differentiating functions. Students present their problem solving
strategies to the class almost daily. Students work alone or in pairs at the white board to present the
solutions to the class. They explain their work to the class and field questions from their classmates.
Resources:
1. In addition to the Larson text, a variety of preview copies of Calculus textbooks are available in the
classroom for reference.
2. Calculus Explorations by Paul A. Foerster, Key Curriculum Press, 1998. The explorations in this
collection are used frequently to promote a more complete understanding of topics and to provide
students with the opportunity to construct their own understanding of a topic through exploration prior
to receiving a detailed lecture or discussion on the topic.
3. The released multiple choice items from the 2003 and 2008 AP Calculus BC exam are used
exhaustively in preparation for the exam.
4. All free response questions from the AP Calculus BC exam as posted on the AP Central website
dating back to 2004 are presented to the students to solve and discuss.
5. In addition to my website, feel free to use other web resources such as:
http://www.hippocampus.org/, http://www.mathvids.com/topic/mathhelp/5-calculus ,
http://www.calculus-help.com/tutorials ,
http://apcentral.collegeboard.com/apc/members/exam/exam_questions/1997.html ,
http://www.mathforum.com/
Assessment:
Students are assessed on tests administered 3-4 times per marking period, on nightly homework, and
on various Exploration activities throughout the year. Chapter tests involve open ended and multiplechoice items, in keeping with the format of the AP Exam and with formats of various universities
around the country. Throughout the 2nd semester students are given specific open-ended questions
from previous years’ exams. In preparation for the AP Exam all students complete the entire set of
2003 and 2008 released test items in multiple-choice format and selected open-ended items from
2004 to the present. Their solutions must include all mathematical calculations necessary to correctly
answer the question as well as a written explanation of their solutions. Quarterly exams are
administered following the 1st, 2nd and 3rd quarters. A final exam is administered for students not
taking the AP Exam.
After the AP Exam, students are assessed on a major project that varies from year to year.
Topics from Precalculus:
Appendix and Preparation
Covered in the summer review packet and
assessed during the first week of classes.
Review of Precalculus and Trigonometry
Skills
1) Describe intervals on a number line using interval notation.
2) Solve inequalities in one variable, two variables and absolute value.
3) Calculate the slope of a line and use it to write the equation of the line.
4) Find parallel and perpendicular lines.
5) Determine equations of circles.
6) Graph a variety of functions including polynomial, exponential, logarithmic, trigonometric,
piecewise and rational, and determine intercepts, symmetry, roots, and points of
intersection.
7) Use graphs to form a mathematical model.
8) Classify and combine functions.
9) Use correct function notation.
10) Use the distance and midpoint formulas.
11) Evaluate trigonometric functions.
12) Solve and simplify trigonometric identities.
13) Solve trigonometric equations.
14) Graph trigonometric functions.
Chapter 1: Limits and Their Properties
1.1 Preview of Calculus
Students read on their own
1.2 Finding Limits Graphically and
Numerically
Covered in Precalculus, reviewed in the first
1.3 Evaluating Limits Analytically
week of Calculus.**
1.4 Continuity and One-Sided Limits
Taught in the first week of Calculus.*
1.5 Infinite Limits
Skills
1) Find limits using a graphical, numerical, or analytical approach.**
2) Use the TI-84 graphing calculator to graph a function and determine the limit of the function
at a particular value of x. Determine the limit graphically and also numerically by using the
table of values.**
3) Construct a formal definition of a limit.**
4) Know the properties of limits.**
5) Determine limits of algebraic, trigonometric and rational functions.**
6) Apply the Squeeze Theorem to determine limits.*
7) Apply the definition of continuity.**
8) Evaluate indeterminate forms.**
9) Evaluate one-sided limits.**
10) Apply the Intermediate Value Theorem.*
11) Apply properties of continuity and properties of infinite limits.**
12) Determine infinite limits from a graph.**
13) Find vertical asymptotes.*
Chapter 2: Differentiation
2.1 The Derivative and the Tangent Line
Problem
2.2 Basic Differentiation Rules and Rates
of Change
2.3 The Product and Quotient Rules and
Higher Order Derivatives
2.4 The Chain Rule
2.5 Implicit Differentiation
2.6 Related Rates
1 day
2 days
2 days
3 days
1 day
3 days
Skills
1) Determine the tangent line to graph of a nonlinear function.
