AP Calculus AB
Course Title
Advanced Placement Accelerated
Student Level
Course #: 2102
Grade: Senior High
Time/Credit: 27 Weeks (1 Block)/1.5 Credits
Development/Revision Dates: Summer, 1997/Summer, 2000
Central Bucks School District
315 West State Street
Doylestown, PA 18901
Adopted by School Board
Table of Contents
Central Bucks Student Learning Goals and State/CB Standards
............................................ p. 1
Course Level Goals and Student Learning Outcomes
............................................................ p. 2
Course Level Assessment (Description) ...............................................................................
p. 3
Content and Instructional Times
............................................................................................. p. 4-5
Units ......................................................................................................................................
p. 6-19
Instructional Materials
........................................................................................................... p. 20
Course Description
This course provides a complete study of differential and integrated calculus. It is
designed to prepare
students for the Advanced Placement Calculus AB Examination. The course outline
completes the
recommended topics described by the College Board at a fast and intense pace to
guarantee time for
practicing testing exercises.
Students selecting this course should plan to take the Advanced Placement Calculus
AB
Examination. Weighted-grade course.
Prerequisite: PreCalculus/Trigonometry 3 or Advanced Mathematics Concepts, B grade
or better.
Committee Members: Ellen Christoff
Carolyn Kidd
Gail Tannery
John Wexler
Paul Wilson
Sarah Eustis, Secondary Math Supervisor
The Central Bucks Schools will provide all students with the academic and
problem-solving skills essential for personal development,
responsible citizenship, and life-long learning.
Central Bucks Student Learning Goals
Central Bucks Academic Standards
Mathematics
Central Bucks schools shall teach, challenge and support every student to realize his or
her maximum
potential and acquire the knowledge and skills needed to:
2.1 Understand and Apply Concepts Related to Numbers, Number Systems, and
Number
Relationships
A. Types of numbers (whole, prime, irrational, complex, etc.)
B. Equivalent forms (fractions, decimals, percents, etc.)
2.2 Understand and Apply Concepts Related to Computation and Estimation
A. Basic functions (addition, subtraction, multiplication, and division)
B. Reasonableness of answers
C. Calculators
2.5 Select and Communicate Appropriate Problem Solving Strategies
A. Problem solving strategies
B. Representing problems in various ways
C. Interpreting results
2.8 Use Algebraic Methods to Describe Patterns and Model Functions
A. Equations
B. Patterns and functions
2.9 Understand and Apply the Space and Dimensionality Concepts of Geometry
A. Shapes and their properties
B. Using geometric principles to solve problems
C. Three dimensional geometry
2.10 Understand and Apply Concepts Related to Trigonometry
A. Right triangles
B. Angle measurements
C. Measuring and computing with triangles
D. Calculators and graphing calculators
2.11 Understand and Apply Concepts Related to Calculus
A. Study and application of limits
B. Comparing quantities and values
C. Graphing rates of change
D. Continuing patterns infinitely
-1- Revised 3/20/2001
Knowledge
Base
Complex
Thinker
Self-Directed
Learner
Effective
Communicator
Informed
Citizen
Collaborative
Worker
Quality
Producer
Course/Student Outcomes
After completing AP Calculus AB, students will be able to:
• apply the concept of function and represent the concept algebraically, graphically,
verbally,
and through the use of charts
• understand the use and application of the derivative
• understand the use and application of the definite integral
• understand the Fundamental Theorem of Calculus and the relationship between the
derivative and the definite integral
• use correct calculus techniques to solve physical problems
• use technology to help solve problems, interpret results, and verify conclusions
• develop an appreciation of the interrelationship of the previously learned branches of
mathematics
-2Course outcomes will be assessed by:
• District assessment in the form of AP questions (30 % of final exam grade) available
from
the math building department coordinator
• Final exam (70 % of final exam grade)
Additional assessments should be developed by teachers and should include a variety of
assessment
strategies.
