# AP Calculus AB Curriculum 2008-09 by qbp14515

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```									                                           Course name:   AP Calculus AB
Revised: Aug 11, 2008
Course Description: In AP Calculus AB, students will explore four main ideas from calculus: limits, derivatives, indefinite
integrals, and definite integrals. Problems will be approached through a balance of multiple representations including:
graphically, numerically, analytically/algebraically, and verbally. Wherever practical, concepts will be applied to analyze
real-world situations.
Graphing calculators and/or computers will be used on a regular basis to help solve problems, experiment, interpret
results, and support conclusions.
Students will learn to communicate mathematics through the use of a math journal and having time each day to work in
groups to exchange ideas and approaches and to reflect on homework assignments.
A variety of assessments will be used to including paper based tests and quizzes, homework, group and individual
laboratory work. In order to prepare students for the AP exam, tests and quizzes will contain questions similar to those
that appear on the AP exam in terms of content, difficulty, and structure.
Primary text(s) and other major resources:
Calculus of a single variable Larson, Hostetler, and Edwards
(D.C. Heath and Company, 1994)
Page 1 of 6
o   unit number & title                            Objectives                              Essential Concepts                          Assessment
(specific skills and knowledge students will have)                                             Students must show proficiency
with MLR skills and knowledge in
assessments marked Essential in
order to progress to the next course
level.
U1                              Objective set 1:
Prerequisites and Analysis of                                                                                                 x   Homework and quizzes
Graphs                          2-3 Weeks                                             How are real numbers and sets of            will be given on a regular
Student will:                                         real numbers be represented,                basis throughout the
x   review—the real number system                     classified, and ordered?
course
x   review—the cartesian plane                        What algebraic techniques are
x   review—graphs of equations                        commonly used in calculus?
x   review—lines in the plane                         How can data and relationships be
x   review—functions                                  organized and represented
Project #1
x   review—trigonometric functions                    graphically?
Test #1
U2                              Objective set:
Limits and Continuity           4-5 weeks
Students will:                                        What is a limit?
x   Evaluate Limits at a Point                        How do you find a limit with a table,
o 1 sided limits                              graph, or analytically?
o 2 sided limits                              When does a limit not exist?
o Sandwich Theorem                            What is the definition of continuity
x   Evaluate Limits involving Infinity                for a function on an open or closed
o Asymptotic Behavior                         interval?
o End Behavior models
o Properties of Limits (Algebraic
Analysis)
o Visualizing Limits (Graphic Analysis        What is a continuous function?
x   Determine Continuity of a function                How can I determine if a function is
o Continuity at a Point                       continuous?
o Continuous Functions
o Discontinuous Functions
    Removable Discontinuity
    Jump Discontinuity
    Infinite Discontinuity        What is meant by rates of change?
x   Determine and Analyze Rates of Change and         How is the rate of change
Tangent Lines                                     determined?
o Average rate of change                      How are tangent lines and a rate of
o Tangent line to a curve                     change related?
o Slope of a curve (algebraically and         What is a normal line and how can I
graphically)                           determine the formula for a normal
Page 2 of 6
o   unit number & title                        Objectives                                 Essential Concepts                     Assessment
(specific skills and knowledge students will have)                                             Students must show proficiency
with MLR skills and knowledge in
assessments marked Essential in
order to progress to the next course
level.
o       Normal line to a curve (algebraically   line?
and graphically)
o       Instantaneous rate of change                                                  Project #2
Test #2
U3                        Objective Set
The Derivative            5-6 Weeks
Students will:
x   Determine Rates of Change                           How do I determine a rate of
o Average Speed                                change?
o Instantaneous Speed                          What is the difference between an
x   Use a variety of methods to determine the           average rate of change and an
Derivative of a Function                            instantaneous rate of change?
o Definition of the derivative (difference     How I determine the derivative of a
quotient)                                function?
o Derivative at a Point
o Relationships between the graphs of f
and f’
o Graphing a derivative from data
o One sided derivatives
x   Determine if a function is differentiable           How do I determine if a function is
o Cases where f’(x) might fail to exist        differentiable?
o Local linearity                              How do I determine if a function is
o Derivatives on the calculator                continuous?
(Numerical derivatives using NDERIV)     What facts can we conclude about
o Symmetric difference quotient                a continuous function?
o Relationship between differentiability
and continuity
o Intermediate Value Theorem for
Derivatives
x   Use Rules for Differentiation to determine
derivatives of functions
o Constant, Power, Sum, Difference,
Product, Quotient Rules
o Higher order derivatives
x   Apply concepts of the derivative to analyze         How can we use derivatives to
motion problems                                     analyze motion?
o Position, velocity, acceleration, and
Page 3 of 6
o   unit number & title                               Objectives                               Essential Concepts                           Assessment
(specific skills and knowledge students will have)                                              Students must show proficiency
with MLR skills and knowledge in
assessments marked Essential in
order to progress to the next course
level.
jerk
o Particle motion
o     L’HÔpital’s Rule
x     Apply concepts of the derivative to analyze        How can we use derivatives to
economic problems                                  analyze economic problems?
o Marginal Cost
o Marginal Revenue