2) Demonstrate linearization of a function by using the zoon feature or table feature
of the graphing calculator.
3) Define and apply the definition of the derivative.
4) Examine the relationship of differentiability and continuity.
5) Use derivatives to find velocity, acceleration, and other rates.
6) Apply the following rules for derivatives: constant rule, power rule, constant
multiple rule, sum and difference rules, product rule, quotient rule.
7) Find derivatives of trigonometric functions.
8) Solve simple applications of the derivative.
9) Write about the meaning of the derivative as it applies to a specific application
such as velocity, acceleration, and rate of change.
10) Find higher order derivatives.
11) Compose and decompose composite functions.
12) Find derivatives using the chain rule and the general power rule.
13) Simplify derivatives via factoring, common denominators and other algebraic
manipulations.
14) Find derivatives of implicit and explicit functions.
15) Solve related rate problems.
16) Given the graph of a function, sketch the derivative of that graph. Explain how
the derivative graph is determined.
17) Given a table of values of two functions and their derivatives, students will
compute the value of the derivative of the product, quotient, or composition of the
two functions.
18) Use a graphing calculator to determine the derivative of a function at a
particular value.
19) Use Geometer’s sketchpad to graph a function and its derivative.
Chapter 3: Applications of
Differentiation
3.1 Extrema on an Interval
3.2 Rolle's Theorem, 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 Curve Sketching
3.7 Optimization
3.8 Newton’s Method
3.9 Differentials
1 day
1 day
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3 days
1 day
2 days
Skills
1) Define and locate local and absolute extrema.
2) Define and apply Rolle’s Theorem and the Mean Value Theorem. Students will
summarize in writing the conditions of Rolle’s Theorem and the Mean Value
Theorem and demonstrate whether or not the conditions have been met.
3) Define and test for intervals where a function is increasing and/or decreasing.
4) Apply the First and Second Derivative Tests. Students use a sign chart to
determine the maximum and minimum values and summarize the results in a
written statement to justify the existence of a max or min.
5) Test for points of inflection and concavity. Summarize with a written statement.
6) Evaluate a limit at infinity.
7) Determine vertical, horizontal and slant asymptotes.
8) Sketch functions using the curve sketching techniques.
9) Solve optimization applications. Write a summary statement explaining the
determination of maximum and minimum values in the context of the application.
10) Use Newton’s Method to approximate the zeros of a function.
11) Use the differentials to solve applications involving propagated error.
12) Graph a given function using the TI-84 graphing calculator to confirm maximum
and minimum values, points of inflection, x-intercepts and asymptotes.
Chapter 4: Integration
4.2 Upper, Lower and Midpoint Sums
4.6 Trapezoid Rule, Simpson’s Rule
4.1 Antiderivatives and Indefinite
Integration
4.3 Definite Integral
4.4 Fundamental Theorem of Calculus,
2nd Fundamental Theorem of Calculus
4.5 Integration by Substitution
1 day
1 day
2 days
1 day
3 days
3 days
Skills
1) Find the area of a plane region using Riemann sums: upper, lower, midpoint, left
and right sums.
2) Find the area of a plane region using the Trapezoid Rule and Simpson’s Rule.
3) Students will determine which sum best approximates the area of a plane region.
Students will compare and contrast the accuracy of the various sums and
summarize their comparison in a written statement.
4) Define the anti-derivative
5) Apply basic integration rules.
