AP Calculus AB
Mr. Wren’s Syllabus
Course Overview
It is the intent of this course to present calculus as a singular body of
knowledge. James Stewart in his text, Calculus Concepts and
Contexts, 2nd edition, remarks “the two branches of calculus and
their chief problems, the area problem and the tangent problem
appear to be very different, but it turns out that there is a very close
connection between them.”(4) Showing the connection between
these two branches of calculus is a chief aim of this course.
In order to help students see and appreciate the connections between both
integral and differential calculus I employ a “multi-representational” model of
instruction and practice. Students encounter on a daily basis functions that are
represented verbally, visually (graph form), analytically (algebraic functions),
and numerically (in table form). By representing functions in any one of these
four forms students learn to apply calculus concepts to solve the problem at
hand.
The TI-83 and TI- 89 graphing calculators facilitate the exploration of
functions and enable students to give meaningful approximate answers
to problems when an analytic solution is found wanting. The calculator
allows for a fluid movement between the various ways in which functions are
represented. Students for instance can employ the calculator to create
a regression equation from a table of values and then integrate the function
using known techniques of integration (or simply use the calculator to
integrate the function for them). Similarly, students can take data from
a graph measuring the velocity of a particle and use the calculator to find an
approximation of the distance the particle traveled using the trapezoid rule.
The course covers topics presented by the College Board in the
Teacher’s Guide – AP Calculus. The tangent problem and differentiation
are considered primarily in the first semester of the course and the area
problem and integration are taught primarily in the second semester
of the course. How the Fundamental Theorem of Calculus relates the
integral to the derivative is a major idea in the second semester.
C1 The course provides
students with the
opportunity to work with
functions represented in
a variety of waysalgebraically, and
verbally- and emphasizes
the connections among
these representations
C7 The course teaches
Students how to use graphing
calculators to help solve
problems, experiment,
interpret results, and support
conclusions
AP Calculus AB Course Outline
Unit 1 Precalculus Review: Functions and Models (2-3
weeks)
This unit prepares the way for calculus by reviewing the basic concepts of functions
learned in precalculus. Students are introduced to the four ways of representing a
function. The graphing calculator as a tool for approximating values is discussed
A. Representing Functions
1. Four ways to represent a function
2. Piecewise Functions
3. Symmetry; increasing and decreasing functions
B. Mathematical Models
1. The function as a model for understanding data
2. A review of algebraic and transcendental families
of functions
3. Using the calculator to create regression equations
to model data
C. Transformations and Compositions of Functions
D. Using the graphing calculator
E. Exponential and Logarithmic Functions
C1 The course provides
students with the
opportunity to work with
functions represented in
a variety of waysalgebraically, and
verbally- and emphasizes
the connections among
these representations
1. Exponential growth and decay
2. Inverse functions
3. Logarithmic functions and the properties of logarithms
F. Trigonometric Functions
1. Graphs of basic trigonometric functions
and their inverses
2. Transformations of functions
3. Applications of trigonometric functions
G. Review and testing
Unit 2 Limits and Derivatives
(4 weeks)
This unit explores the concept of the “limit” and the properties
of limits. Of particular interest is the use of the “limit” in
developing the definition of the derivative . Derivatives as rates
of change in different situations are explored . Students also learn
how to use the derivative as a function to predict how the original
function behaves.
A. The tangent line problem: an introduction to a “limit”
1. Tangent line viewed as the limit of secant lines
2. Concepts of average versus instantaneous velocity
described numerically, graphically, and in physical
terms.
3. Local linearity: “Zooming” in on a smooth function
C7—The course teaches
students how to use graphing
calculators to help solve
problems, experiment,
interpret results, and support
conclusions
B . Limits at a point
1. Informal definition of the limit of a function
2. The definition applied when examining a table
of points and when examining a graph
3. One sided limits
4. Pitfalls of using the graphing calculator to determine
a limit from a table of points or from a graph
E. Calculating limits using algebra
1.
2.
3.
4.
