AP Calculus BC (1202320)
Curriculum and Pacing Guide - Mathematics
AP Calculus BC (1202320)
(Every day period)
Revised June 2011
AP Calculus BC (1202320)
I.
AP Calculus AB Course Overview and Philosophy
In developing and applying the topics of differential and integral calculus, students have the opportunity to solidify all their previous high school
math experience, since all the major ideas from the prerequisite math courses are necessary for calculus success. These prerequisite courses
include algebra I and II, geometry, and precalculus - which consist of the study of transcendental functions and their graphs, with a strong
emphasis on trigonometry and its applications. Consequently, there are high expectations which emphasize concepts reinforced through
development and application rather than just procedure. The goal is that the successful student of Advanced Placement Calculus will validate
his/her success with adequate performance on the AP Exam and place out of a comparable college course. However, beyond this, it is desired that
the successful student appreciates the high level of their mathematical success and at the same time realizes that this accomplishment is not an
end in itself, but a beginning point for many other opportunities in various fields of study and work. The hope is that through the study of this
course, students gain a genuine appreciation for the intricacies and beauty of mathematics, as well as proficiency with the subject matter of
calculus.
II. Use of Technology
Students learn to use the TI-89 graphing Calculator in the following ways:
1. The Use of Tables to analyze data and behavior of functions.
2. The use of graphs to determine behavior of functions and their limits.
3. The use of various types of solution capabilities of the calculator for various functions.
4. The use of the symbolic manipulation capabilities of the calculator in order to determine information about functions for which they have no
analytical techniques.
AP Calculus BC (1202320)
III. Strategies for Success
Since communication, both verbal and written is a major component of the course, the main classroom focus, besides guided practice for
instruction of techniques, is cooperative group work for learning concepts and problem solving.
1. Almost every section of our adopted text, which is an AP Edition textbook, includes problem sets of Standardized Test Questions, Explorations,
and Extensions. I use these exercises after students have done an assignment on the section, beginning at Chapter 1. Students discuss and defend
their answers before submitting one set of answers per group. Student interaction is a necessity if they are to know the depth to which they
understand a concept. Having to explain to each other and to the class gives the opportunity to do this, whether they are questioning or explaining.
2. After 2 or 3 sections of each Chapter, the text includes a Quick Quiz for A.P. Preparation. These along with College board released exam items
are invaluable in giving students the “A P Experience” throughout the course.
3. Students accumulate a set of note cards on important theorems as they progress through the course. In review for the AP test, students are
randomly given a theorem to present and explain on the spot, so they know them well for justifications.
4. The goal for the AP Calculus student is to think creatively in solving problems that apply “familiar concepts in unfamiliar settings” and those
which combine concepts. Also, students must learn to solve multi-part problems in which they often have to explain not only what they have done,
but the theory behind why it works. In my view, the group setting is essential for brain-storming to acquire attack skills on such problems and to
gain confidence in solving problems that seem totally unfamiliar.
5. Students gradually become independent problem-solvers through the following progression. When the AP free response questions are first
presented, students are in groups of 4 which change 3 times during the first term. At this time, questions are used a few at a time and grouped by
the topic being studied. During the second term, students work with one partner, and then finally, for the last two weeks before the AP test, each
works alone. At this time, the questions are presented in their original test format, beginning with the most recent and working back in time. One
week before the actual exam, students also have the opportunity to take a complete released exam, and then given the opportunity to score it,
according to the released rubric, and discuss the solutions.
6. District will provide licenses for all students to access Study Island website located at http://www.studyisland.com/. AP Calculus can be found
under the US Programs tab on the left side of the webpage. We will use SI in the suggested approaches column to note Study Island.
IV.
Course Outline for AP Calculus BC
The following is from the document, “Curriculum and Pacing Guide – Mathematics: Calculus BC.” This document is relatively dynamic in that is
regularly reviewed and updated. The columns in the pacing of each unit provide the general time line, sections of the text book, topics, and major
concepts covered along with suggested approaches and activities for some of the concepts. The TI-89 graphing calculator is an integral part of
concept investigation for this course and one is provided to the students who do not own one. However, it is not used for its algebraic
manipulation capabilities, except as a check during term 2 after the mastery of analytical skills.
