Curriculum and Pacing Guide - Mathematics
AP Calculus AB (1202310)
(Every day period)
Revised June 2011
AP Calculus AB (1202310)
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, including numerical derivatives and definite integrals.
III. Strategies for Success
AP Calculus AB (1202310)
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. These exercises are used 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 in free response items.
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. For this, 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 which change multiple 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 denote Study Island.
IV. Course Outline for AP Calculus AB
The following is from the document, “Curriculum and Pacing Guide – Mathematics: Calculus AB.” 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 during term 1; then during term 2, after the mastery of analytical skills, it is used for checking and determining numerical
derivatives and definite integrals.
AP Calculus AB (1202310)
Curriculum and Pacing Guide - Mathematics
AP Calculus AB (1202310)
Revised June 2011
Students who successfully complete this course will receive one credit AP Calculus AB and will take the AP Calculus AB 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
1.1
Lines
1
1.2
Functions and graphs
1
1.3
Exponential Functions
1
1.5
Functions and
Logarithms
2
1.6
Trigonometric Functions
1
1
8
Review
Test
Total Days
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 AB (1202310)
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
2
2.1
Rates of Change and Limits
Average v. Instantaneous Speed,
Properties of Limits, One sided v.
Two sided Limits, Sandwich Theorem
2
2.2
Limits Involving Infinity
Horizontal Asymptote, Properties of
Limits as x approaches infinity,
Vertical Asymptote, End Behavior
Model
2
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,
2
1
11
Review and Quiz
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 AB (1202310)
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 differentiation 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 investigated along with their corresponding derivatives. Various
derivative techniques are developed for natural logarithm functions as needed for techniques for rational form functions, including trigonometric
functions.
Chapter 3: Derivatives
Days Section
AP Calculus AB (1202310)
Topic
Concepts
Derivative, Derivative at a Point, Notation,
Relationship between the Graphs of F and F
Prime, One-Sided Derivatives
Non-Differentiability, Local Linearity
Differentiation Rules, Higher Order Derivatives
3
3.1
Derivative of a Function
2
3
3.2
3.3
Differentiability
Rules for Differentiation
3
3.4
Velocity and Other Rates
of Change
3
3.5
Derivatives of
Trigonometric Functions
1
3
3.6
Mid-Chapter Test
Chain Rule
2
3.7
Implicit Differentiation
2
3.8
3
3.9
1
1
27
Review
Test
Total
Instantaneous Rates of Change, Instantaneous
Velocity, Speed, Acceleration, Linear Motion,
Free-Fall Motion
Derivative of Trig Functions, Jerk,
Suggested Approaches
(T) Technology, (V) Verbal, (G) Graphical, (N)
Numerical
(SI) Study Island
(VGN) (T)
http://www.ima.umn.edu/~arnold/graphics.html
(SI) 2.a.
(VGN)
(VN)
(T) Chapter 2: Lesson 2-4 http://www.calculushelp.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.
Derivative of Composite Functions, Chain Rule,
Finding dy/dx Parametrically
(Skip Derivatives of Parametric Defined
Functions)
Implicit Function, Implicit Differentiation, Power
Rule for Rational Powers
(VN)
(T) http://www.calculus-help.com/the-chain-rule/
(SI) 2.e.
Derivatives of Inverse
Trigonometric Functions
Inverse, Derivatives of Inverse Trigonometric
Functions
Derivatives of Exponential
and Logarithmic Functions
Derivative of Exponential Functions base e and
base a, Derivative of Log Functions base e and
base a
(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.
(SI) 2.f.
(VN)
(SI) 2.k.
AP Calculus AB (1202310)
4. This unit continues the study of derivatives by investigating their applications. 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 concavity intervals. 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 their relationship 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 also investigated in this
unit.
Chapter 4: Applications of Derivatives
Days
Section
Topic
4
4.1
Application of Derivatives
2
4.2
Mean Value Theorem
4
4.3
Connecting f prime and f
double prime with the
graph of f
4
4.4
3
4.5
Modeling and
Optimization
Linearization
6
4.6
Related Rates
2
1
26
Review and Quiz
Test
Total
Concepts
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,
Max-Min Problems, Optimization
Linear Approximation, Differentials, Absolute
Relative and Percent Change,
(Skip Newton’s Method)
Related Rates
Suggested Approaches
(T) Technology, (V) Verbal, (G)
Graphical, (N) Numerical
(SI) Study Island
(VGN)
(VN)
(G)http://www.univie.ac.at/future.media
/moe/tests/diff1/ablerkennen.html
(G)http://archives.math.utk.edu/visual.c
alculus/3/graphing.3/index.html
(T) www.unitedstreaming.com :
“applications of derivatives”
(SI) 2.j.
(V)
(SI) 2.g.
(VN)
(VN)
AP Calculus AB (1202310)
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 is then given as a limit of 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 and 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 by approximation, including the analysis of whether the approximation is an over or under estimate. This unit also
includes estimation of plane regions by trapezoidal approximation.
Chapter 5: The Definite Integral
Days
Section
Topic
Concepts
3
5.1
Estimating with Finite
Sums
Distance Traveled, Rectangular Approximation
Method
4
5.2
Riemann Sums
Definite Integral, Integrability,
6
5.3
4
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
3
5.5
Trapezoidal Rule
Trapezoid Rule
(Skip Other Algorithms, Simpson’s Rule and Error
Bounds)
2
1
23
Review and Quiz
Test
Total
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 AB (1202310)
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. 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
3
6.1
Slope Fields
Slope Fields, Differential
Equations (Skip Euler’s Method)
3
6.2
Antidifferentiation by
Substitution
1
6.3
Antidifferentiation by Parts
4
6.4
Exponential Growth and Decay
3
6.5
Logistic Growth
Properties of Indefinite Integrals,
Indefinite Integrals, Leibniz
Notation, u du substitution
Integral forms whose
antiderivatives are inverse trig
functions (Skip Antidifferentiation
by Parts)
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
16
Review and Quiz
Test
Total
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 AB (1202310)
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 dimensional shapes of
uniform cross sections. Next included are volumes of rotation using the disk, and washer method, also incorporating the change of axis.
Chapter 7: Applications of Definite Integrals
Days
Section
Topic
4
7.1
Integral as Net Change
3
7.2
Areas in the Planes
7
7.3
Volume
2
1
17
Review and Quiz
Test
Total
Days
Concepts
Suggested Approaches
(T) Technology, (V) Verbal, (G) Graphical,
(N) Numerical
(SI) Study Island
(VN)
Displacement, Consumption over
time, Net Change from Data
Area between Curves, Area Enclosed
(VGN)
by Intersecting Curves, Integrating in
(T) Calculus in Motion (Fee)
respect to y, Integration by Geometric
(T)
Formula
http://clem.mscd.edu/~talmanl/MathAnim.html
(SI) 3.f.
Cross Sections, Washers
(Skip Shells)
(VGN)
(T) Calculus in Motion (Fee)
Supplement from Larson for Disk and Washer
Methods
(T)
http://clem.mscd.edu/~talmanl/MathAnim.html
(SI) 3.g.
Total Instructional days: 128
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
VI.
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.