Properties of Limits
Direct Substitution
Rationalization technique
Squeeze Theorem
C8. Students should
be able to determine
the reasonableness
of solutions,
including sign, size,
accuracy, and units
of measurement
F. Continuity
1. Geometric and mathematical definitions of
continuity
2. Discontinuous functions
a. removable discontinuity
b. jump discontinuity
c. infinite discontinuity
3. Determining the continuity of composite functions
4. The Intermediate Value theorem
G. Limits involving infinity
1. Infinite limits and vertical asymptotes:
2. Limits at infinity and horizontal asymptotes
3. Dangers of using calculators to check limits
(numerically and graphically)
H. Instantaneous Rates of Change
Unit 3 The Derivative
(5 weeks)
A. The definition of the derivative as a limit of a difference
quotient
1. The derivative notation
2. Interpreting the derivative as the slope of a tangent
line
3. The derivative as an approximate rate of change
when working with discrete data
B. The Derivative as a function
1. The concept of a differentiable function, interpreted
graphically, numerically, and descriptively
2. Differentiability and Continuity: How can a
function fail to be differentiable?
3. What the first derivative says about the function “f”
4. Higher order derivatives. What does f′′ say about f′?
C. Linear Approximations
C2 Students
should understand
the derivative in
terms of a rate of
change and local
linear
approximation
and should be able
to use derivatives
to solve a variety
of problems
1. Computing linear approximations
2. Using linear approximations to approximate
functions
D. Differentiation of Algebraic and Transcendental
Functions.
In this section the definition of the derivative is used
to derive rules for differentiation. Students are “walked”
through” proofs for the power rule, constant multiple rule
and the product rule. These rules and others not proved
reinforce the idea that calculus is a coherent and
logical body of knowledge.
C9 Students
should develop an
appreciation of
calculus as a
coherent body of
knowledge and as
a human
accomplishment
1. Differentiation of algebraic functions
2. Derivative Rules when combining functions
3. Applications of the derivative to rates of change
in the natural and social sciences.
Group presentations. Students are assigned to work in small
groups to present their solutions to a problem specific to
their field of interest – social science, physical science, etc.
When calculating the derivative students are required to explain the
meaning of the derivative as an instantaneous rate of change.
Presentations include a statement of the problem, the data or funtion from which data is derived, method and calculation by which the
rate of change is determined (derivative of a known function, limit
from a table) and a visual aid to help illustrate the problem.
4. Applications to velocity and acceleration – linear
motion problems
5. Derivative of ex
6. Derivatives of trigonometric functions
7. Chain Rule
8. Implicit Differentiation
9. Derivatives of inverse trigonometric functions
10. Derivative of Logarithmic functions
11. Linear Approximations and Differentials
Unit 4 Applications of Differentiation (6 weeks)
Students are given the opportunity to pursue the applications of
differentiation. Students discover the value of determining “extreme values”
of a function when faced with solving practical problems that occur in real
C5. Students should
be able to
communicate
mathematics both
orally and in well –
written sentences
and should be able
to explain solutions
to problems.
life. Strategies for solving related rates and optimization problems are
discussed
A. Related Rates
Individual Lab Activity: “How Many Licks” –Lab Activity 7
Students determine the rate of change of volume of a
Tootsie Roll Pop as you consume it
(see Kamische, Ellen., A Watched Cup Never Cools 1999. Key
Curriculum Press), pp.23-24)
B. Extreme Values
1. Local extrema and critical points
2. Absolute extrema and the Extreme Value Theorem
C. Derivatives and the shapes of Curves
1. Mean Value Theorem
2. Increasing/Decreasing Test
3. First Derivative Test
4. Concavity
5. Second Derivative Test
6. Curve Sketching
7. Curve Sketching and the use of the graphing
calculator
Students use the their knowledge of calculus to refine the
display of a function graphed on their calculators. Students
discover that for some functions two or three different
viewing windows are necessary to see all the salient features
of the function. Students also use the calculator to determine
approximate values of maxima and minima, inflection points,
and roots for functions that are too difficult to consider without
technology.