AP Calculus BC (1202320)
Curriculum and Pacing Guide - Mathematics
AP Calculus BC (1202320)
Revised June 2011
Students who successfully complete this course will receive one credit AP Calculus BC and will take the AP Calculus BC Exam.
1.
Review of the concepts from algebra and pre-calculus concerning relations /functions will probably be necessary to some extent. These
include linear, trigonometric, exponential, logarithmic, the generic functions, and piecewise functions. Topics relative to these are; slopes, domain
and range, symmetry, intercepts, intersections, and the graphs of these relations/functions. The graphing calculator (TI-89) is used to reinforce
these concepts, referring to the solving, table, and graphing capabilities of the calculator.
Chapter 1: Prerequisites for Calculus
Section
Topic
Days
*
1.1
Lines
*
1.2
Functions and graphs
*
1.3
Exponential Functions
*
1.5
Functions and
Logarithms
*
1.6
Trigonometric Functions
Concepts
Increments, Slope, Point-slope Form,
Slope-intercept, Standard Form
(Skip Regression Analysis)
Function, Even and Odd Functions,
Composition, Piecewise Functions,
Recognizing Graphing Failure
Exponential Function, Exponential
Growth and Decay, Half Life
One to One, Logarithms Functions,
Inverse Functions, Identity Functions,
Change of Base Formula
Periodic Function, Period,
Trigonometric Functions, Inverse
Trig, Even and Odd
Suggested Approaches
(T) Technology, (V) Verbal, (G)
Graphical, (N) Numerical
(NV)
(TGV) Graphing Features of
calculator, Graph Viewing Skills
(TG) Graphing and Table Features
(VGN)
(TVGN)
AP Calculus BC (1202320)
2.
The concepts of a limit, as the independent variable approaches a particular value, are addressed numerically, analytically, and graphically.
This involves developing the ideas of the existence of a limit. The issues are, if the limit exist, what is it, and if it does not, why not? This
necessitates the understanding of “broken graph” oscillating, and asymptotic behaviors. Examination of limits, including one sided limits, is done
using the various algebra techniques for the types of functions, the properties of limits and special techniques for rational trig related functions.
Continuity and the Intermediate Value theorem and their applications are also part of this unit. Areas of application in this unit are finding equations
of tangent lines (and normal lines) at a point and beginning motion problems.
Chapter 2: Limits and Continuity
Days
Section
Topic
Concepts
1
2.1
Rates of Change and Limits
Average v. Instantaneous Speed,
Properties of Limits, One sided v.
Two sided Limits, Sandwich Theorem
1
2.2
Limits Involving Infinity
Horizontal Asymptote, Properties of
Limits as x approaches infinity,
Vertical Asymptote, End Behavior
Model
1
2.3
Continuity
Continuity at a Point, Properties of
Continuous Functions, Composition of
Continuous Functions,
Discontinuities, Intermediate Value
Theorem, Jump Discontinuity
2
2.4
Rates of Change and Tangent
Lines
Average Rate of Change, Slope of a
Curve at a Point, Tangent Line,
Normal to a Curve,
1
1
7
Review
Test
Total
Days
Suggested Approaches
(T) Technology, (V) Verbal, (G) Graphical, (N)
Numerical (SI) Study Island
(T) www.calculus-help.com
(T) table features of calculator
(T) trace features of calculator
(T) Chapter 1: Lessons 1-3 http://www.calculushelp.com/tutorials/
Supplement Algebraic Limit Techniques from
Larson
(SI) 1.a. / 1.b.
(T) Chapter 1: Lesson 4 http://www.calculushelp.com/tutorials/
(T) www.calculus-help.com
(G) Graphical
(SI) 1.c / 1.d.
(T) Chapter 1: Lesson 5-6 http://www.calculushelp.com/tutorials/
(T) www.calculus-help.com
(G) Graphical
(VN)
(SI) 1.c.