o Marginal Profit
x     Determine derivatives of Trigonometric             How do we determine the
Functions                                          derivative of trigonometric
x     Use the Chain Rule to derive composite             functions?
functions                                          What are composite functions and
x     Determine rate of change using Implicit            how do we determine the
Differentiation                                    derivative?
o Differential Method                         What are implicit functions and how
o Y’ Method                                   do we determine the derivative?
x     Determine derivatives of Inverse Trigonometric     What are inverse trigonometric
Functions                                          functions and how do we determine
x     Determine derivatives of Exponential and           the derivative?
Logarithmic Functions                              What are exponential functions and       Project #3
logarithmic functions and how do         Test #3
we determine the derivative
U4                               Objective Set
Applications of the Derivative   5-6 Weeks
Students will:
x     Locate Extreme Values of a function                What are extrema and where do
o Relative Extrema                            they occur?
o Absolute Extrema
o Extreme Value Theorem
o Definition of a Critical Point
x     Interpret Implications of the Derivative           What can we interpret about a
o Rolle’s Theorem                             function from the derivative?
o Mean Value Theorem
o Increasing and Decreasing functions
x     Produce accurate graphs by using the
relationships of f’ and f” with f(x)               How is a function related to its first
Page 4 of 6
o   unit number & title                        Objectives                               Essential Concepts                        Assessment
(specific skills and knowledge students will have)                                           Students must show proficiency
with MLR skills and knowledge in
assessments marked Essential in
order to progress to the next course
level.
o First derivative test for relative           and second derivatives?
max/min
o Second Derivative
    Concavity
    Inflection Points
    Test for relative max/min
x     Analyze optimization problems using calculus       What is an optimization problem
x     Estimate functions using Linearization models      and how can we use derivatives to
o Local Linearization                          solve them?
o Tangent Line approximation                                                         Project #4 – Saving material by
o Differentials                                How can we produce estimates of       improving the design of cereal
x     Analyze Related Rate problems                      the value of a curve using            boxes
linearization models?                 Test #4
What is a related rate problem and
how can we use derivatives to solve
them?
U5                        Objective Set
The Definite Integral     3-4 Weeks
Students will:
x     Approximate Areas between curves using             What methods can be used to
summations                                         approximate areas underneath
o Riemann sums                                 curves?
    Left                            How can we modify our
    Right                           approximation methods to improve
    Midpoint                        accuracy?
    Trapezoidal
x     Compare and contrast definite integrals to
Riemann sums
x     Use Properties of Definite Integrals               What are the basic properties of
o Power Rule                                   definite integrals?
o Mean Value Theorem for Definite
Integrals                                                                       Project #5
x     Understand how The Fundamental Theorem of          How are derivatives and integrals     Test #5
Calculus describes the inverse relationship        related?
between the integral and derivative
Page 5 of 6
o   unit number & title                            Objectives                              Essential Concepts                         Assessment
(specific skills and knowledge students will have)                                            Students must show proficiency
with MLR skills and knowledge in
assessments marked Essential in
order to progress to the next course
level.
o     Part 1
o     Part 2
U6                           Objective Set
Differential Equations and   3 Weeks
Mathematical Modeling
Students will:
x     Produce Slope Fields for differential equations    What is a differential equation? How
x     Evaluate Antiderivatives using common              can we visualize the solutions for a
formulas                                           differential equation?
o Indefinite Integrals                         What are some common
o Power Formulas                               antiderivatives?
o Trigonometric Formulas                                                              Project #6
o Exponential and Logarithmic formulas                                                Test #6
x     Analyze Logistic Growth models                     What are logistic growth models
and how do I analyze them using
calculus?
Unit 7                       Objective Set
Applications of Definite     3-4 Weeks
Integrals
Students will explore the following topics:
x     Evaluate and interpret Integrals as net change     How can integrals be used to
o Calculating distance traveled                describe motion or other changing
o Consumption over time                        variables?
o Net Change from data
x     Calculate areas between curves                     How can we use integrals to
o Integrating with respect to x                calculate areas between curves?
o Integrating with respect to y
x     Calculate areas between intersecting curves
o Integrating with respect to x
o Integrating with respect to y
x     Calculate Volume of solids                         How can we use integrals to
o Cross sections                               calculate volumes of solids?           Project #7 – Determine distance of
o Disc Method                                                                         a trip using speed and time data.
o Shell Method                                                                        Test #7
Page 6 of 6
Line Graphs
Notes
Function
This is a sample of a sheet that we
typically use to analyze various calculus                             1st Derivative
problems using a multiple approaches.
We use it to compliment our work with                                 2nd Derivative
the graphing calculators. The "notes"
section is used for a written and           Graph of Function
analytical approach. The "table" section
below is used for numerical approach
with enough room to examine the
function, first derivative, and second
derivative. The "graphs" section is for a
graphical approach allowing the student
to see function and derivative behavior
either on the coordinate plane and/or
using intervals along a number line.