D. Optimization Problems
E. Linearization Models – Newton’s Method
F. Antiderivatives
Students are introduced at this point to antiderivatives to
to prepare them for finding the integral of a function in the
unit ahead. Students are introduced to a few “antiderivative”
formulas. It is not enough however to show the students that
the concept of the antiderivative reverses the process of finding
the derivative. What is important is that the students realize
the value in finding the antiderivative. Practice is thus given
to finding a position function from a velocity function first by
a guess and check method, and then learning a few basic antidifferentiation rules to speed up the process. The idea that
a physicist can learn a great deal about the position of a
particle at a given time because the velocity of the particle is
C7. Students
should be able to
use technology to
help solve
problems,
experiment,
interpret results,
and verify
conclusions.
known is drilled home with a return to a study of rectilinear
motion. Students are also introduced to slope fields here to
allow students a chance to guess the shape of a family of antiderivative curves from the given slope field.
1. Finding the most general antiderivative of a given
function
2. Using slope fields to determine the shape of an
unknown function. This exercise anticipates the
lesson on separable equations: solving a differential
equation for an unknown function that is a
solution to the differential equation.
Unit 5 The Definite Integral
(5 weeks)
The area and distance problems serve as the launching point for introducing
the idea of the definite integral. Emphasis is placed on finding the limit of the
sum of rectangles of equal width to determine an approximate area under a
curve. However, subdividing the area under a curve into unequal partitions is
discussed so that taking the limit of the normal to zero can be understood.
Several days are given over to calculating the integral through Riemann
sums. A day each is given over to approximating the integral by the midpoint
and trapezoidal rules. The proof for the Evaluation Theorem of the
Fundamental Theorem is “walked” through, and students are made
accountable for reproducing the same on a test. The rationale for this is the
inherent value of recognizing the value of applying the Mean Value Theorem
within the proof. (Those existence theorems don’t exit for nothing!)
Finally, several days are taken to studying the Fundamental Theorem to
discern the subtle idea integration and differentiation are inverse processes.
C4. Students should
understand the
relationship between
the derivative and
the definite integral
in both parts of the
Fundamental
Theorem of
Calculus
A. Approximating Areas
1. Riemann sums (left end, right end and midpoint
sums) to approximate the area of a region
2. Calculating approximate distances from a table
or graph of discrete data of varying velocities
3. Evaluating limits of Riemann sums with partitions
of equal width to determine the area of a region
under polynomial functions.
4. The definite integral as a limit of Riemann sums
5.. Properties of the definite integral
B. The Fundamental Theorem of Calculus (part 2)
1. Proof of the Evaluation Theorem
2. Calculating the definite integral using known
C3. Students should
understand the
meaning of the
definite integral
both as a limit of
Riemann sums and
as the net
accumulation of
change and should
be able to use
integrals to solve a
variety of problems
formulas of indefinite integrals and the Evaluation
Theorem. Here integrands are chosen to be
velocity functions so that accumulating rates of
change in distance can be seen to accrue to a total
distance
C.
D.
E.
F.
The Fundamental Theorem of Calculus (Part 1)
Substitution Rule for Integrals
Integration by parts ( simple examples)
Additional technique of integration
1. Trigonometric substitutions
2. Partial fractions (linear and linear -quadratic
factors
G. Approximating the exact value of a definite integral
using the Trapezoid Rule.
Unit 6 Applications of Integration
(5 weeks)
The main idea in this unit is for students to be able to divide a quantity –
whether it be an area between two curves or the length of an arc – into small
pieces and estimate the answer with Riemann sums, and then recognize the
limit of such sums as an integral. Once the integral is represented, students
will use the Evaluation theorem or their calculators to evaluate the integral.
A. Area between curves
B. Using areas to determine distances – Students represent
the area between two velocity curves to compute the
the distance between two objects over an interval of time
“t” after starting from the same point.