(VN)
(T) http://www.calculus-help.com/the-differencequotient/
AP Calculus BC (1202320)
3.
The derivative is developed in this unit which involves the geometric interpretation of the tangent line at a point, leading to the limit definition
of the derivative of a function. The limit definition is used both to find derivatives at a point and to develop the basic derivative rules. Basic
derivative rules are used for first, second, and higher order derivatives and also for implicit derivatives. The derivative is also investigated in the
relationship between position, velocity, acceleration and jerk. Inverse functions are addressed with particular attention given to the natural
exponential function as the inverse of the natural logarithm function. The derivatives of the natural exponential function and the natural log
function are given. Logarithmic and exponential functions of any base are also given along with their corresponding derivatives. Various derivative
techniques are developed for natural logarithm functions techniques for rational form functions, including trigonometric.
AP Calculus BC (1202320)
Chapter 3: Derivatives
Days Section
Topic
Concepts
2
3.1
Derivative of a
Function
Derivative, Derivative at a Point, Notation, Relationship
between the Graphs of F and F Prime, One-Sided
Derivatives
2
2
3.2
3.3
Differentiability
Rules for
Differentiation
Non-Differentiability, Local Linearity
Differentiation Rules, Higher Order Derivatives
3
3.4
2
3.5
Instantaneous Rates of Change, Instantaneous Velocity,
Speed, Acceleration, Linear Motion, Free-Fall Motion
Derivative of Trig Functions, Jerk,
2
3.6
Velocity and Other
Rates of Change
Derivatives of
Trigonometric
Functions
Chain Rule
2
3.7
2
3.8
Implicit Function, Implicit Differentiation, Power Rule
for Rational Powers
Inverse, Derivatives of Inverse Trigonometric Functions
3
3.9
Implicit
Differentiation
Derivatives of Inverse
Trigonometric
Functions
Derivatives of
Exponential and
Logarithmic Functions
2
1
23
Review
Test
Total
Derivative of Composite Functions, Chain Rule
(Skip Slopes of Parameterized Curves)
Derivative of Exponential Functions base e and base a,
Derivative of Log Functions base e and base a
Suggested Approaches
(T) Technology, (V) Verbal, (G) Graphical,
(N) Numerical
(SI) Study Island
(VGN) (T)
http://www.ima.umn.edu/~arnold/graphics.ht
ml
(SI) 2.a.
(VGN)
(VN)
(T) Chapter 2: Lesson 2-4
http://www.calculus-help.com/tutorials/
(T) www.calculus-help.com
(SI) 2.b. / 2.c. / 2.h.
(VG)
(SI) 2.i.
(VG) Graphical
(T) www.calculus-help.com
(SI) 2.d.
(VN)
(T) http://www.calculus-help.com/the-chainrule/
(SI) 2.e.
(VN)
(SI) 2.k.
(VGN) Supplement Theorem 5.9 From Larson
for Derivative of Inverse Functions
(SI) 4.c.
(VN) Supplement from Larson Implicit
Differentiation with Exponentials and
Logarithmic
(SI) 4.a. / 4.b.
AP Calculus BC (1202320)
4. This unit continues the application of derivatives. Also three principle theorems are developed and used – the Extreme Value Theorem, Rolle’s
Theorem, and the Mean Value Theorem. First and second derivatives are used to determine for a given function the critical values, intervals of
increase and decrease, relative maxima and minima, points of inflection, and intervals concave up and concave down. This application is done with
and without graphing calculators. Included with this application is the examination of the relationships of the graphs of a function, the graph of its
1st derivative, and the graph of its 2nd derivative and through the use of tables. The very useful and important derivative application of solving
optimization problems, as well as linear approximations, differentials, and related rates are in this unit.