-3-
Course Level Assessment
Content and Instructional Time
(Based on 135 Days)
Content Instructional Time Presentation
Functions and Models 10-12 Days
(Standards 2.5, 2.8)
Evaluation of Functions R
Analysis of Graphs R, E
Exponential Functions R
Inverse Functions R
Logarithmic Functions I
Models and Curve Fitting I
Limits 10-12 Days
(Standards 2.5, 2.8, 2.10, 2.11)
Slope of Tangent Line I, E
Limit of a Function I, E
Calculating Limits (using limit laws) I, E
Estimating Limits from Graphs I, E
Continuity I, E
Intermediate Value Theorem I, E
Extreme Value Theorem I, E
Derivatives 8-10 Days
(Standards 2.8, 2.11)
Definition of the Derivative I, E
Derivative at a Point I, E
Derivative as a Function I, E
Second Derivative I, E
Differentiation Rules 10-12 Days
(Standards 2.5, 2.8, 2.10)
Derivatives of Polynomials and Functions I, E
Product and Quotient Rules I, E
Chain Rule I, E
Derivatives of Trigonometric Functions I, E
Derivatives of Exponential Functions I, E
Derivatives of Logarithmic Functions I, E
Implicit Differentiation I, E
Linear Approximation I, E
I - Introduce
E - Emphasize -4R - Review
Content and Instructional Time
(Based on 135 Days)
Content Instructional Time Presentation
Applications of Differentiation 10-12 Days
(Standards 2.5, 2.8, 2.11)
Rates of Change I, E
Related Rates I, E
Velocity, Speed, Acceleration I, E
Optimizing Problems I, E
Inverse Functions I, E
Integrals 18-20 Days
(Standards 2.8, 2.9, 2.11)
Areas and Distance I, E
Definite Integrals I, E
Riemann Sums (Trapezoidal Rule) I, E
Fundamental Theorem of Calculus I, E
Antiderivatives of Basic Functions I, E
Substitution Rule I, E
Separation of Variables I, E
Application of Integration 10-12 Days
(Standards 2.5, 2.9)
Areas I, E
Volumes I, E
Disk
Washers
Shells
Known Cross Section
Average Value I, E
Applications to Physical Models I, E
Use in Modeling (exponential growth) I, E
Review for AP Test 15-20 Days R
Miscellaneous Topics
(Standards 2.8, 2.9, 2.10) 12-18 Days
Techniques of Integration I
Integration by Parts I
Partial Fractions I
Trig. Functions I
Arc Length I
Derivatives I
Parametric Equations I
I - Introduce
E - Emphasize -5R - Review
Unit: Functions and Models
Student Learning Outcomes Content Skills and Knowledge
Suggested Learning Activities
and
Instructional Strategies
Students will know the interplay between geometrical and
analytical information.
Students will be able to use calculus to predict and explain the
observed local and global behavior of a function.
Use graphing calculators
Use AP questions
Use idea of families of graphs from
parent graphs
Note standard form of conics
-6Students will predict and explain the
behavior of a function.
Unit: Functions and Models
UNIT ENRICHMENTS
From instructor’s guide:
1.1 Four ways to represent a function
1.2 New functions from old functions
1.5 Exponential functions
1.6 Inverse functions and logarithms
MATERIALS/TECHNOLOGY SUGGESTED ASSESSMENT TECHNIQUES
Students may:
work with partners on assignments.
use additional time to complete assignments.
Teacher generated assessments
quizzes
homework
test
District Performance Assessments
Graphing calculator
Stewart CD Rom
-7-
UNIT MODIFICATIONS
Unit: Limits
Student Learning Outcomes Content Skills and Knowledge
Suggested Learning Activities
and
Instructional Strategies
Students will be able to:
calculate limits using algebra.
use asymptotes to describe the graphical behavior of a
function.
describe continuity in terms of limits.
describe asymptotic behavior in terms of limits
involving infinity.
Students will know:
that limits can be estimated from graphs and tables of
data.
the geometric relationships of graphs of continuous
functions.
Use graphing calculators,
emphasizing trace and tables
Use algebraic methods
Avoid the rigorous definition of limit
Use AP questions
Stress relationship between limits
and asymptotes
-8Students will be able to evaluate limit
functions.
Students will be able to identify all
asymptotic characteristics of a function.
Students will be able to describe the
continuity of a function algebraically and
geometrically.
-9-
Unit: Limits
UNIT ENRICHMENTS
Course material - Application problems
2.1 Slope patterns (worksheet - Stewart’s Instructor’s Guide)
2.5 Infinite limits (worksheet - Stewart’s Instructor’s Guide)
2.6 The Cart and the Horse (Stewart’s Instructor’s Guide)
MATERIALS/TECHNOLOGY SUGGESTED ASSESSMENT TECHNIQUES
Have instructor read questions aloud
Work with partners on assignments
Use additional time to complete assignments
Take tests and quizzes in the resource room.
Teacher generated assessments
quizzes
homework
texts
District Performance Assessment
Small group work
Graphing calculators
Stewart CD Rom
UNIT MODIFICATIONS
Unit: Derivatives
Student Learning Outcomes Content Skills and Knowledge
Suggested Learning Activities
and
Instructional Strategies
Students will know:
the definition of limit.
relationship between differentiability and continuity.
characteristics of graphs of f, f ′ , f ′′ .
relationship between concavity of f and sign of f’.
points of inflection as changes in concavity.