C. Volumes of Solids of Revolution
1. Disc Method
2. Washer Method
D. Volumes of Solids with known cross sections
E. Average Value of a Function
F. Applications to Physics and Engineering
1. Work – variable force
2. Hydrostatic pressure and force
Individual Writing Project: Tile Project
Each student tries to sell the class on a tile design created in class on
the computer or on their graphing calculator. Students use their
calculators or classroom computer to create two functions that enclose
a region. Each region is “painted” a different color thus creating an
aesthetically pleasing pattern for a two-colored ceramic tile. The area
C3. Students should
understand the
meaning of the
definite integral
both as a limit of
Riemann sums and
as the net
accumulation of
change and should
be able to use
integrals to solve a
variety of problems
between two functions is one color and the area of the remaining part
of the tile is a second color. Students integrate the enclosed region (the
bounds of which are determined by the intersection of the curves).
tudents calculate the ratio of the areas of each colored part of the tile.
Students are to duplicate the tiles many times over and arrange the
tiles (a minimum of four) in several formations. Students are required
to create a business letter introducing their company to the class and
point presentation is required for the presentation.
Group Project
C5 Students
should be able
to communicate
mathematics
both orally and
in well-written
sentences and
should be able
to explain
solutions to
problems
“As Easy as п” - Lab Activity 15
Students determine the volume of a pan (circular object)
(see Kamische, Ellen., A Watched Cup Never Cools 1999. Key
Curriculum Press), page 57.)
Unit 7 Modeling with Differential Equations (2
weeks)
Students are introduced to differential equations and learn to solve these
equations through the technique of separable equations. Students examine
data on population growth, radioactive decay, and continuous compounded
interest. The unit is introduced with the idea that the growth rate of a
population is proportional to the population’s size (for early growth of the
population). This example allows the student to see what constitutes a
differential equation and what solution is being sought. Direction or slope
fields are introduced to show students that the field represents general
solutions to the differential equation. Finally the actual technique of solving
simple differential equations is given.
A.
B.
C.
D.
Modeling with Differential equations
Direction Fields
Separable Equations
Exponential Growth and Decay
Group Project “A Watched Cup Never Cools” - Lab
Activity 10
Students determine a mathematical model that describes how a hot cup
of water cools.
(See Kamische, Ellen., A Watched Cup Never Cools 1999. Key
Curriculum Press), pp 34-36.)
C6 Students should be
able to model a written
description of a
physical situation with
a function, a
differential equation,
or an integral
Review of Specific Concepts
A.
B.
C.
D.
E.
(1 week)
Logarithmic differentiation
Inverse Functions and their derivatives
Inverse Trigonometric functions and their derivatives
Related Rates
Optimization
Preparation for the AP Test (3 weeks)
Students are given over the course of three weeks three complete simulated
AP tests from the spiral booklet Preparing for the (AB) AP Calculus
Examination 2006 by George Best and J. Lux . Students also use the
companion solution manual volume accompanying the spiral text. After
school tutoring sessions are arranged for students who have questions
surrounding the practice test. Class time is not taken for reviewing the
practice tests
Post AP Test
(2 weeks)
The remaining days after the AP Test are given over to studying further
concepts like arc length and surface area so students can complete one last
project called the “Cup”and then prepare for their final test in the class. The
project is described below:
Group Project: “The Cup” – Lab Activity 19
Students are given a cup and challenged to find the volume of the cup
and the inner surface area of the cup. They are also to find at what rate
the water level is rising in the cup when the cup contains half its
volume and the water is being poured into the cup at a rate of 2 cubic
centimeters per second (see Kamische, Ellen., A Watched Cup
Never Cools 1999. Key Curriculum Press), pp. 66-67)
Students must use the regression capabilities of a graphing calculator
to find a model curve which they can then integrate on the graphing
calculator
Teaching Practices
I encourage my students to be patient with themselves and to enjoy thinking.
Math has its own beauty and elegance. By taking time to think students will
more readily see the connections present within this discipline we call
calculus and find the joy that is the inevitable reward of study.
I begin most class periods with a statement of what we are about to study. I
try to reveal the objective for the day in the clearest and simplest language
C7. Students
should be able to
use technology to
help solve
problems,
experiment,
interpret results,
and verify
conclusions.
possible. Whenever possible I like to share the purpose for the lesson hoping
to instill some excitement that naturally comes when one perceives some
relevance to the lesson at hand.