Chapter 4: Applications of Derivatives
Days
Section
Topic
Concepts
2
4.1
Application of Derivatives
1
4.2
Mean Value Theorem
2
4.3
Connecting f prime and f
double prime with the graph
of f
Absolute Extrema, Local Extreme Values, Critical
Point, Extreme Value Theorem,
Mean Value Theorem, Increasing Decreasing
Intervals, Antiderivative, Rolle’s Theorem,
Monotonic Functions
First Derivative Test, Definition of Concavity,
Concavity Intervals, Point of Inflection, Second
Derivative Test,
3
4.4
Modeling and Optimization
Max-Min Problems, Optimization
1
4.5
Linearization
3
4.6
Related Rates
Linear Approximation, Differentials, Absolute
Relative and Percent Change,
(Skip Newton’s Method)
Related Rates
2
1
15
Review
Test
Total
Suggested Approaches
(T) Technology, (V) Verbal, (G)
Graphical, (N) Numerical
(SI) Study Island
(VGN)
(VN)
(G)http://www.univie.ac.at/future.m
edia/moe/tests/diff1/ablerkennen.ht
ml
(G)http://archives.math.utk.edu/visu
al.calculus/3/graphing.3/index.html
(T) www.unitedstreaming.com :
“applications of derivatives”
(SI) 2.j.
(V)
(SI) 2.g.
(VN)
(VN)
AP Calculus BC (1202320)
5.
The definite integral is developed by first examining estimates of the areas of plane regions as sums of rectangles constructed by using a
partitioning of an interval and the right, left, midpoint or any point of the partition. The definition of a definite integral can then be given as a limit
to an infinite Riemann Sum, the exact area of the plane region. The Fundamental Theorem of Calculus is developed along with the Mean Value
Theorem for Integrals leading to the Average Value of a function on an interval. The Second Fundamental Theorem is also given. The main
applications here are areas of simple plane regions, and average-value-of-a-function problems. This unit also includes estimation of plane regions
by using trapezoidal approximation.
Chapter 5: The Definite Integral
Days
Section
Topic
Concepts
2
5.1
Estimating with Finite
Sums
Distance Traveled, Rectangular Approximation
Method
3
5.2
Riemann Sums
Definite Integral, Integrability,
5
5.3
3
5.4
Definite Integrals and
Antiderivatives
Fundamental Theorem
of Calculus
Integral Properties, Average Value, Mean Value
Theorem for Integrals
Fundamental Theorem of Calculus Part 1 and 2,
Connection to Area
1
5.5
Trapezoidal Rule
Trapezoid Rule
(Skip Other Algorithms, Simpson’s Rule and Error
Bounds)
1
1
16
Review
Test
Total days
Suggested Approaches
(T) Technology, (V) Verbal, (G)
Graphical, (N) Numerical
(SI) Study Island
(VN)
http://designatedderiver.wikispaces.co
m/Integrals+Def+and+Ind
(Click on: Calc-SM-Approximating
Under the Curve. Docx)
(SI) 3.e.
(T, N) Graphing calculator: table
feature
(SI) 3.c.
(VN)
(SI) 3.d.
(VGN)
(T)
http://clem.mscd.edu/~talmanl/HTML/
FTOC.html
(VN)
AP Calculus BC (1202320)
6.
This unit introduces slope fields, and the solving of differential equations, leading to the concept of an antiderivative. Indefinite integrals are
solved through various techniques including u du substitution and pattern recognition. Integral techniques are expanded to include integration by
parts, trig identities (including using identities to find integrals of powers of trig functions), and partial fractions. Exponential growth and decay
model is developed from integration of separation of variables. The Logistic Growth model is also included.
Chapter 6: Differential Equations and Mathematical Modeling
Days
Section
Topic
Concepts
4
6.1
Slope Fields
Slope Fields, Differential
Equations, Euler’s Method
3
6.2
Antidifferentiation by
Substitution
2
6.3
Antidifferentiation by Parts
4
6.4
Exponential Growth and Decay
4
6.5
Logistic Growth
Properties of Indefinite Integrals,
Indefinite Integrals, Leibniz
Notation, u du substitution
Product Rule in Integral Form,
Solving for the Unknown Integral,
Tabular Integration, Inverse
Trigonometric and Logarithmic
functions
Separable Differential Equation,
Exponential Change, Continuous
and Compound Interest, Modeling
Growth and Decay
Logistic Differential Equations,
Logistic Curve, Logistic Growth
Model (Skip Partial Fractions)
2
1
20
Review
Test
Total
Days
Suggested Approaches
(T) Technology, (V) Verbal, (G) Graphical,
(N) Numerical (SI) Study Island
(VN)
(G) www.apcentral.com :
Teacher Resource: Slope fields
(SI) 3.b. / 4.d.