6) Find general and specific solutions of the integral. Describe general solutions as
a “family” of functions as compared to a specific solution which is a numerical
value.
7) Define and evaluate definite and indefinite integrals.
8) Use the TI-84 graphing calculator to evaluate a definite integral.
9) State and apply the Fundamental Theorem of Calculus, Mean Value Theorem,
and the Second Fundamental Theorem of Calculus.
10) Write about the meaning of the definite integral, the Mean Value Theorem and
the Second Fundamental Theorem of Calculus as each applies to particular
applications.
11) Solve application problems involving an accumulation function. Explain verbally
and in writing the meaning of accumulating area under a curve for a specific
application.
12) Find the average value of a function. Explain verbally and in writing the meaning
of the average value of the function for specific applications.
13) Recognize special integration patterns.
14) Integrate using u-substitution and/ or general power rule.
15) Given the graph of a first or second derivative, sketch the graph of the function.
Chapter 5: Logarithmic, Exponential
and other Transcendental Functions
Review of Logarithms
5.1 Differentiation of Natural Logarithm
5.2 Integration of Natural Logarithm
5.4 Exponential Functions - Derivatives
and Integrals
5.5 Bases other than e
5.6 Inverse Trig Functions and
Differentiation
5.7 Integration of Inverse Trig Functions
1 day
2 days
2 days
3 days
2 days
1 day
2 days
Skills
1) Define and know the properties of the natural logarithmic function.
2) Define e in terms of the definite integral of y = 1/x.
3) Differentiate logarithmic functions and natural log functions.
4) Apply log and natural log rules for integration.
5) Integrate trigonometric functions.
6) Define and know the properties of inverse functions.
7) Find the inverse of a function if it exists.
8) Define and know the properties of the exponential function.
9) Differentiate exponential functions.
10) Define and know the properties of bases other than e.
11) Differentiate and integrate functions of bases other e.
12) Solve application problems of bases other than e such as exponential growth
and decay and explain, in writing and verbally, the use of Calculus to solve such
problems.
13) Define and know the properties of the inverse trigonometric functions.
14) Solve equations using inverse trigonometric functions.
15) Differentiate inverse trigonometric functions.
16) Integrate functions resulting in arcsin, arcos, and arctan functions.
17) Review all basic integration formulas.
Chapter 6: Differential Equations
6.1 Slope Fields and Euler’s Method
6.2 Differential Equations: Growth and
Decay
6.3 Separation of Variables and the
Logistic Equation
6.4 First-Order Linear Differential
Equations
1 day
2 days
2 days
1 day
Skills
1) Use slope fields to approximate solutions of differential equations.
2) Solve differential equations using initial conditions to find particular solutions.
3) Solve basic differential equations using separation of variables.
4) Model growth and decay applications using exponential functions.
5) Solve first-order linear differential equations.
Chapter 7: Applications of Integration
7.1 Area of a Region Between Two Curves
7.2 Volume: Disk Method
7.2 Volume: Washer Method
7.2 Volume: Revolve around another axis,
Solids of known Cross Sections
7.3 Shell Method
1 day
1 day
1 day
2 days
1 day
7.4 Arc Length
7.4 Surface of Revolution
7.5 Work
1 day
1 day
1 day
Skills
1) Find the area of the region between two curves.
2) Use the disc and washer methods to determine the volume of a solid of
revolution.
3) Determine the volumes of figures formed by known cross-sections on a base.
4) Use the shell method to find the volume of a solid of revolution.
5) Compare the disc, washer and shell methods to determine relative
appropriateness of each method. Explain verbally and in writing which methods
are most appropriate for specific cases.
6) Find the length of arcs of smooth curves.
7) Find the area of a surface of revolution.
8) Use the TI-84 graphing calculator to set up and evaluate the correct integral to
determine a volume of revolution.
9) Find the work done by a constant or variable force.