Students will be able to:
interpret geometrically, analytically, and numerically.
interpret as instantaneous rate of change.
find the slope of a curve at a point.
find the tangent line to a curve at a point and by local
linear approximation.
relate the instantaneous rate of change as limit of
average rate of change.
approximate rate of change from graphs and tables.
determine the characteristics of graphs of f and f ′ .
determine the relationship between the increasing and
decreasing behavior of f and the sign of f ′ .
Teach limit algebraically and
intuitively through use of the
graphing calculator.
Stress derivative as instantaneous
rate of change in a variety of
things.
Stress graphical interpretations of
continuity and discontinuity.
Use the difference quotient for
f ′ at the specific point.
Stress local linearity.
Use examples given different time
intervals of change then calculate
instantaneous rate of change for
same function.
Given a table and or a graph,
interpret the derivative(s) at
various points.
Given the graphs of f ′ and f ′′ ,
find the graph of f.
Make charts using critical values to
note changes in concavity.
Use graphing calculator.
-10Student will be able to:
demonstrate understanding the
derivative geometrically,
analytically, and numerically.
apply the rules of limits to finding
derivatives.
approximate the derivative value
from local linearity graphs and
tables.
relate the graphs of f, f ′ , and
f ′′ .
use the second derivative,
determine concavity, and points of
inflection.
Unit: Derivatives
UNIT ENRICHMENTS
Course material - Application problems
3.2 Sparse Data (Stewart’s Instructor’s Guide)
3.5 Unbroken Chair (Stewart’s Instructor’s Guide)
MATERIALS/TECHNOLOGY SUGGESTED ASSESSMENT TECHNIQUES
Take tests and quizzes in resource room
Work with partners on assignments
Tests, quizzes, and homework
Practice AP problems & tests should be required
Two Performance Assessments required
A graphing calculator is required for this course.
Stewart CD Rom
-11UNIT MODIFICATIONS
Unit: Differentiation Rules
Student Learning Outcomes Content Skills and Knowledge
Suggested Learning Activities
and
Instructional Strategies
Students will be able to:
find derivatives of polynomials.
find the derivatives of trigonometric functions.
find the derivatives of exponential functions.
find the derivatives of logarithmic functions.
find the derivative of implicitly defined functions.
use linear approximation to estimate values of a
function.
model rates of change including related rates problems.
Students will be able to find the derivative using:
Product rule
Quotient rule
Chain rule
Power rule
Stress rules verbally (oral
recitation).
Understand the chain rule as the
derivative of composite functions.
Memorizing these derivatives is
imperative to the rest of the
course.
Compare and contrast rates of
change for exponential growth,
polynomial growth, and logarithmic
growth.
-12Students will be able to:
use the various rules for finding the
derivatives of algebraic and
transcendental functions.
take derivatives implicitly.
use and apply linear
approximations.
Unit: Differentiation Rules
UNIT ENRICHMENTS
From instructor’s guide:
3.1 Derivatives of Polynomial and Exponential Functions
3.2 The Product and Quotient Rules
3.3 Rates of Change in the Natural and Social Sciences
3.5 The Chain Rule
3.6 Implicit Differentiation
3.8 Linear Approximations and Differentials
MATERIALS/TECHNOLOGY SUGGESTED ASSESSMENT TECHNIQUES
Students may do any of the following:
Work with partners on assignments.
Use additional time to complete assignments.
Tests, quizzes, and homework
Practice AP problems and tests should be required
Performance Assessments are required (2)
A graphing calculator is required for this course.
Stewart CD Rom.
-13UNIT MODIFICATIONS
Student Learning Outcomes Content Skills and Knowledge
Suggested Learning Activities
and
Instructional Strategies
Students will be able to:
solve problems with higher order derivatives.
find tangents and normals to a line.
analyze curves, using monotonicity and concavity.
solve optimization problems including absolute and
relative extrema.
solve related rate problems.
translate verbal descriptions into equations involving
derivatives and vice versa.
Students know the process of curve sketching.
Students will use implicit differentiation to find derivative of an
inverse function.
Students will solve applications problems involving position,
velocity, acceleration, and other applied contexts.
Students will know and apply the Mean Value Theorem and
its geometric consequences.
Use graphing calculator.
When finding absolute maximums
and minimums, do not forget to
check the end points.
Stress drawing diagrams and label
to illustrate related rates problems
and optimization problems.
-14Students will be able to solve real world
problems by applying the rules of
derivatives.