(VN)
(SI) 5.a.
Supplement from Larson
(VN)
(VN)
AP Calculus BC (1202320)
7.
This unit involves the interpretation of the integral as an accumulator and applications of finding areas and volumes. The definite Integral is
used to find the areas of regions between curves using all types of functions. These are areas on an interval, areas between curves including
curves with more than two intersections, also incorporating change of axis. The next application is volume beginning with three diminution shapes
also knows as cross sections. Also included are volumes of rotation using the disk, and washer method incorporating the change of axis.
Chapter 7: Applications of Definite Integrals
Days
Section
Topic
Concepts
3
7.1
Integral as Net Change
1
7.2
Areas in the Plane
3
7.3
Volume
Cross Sections, Washers
(Skip Shells)
2
7.4
Lengths of Curves
Optional
7.5
Applications from Science
and Statistics
Review
Test
Sine Wave, Length of a smooth curve,
Vertical tangents, Corners, and Cusps
Work, fluid force, fluid pressure, normal
probabilities
1
1
11
Total
Days
Displacement, Consumption over time,
Net Change from Data
Area between Curves, Area Enclosed by
Intersecting Curves, Integrating in respect
to y, Integration by Geometric Formula
Suggested Approaches
(T) Technology, (V) Verbal, (G)
Graphical, (N) Numerical
(SI) Study Island
(VN)
(VGN)
(T) Calculus in Motion (Fee)
(T)
http://clem.mscd.edu/~talmanl/MathAni
m.html
(SI) 3.f.
(VGN)
(T) Calculus in Motion (Fee)
Supplement from Larson for Disk and
Washer Methods
(T)
http://clem.mscd.edu/~talmanl/MathAni
m.html
(SI) 3.g.
AP Calculus BC (1202320)
9.
This unit starts with developing the concepts and notation of sequences followed by what it means for the sequence to converge or diverge.
Attention is given to types of sequences such as bounded, monotonic and oscillating. Limits are briefly reviewed focusing on indeterminate forms
and a reminder of the methods used for improper integrals is also included. Previous math courses have dealt with part of these concepts.
The main purpose for the time spent dealing with the above is to set a foundation for the study of Series. The concepts of Series and the
notation can now be developed and explained with the goal that students understand a Series is a sequence of partial sums. Whether or not this
sequence of partial sums has a limit determines the convergence or divergence of the Series. This is begun by examining geometric and
telescoping Series and the decision process used for convergence and divergence. The repeating decimal is a primary example for a convergent
geometric Series. Other Series are then addressed such as harmonic and p-series and alternating including alternating series estimation. Various
test for convergence and divergence are developed and used including the integral test, p-series convergence, direct comparison, limit comparison,
alternating series test, ratio, and root test.
Next is a transition to polynomial approximations of elementary functions and the Taylor and MacLaurin Series. The polynomial approximations
are built for elementary functions such as sin x, cos x, e^x, and 1/(1-x), first centered at 0 (MacLaurin) and then centered at other values, “c”,
(Taylor). Use of the TI – 89 graphing utilities is indispensable at this juncture for students to understand the approximation process. These
polynomials are used to find a function approximate value and Taylor’s Theorem (Lagrange remainder) used involving the accuracy of the
approximation. We then address general power series of functions and the examination of their domains, convergence, and radii and intervals of
convergence. The rest of this unit goes into the manipulation of power series and differentiation and integration of power series.