James Stewart’s Instructor’s Guide for Single Variable Calculus, Concepts
and Contexts 2001, Brooks Cole publishers has been incredibly helpful for
structuring lectures and planning group discovery activities. Steward stresses
the need for variety. Some days the lecture approach is most suitable
particularly when laying out a proof! I often follow up a lecture with an
extended activity where students will work in groups and solve a singular
problem posed on a worksheet. On other occasions I introduce a concept by
referring the students to their text and posing a thoughtful question related to a
major concept presented in the text. “What is the difference between the
statements f(a) = L and lim f(x) = L as x approaches ‘a’”? “What does it
mean to say “approaches”? “How close can must we get to /a’ to determine
its limit?”
Stewart has impressed on me the need keep students from seeing calculus as a
string of disparate skills to be learned. Always in the forefront of my
approach is to lead the students to understand the concept, the big idea and
encourage them to appreciate the power of calculus to make sense of tables of
data.
The graphing calculator is a wonderful tool for exploring functions. The
calculator is used every day in class. Again, Stewart has much to say about
the employing a proper respect for the calculator’s sometimes apparent
deficiencies. Knowing where a maximum or minimum value may occur on a
graph of a function allows the students to rethink a proper “window” and see
the graph as it should appear. Students are often surprised to out how much
the calculator “hides” when they are not using the proper viewing window!
Student Evaluation
Assessments are given weekly as quizzes and only once every four to five
weeks when the unit of study is complete. Each morning or nearly so a
question from a past AP test is put before the students as a morning warm up.
I see students gaining confidence throughout the course as they see that they
are learning the material and have a fighting chance to pass the AP test. I find
that beginning the period with such a question enables me to apprise the class
of what is considered a “complete” response – fully answering the question
posed and to the correct accuracy.
Students spend a good deal of time working in small groups and regularly are
challenged to work their solutions (all groups simultaneously) on the white
board. This approach gives me the opportunity to see how the students
approach their work and who is really doing the work! On a regular basis
students are encouraged to present a solution to the whole class and are
rewarded with an extra credit point for their effort. Students are quite quick to
rise to the challenge take the “vis-a vis” marker and share their solution on the
overhead to that seemingly esoteric problem.
Although time is fleeting in teaching this course, I make time for a few in
class projects which I use as an alternative means of assessment. I wish I had
more time for more projects and hope in time to learn how to make more use
of projects that better embed the objectives of the course. Following the AP
Exam est, students complete the course with an end of course final exam.
Teacher Resources
Primary textbook
Stewart, James. Calculus, Concepts and Contexts, Single Variable, 2nd ed.
Pacific Grove California: Brooks/Cole Publishing Co. 2001.
Supplementary Resources
Kamische, Ellen. A Watched Cup Never Cools. Emeryville, California: Key
Curriculum Press. 1999.
Best, George and Lux, J.Richard. Preparing for the AB Advanced Placement
Calculus Examination. Andover MA. Venture Publishing, 2006.
Technology Resources
Students are required to use a TI-83 or TI-83 Plus graphing calculator.
Calculators are used on a daily basis both to explore new concepts, solve
problems, approximate integrals that cannot be easily integrated, interpret
solutions and to write and use simple programs.
Students also make use of several Dell computers in the classroom for
creating power-point presentations for classroom projects, searching math
websites, and for graphing functions using a TI-Graph Link or an on-line
website. Two programs that have proven helpful in illustrating class lectures
and in providing ancillary materials to support the class lecture are
Tools for Enriching Learning Calculus, CD-ROM. Harvey Keynes and
James Stewart, Brooks/Cole. 2001
AP Calculus Teachers Tools Teacher CD, CD-ROM. Frane Media LLC.
2005. Stewart’s CD covers topics that correlate nicely with the text and
provide helpful visuals and discussions. The Frane Media CD-ROM is a
collection of “notes” and practice sheets on all the major calculus topics.