Chapter 8: Integration Techniques
8.1 Basic Integration Formulas
8.2 Integration by Parts
8.3 Trigonometric Integrals
8.4 Trigonometric Substitution
8.5 Partial Fractions
8.6 Integration Tables
8.7 Indeterminant Form & L’Hôpital’s Rule
8.8 Improper Integrals
2 days
2 days
2 days
2 days
2 days
1 day
1 day
1 days
Skills
1) Decompose rational functions into partial fractions to facilitate integration.
2) Integrate using partial fractions. Use a tabular method where appropriate for
integration by parts.
3) Solve integrals involving powers of sine, cosine, tangent and secant.
4) Solve integration problems using tables.
5) Solve integration problems using trigonometric substitutions.
6) Identify indeterminate forms.
7) State and apply L’Hopital’s Rule to integration of indeterminate forms.
7) Recognize various type of improper integrals and perform integration.
8) Solve various types of integrals by fitting an integrand to one of the basic
integration rules.
Chapter 9: Infinite Series
9.1 Sequences
9.2 Series and Convergence
9.3 The Integral Test and p-Series
9.4 Comparison 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
1 day
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2 days
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2 days
Skills
1) List the terms of a sequence and a formula for the nth term.
2) Determine whether a sequence is convergent or divergent.
3) Define a convergent infinite series. Apply the properties of an infinite geometric
series.
4) Use the nth Term Test for Divergence of an infinite series.
5) Determine whether an infinite series is convergent or divergent using the Integral
Test.
6) Determine whether a series is convergent or divergent using the Direct
Comparison Test and the Limit Comparison Test.
7) Determine whether an infinite series converges and if so, approximate the sum of
an alternating infinite series using the Alternating Series Test. Determine if a
convergent series is absolutely or conditionally convergent.
8) Rearrange an infinite series determine a different sum.
9) Use the Ratio Test and the Root Test to test for convergence or divergence of a
series.
10) Find Taylor and Maclaurin polynomial approximations of elementary functions.
11) Define a power series. Find the radius and interval of convergence, and
endpoint convergence of a power series.
12) Differentiate and integrate a power series.
13) Represent a function using a geometric power series.
14) Use a basic list of Taylor series to find other Taylor series.
Chapter 10: Conics, Parametric Equations and Polar Coordinates
10.2 Plane Curves and Parametric
Equations
2 days
10.3 Parametric Equations and Calculus 2 days
10.4 Polar Coordinates and Polar
Graphs
2 days
10.5 Area and Arc Length in Polar
Coordinates
2 days
Skills
1) Given a set of parametric equations, sketch the graph.
2) Eliminate the parameter in a set of parametric equations and write the equation
using rectangular coordinates.
3) Write the standard form of the equation of a conic section given the set of
parametric equations representing that conic section.
4) Determine the slope and concavity of a set of parametric equations for a given
parameter.
5) Find the equation of the line tangent at a given parameter.
6) Determine the arc length of the curve defined by a set of parametric equations
over a given interval.
7) Write equations and coordinates in polar form and rectangular form.
8) Find the slope of a tangent line to a polar graph.
9) Classify various types of polar graphs.
10) Determine the area of a region bounded by a polar graph.
11) Find the points of intersection of two polar graphs.
12) Determine the arc length of a polar graph.
13) Find the area of a surface of revolution in polar form.
14) Write polar equations of conics.
Chapter 12: Vector-Valued Functions
12.1 Vector-Valued Functions
12.2 Differentiation and Integration of
Vector-Valued Functions
12.4 Tangent Vectors and Normal
Vectors
2 days
2 days
2 days
Skills
1) Sketch a curve given by vector-valued functions.
2) Apply the concepts of limits and continuity to vector-valued functions.
3) Use a vector-valued function to analyze projectile motion and velocity and
acceleration.
4) Find a unit tangent vector at a point on a curve. Find the tangential and normal
components of acceleration.