AP Calculus BC (1202320)
Chapter 9: Sequences/Infinite Series
Days
Section
Topic
Concepts
Displacement, Consumption over time,
Net Change from Data
Infinites Series, Partial Sums, Converge,
Diverge, Geometric Series, Interval of
Convergence, Power Series, Centered at
x=0, centered at x=a
Taylor Polynomial, Maclaurin and
Taylor Series, Taylor Series Gernated at
x=a, Maclaurin Series Table
Truncation Error, Taylor’s Formula,
Remainder, La Grange Form, La Grange
Error Bounds, Remainder Estimation
Convergence, Radius of Convergence,
Interval of Convergence, Nth term Test,
Direct Comparison Test, Ratio Test,
Absolute Convergence, Telescoping
Series
Integral Test, P-Series, Harmonic Series,
Limit Comparison Test, Alternating
Series Test, Improper Integrals
2
8.1
Sequences
4
9.1
Power Series
4
9.2
Taylor Series
4
9.3
Taylor’s Theorem
3
9.4
Radius of Convergence
5
9.5
Testing Convergence at
Endpoints
2
1
25
Review
Test
Total
Days
Suggested Approaches
(T) Technology, (V) Verbal,
(G) Graphical, (N) Numerical
(VN)
(TVN)
(VGN)
(VN)
(VN)
(VN)
AP Calculus BC (1202320)
8. In this unit we use L’Hopital’s Rule to calculate limits of fractions whose numerators and denominators both approach zero or are unbounded. We
also use L’Hopital’s rule to compare the rates at which functions grow as x becomes large. We also evaluate definite integrals of continuous
functions and bounded functions with a finite number of discontinuities on finite close intervals.
Chapter 8: L’Hopital’s Rule, Improper Integrals, and Partial Fractions
Days
Section
Topic
1
1
2
1
1
6
8.2
8.3
8.4
L’Hopital’s Rule
Relative Rates of Growth
Improper Integrals
Review
Test
Total
Days
Concepts
L’Hopital’s Rule, Indeterminate Form
Transitivity of Growing Rates
Infinite Discontinuities, Integral
Comparison Test
Suggested Approaches
(T) Technology, (V) Verbal,
(G) Graphical, (N) Numerical
(VGN)
(VGN)
(VN)
AP Calculus BC (1202320)
10. In this unit we analyze calculus in three kinds of two variable contexts, parametric, vector, and polar to analyze new kinds of curves. We also
analyze motion that does not proceed along a straight line. This can be done using single variable calculus in some different and interesting ways.
Chapter 10: Parametric, Vector and Polar Functions
Days
Section
Topic
4
10.1
Parametric Functions
3
10.2
Vectors in the Plane
4
10.3
Polar Functions
2
1
14
Concepts
Parametric Differentiation Formulas, Arc
Length
Two Dimensional Vectors, Direction
Angle, Magnitude, Components, Vector
Operations, Unit Vector, Parallelogram
Representation, Displacement, Distance,
Velocity and Acceleration Vectors
Polar Coordinates, Pole, Polar Curves,
Conversion Formulas, Area enclosed by
Polars
Suggested Approaches
(T) Technology, (V) Verbal,
(G) Graphical, (N) Numerical
(VGN)
(VN)
(VGN)
Review
Test
Total
Days
Total Instructional days: 137
The remaining time is for AP Test Review consisting of discussion of release multiple choice and free response questions.
V.
Major Text.
Taken from:
Finney, Damana, Waits and Kennedy. Calculus: Graphical, Numerical, Algebraic (AP Edition), 3rd Ed. Upper Saddle River, New Jersey, 2007
Finney, Damana, Waits and Kennedy. Calculus: A Complete Course, 2nd Ed. Upper Saddle River, New Jersey, 2000
IV.
Supplementary Materials
Calculus in Motion (Must purchase program and must also have geometry sketchpad).
Larson, Ron, Robert P. Hostettler, and Bruce H. Edwards. Calculus with Analytic Geometry. 7th ad. Boston , New York: Houghton Mifflin, 2002
College Board AP Calculus Released Exams.
Study Island: http://www.studyisland.com/. AP Calculus can be found under the US Programs tab on the left side of the webpage. We will
use SI in the suggested approaches column to note Study Island. District will provide licenses for all students.