Unit: Applications of Derivatives
Unit: Applications of Derivatives
UNIT ENRICHMENTS
Course material - Applications Problems
Create and present original related rate application problems
4.1 Related Rates (worksheet - Stewart’s Instructor’s Guide)
4.3 The Graph Game (worksheet - Stewart’s Instructor’s Guide)
4.6 The Waste-Free Box (worksheet - Stewart’s Instructor’s
Guide)
MATERIALS/TECHNOLOGY SUGGESTED ASSESSMENT TECHNIQUES
Work with partners on assignments.
Take tests and quizzes in the resource room.
Use additional time to complete assignments.
Homework, quizzes, tests
Performance Assessment
Small group assignments
Use graphing calculator
Stewart CD Rom
-15UNIT MODIFICATIONS
Students will be able to find the Approximate definite integrals
antiderivative of definite and using Riemann and trapezoidal
indefinite integrals by using the sums represented algebraically,
process and properties of graphically, and by tables of values.
integration.
Use graphing calculator to evaluate
integrals.
Stress separable differential
equations.
Find specific antiderivatives using
initial conditions including applications
to motion along a line.
Solve separable differential
equations using them in studying
the equation Ky y′ = and
exponential growth.
Student Learning Outcomes Content Skills and Knowledge
Suggested Learning Activities
and
Instructional Strategies
-16-
Unit: Integrals
Students will be able to:
find antiderivatives.
solve differential equations by separation of variables.
find geometric interpretations of differential equations in
slope fields and the relationship between slope fields and
solution curves for differential equations.
calculate the definite integral by summation (Riemann sums).
solve indefinite integrals.
solve definite integrals given constants of integration.
integrate by substitution using u, du.
find a definite integral.
given a function using the fundamental theorem of calculus
analyze and answer questions about the function.
Students will know the:
integration of sine and cosine.
properties of definite integrals.
fundamental theorem of integral calculus.
trapezoidal rule for approximating definite integrals.
-17-
Unit: Integrals
UNIT ENRICHMENTS
Course material - Application Problems
5.2 The Area Function (Stewart’s Instructor’s Guide)
5.2 Exploring Definite Integrals (Stewart’s Instructor’s Guide)
MATERIALS/TECHNOLOGY SUGGESTED ASSESSMENT TECHNIQUES
Take tests and quizzes in resource room.
Work with partners on assignments.
Use additional time to complete assignments.
Quizzes
Homework
Tests
Performance Assessments
Use the graphing calculators to solve problems.
Stewart CD Rom
UNIT MODIFICATIONS
Unit: Application of Integrals
Student Learning Outcomes Content Skills and Knowledge
Suggested Learning Activities
and
Instructional Strategies
Students will be able to:
calculate the change in position and distance traveled
by a particle.
find the area of a plane region.
find the volumes of solids of revolution by disks,
washers, and shells.
find the volumes of solids with known cross
sections.
find the average value function
Students will be able to find: (all optional)
the length of an arc.
the area of a surface.
the work done.
fluid pressure on a surface.
the moments and centers of mass.
Optional topics:
arc length
surface area
work
fluid pressure
moments and centers of mass
Use visuals which could be made
from playdoh, styrofoam, or loaves
of bread to show slicing.
Use graphing calculator.
-18Students will be able to apply integration
techniques to model and solve life-science
problems.
-19-
Unit: Application of Integrals
UNIT ENRICHMENTS
Have students develop their own models and explain the relevance to
the applications of integrals.
From Instructor’s Guide:
6.1 More About Areas
6.2 Volume
6.4 Average Value of a Function
MATERIALS/TECHNOLOGY SUGGESTED ASSESSMENT TECHNIQUES
Students may do any of the following:
Work with partners on assignments.
Use additional time to complete assignments.
Quizzes
Homework
Tests
Performance Assessments
Use the graphing calculators to solve problems
UNIT MODIFICATIONS
Textbook Recommendation
Stewart, James (1998). CALCULUS CONCEPTS AND CONTEXTS, California:
Brooks/Cole
Publishing
Instructor’s Guide for Stewart’s Calculus Concepts and Contexts, Single Variable
Calculator Recommendation
The secondary mathematics department of Central Bucks School district recognizes the
use of
calculators as a valuable tool for learning in the mathematics classroom. In certain
advanced courses
graphing calculators with specific capabilities are important for daily classroom
performance and are
required for advanced placement tests. While no specific brands are endorsed, there are
restrictions on the
type of calculators allowed on certain tests and final exams. Calculators which do
operations with variables,
such as the TI-89, TI-92 and HP49G will no be permitted to be used on district final
exams, even though
they may be used on some nationwide tests. Teachers have discretion as to whether these
types can be
used for particular classroom-related purposes.
-20-