We use this sheet for selected
homework problems or in-class
problems along with our Ti graphing
calculators
Graph of 1st Derivative

Graph of 2nd Derivative
Calculus Project
Designing an Environmentally Friendly Package
Introduction

You have been hired by the CEO of a major company to design the packaging for its newest cereal. The
CEO takes environmental issues very seriously and wants to make sure that the box that you design uses
as little material as possible.

Objectives:

   You will be able to calculate the derivative of a function using the product rule
   You will be able to calculate the derivative of a function using the quotient rule
   You will be able utilize derivatives to analyze optimization problems.
   You will be able to show multiple representations of a mathematical solution (algebraic/analytical,
numeric, graphical, written english)

Specifications for the packaging

x   The cereal must be packaged in a rectangular box.
x   The base must be ______ times the width. (number to be assigned by teacher)
x   The volume of the box must hold exactly _____ inches3. (number to be assigned by teacher)
x   The box must use the least amount of cardboard (minimize the surface area) to save money.
x   You must write a clear step by step description of your design process supported by functions,
derivatives, graphs, and data tables to convince the CEO that your design is the most efficient
use of material.
Teacher Notes:
This Box Project is an example of one of our project that requires the student to apply
calculus to solve a real world problem. In this case, the student has to design a cereal box
that meets the volume requirements while minimizing the amount of material used in order to
help minimize the environmental impact. The students must use an analytical, graphical,
numerical, and written approach to describe their solution. Furthermore, to complete the real-
world hands-on experience, the students must actually construct their proposed box.

The next project (Zoo Project) is also a real-world application of calculus. Students use
limited materials to construct a zoo that maximizes living spaces for the animals. Again, the
students use multiple approaches (analytical, written, numerical, and graphical) to help gain
a deeper understanding of the solution. Consistent with the hands-on experience, students
complete the project with a blueprint scaled map of their proposed zoo configuration.

Students will explore these, and several other application of calculus throughout the year.
Calculus
Project 1: Design an environmentally friendly package
4                                 3                                  2                                  1
Name, Date, Class, and Title   All of Name, Date, Class, and     Most of Name, Date, Class,          Some of Name, Date, Class,        None of Name, Date, Class,
Project title are clearly         and Project title are written at   and Project title are written at   and Project title are written at
written at top of the paper.      top of the paper.                  top of the paper.                  top of the paper.
Written Description of your    A very clear step by step         A mostly clear step by step        A step by step description of      A step by step description of
strategy is included. All major   strategy is included. Most,        included. Some, but not all,       included or lacks most major
steps are explained clearly       but not all, major steps are       major steps are explained.         steps. Little or no use of
and precisely. Key                explained clearly and              Little or improper use of          mathematical terms.
mathematical terms are used       precisely. Key mathematical        mathematical terms.
appropriately.                    terms are used appropriately.
Algebraic Representation of    A very clear algebraic solution   A mostly clear algebraic           An algebraic solution to your      An algebraic solution to your
your process                   to your design strategy is        solution to your design            design strategy is included.       design strategy is not
included.                         strategy is included.              This means that proper             included or has many
This means that proper            This means that proper             formulas and equations are         mistakes.
formulas and functions are        formulas and equations are         correctly written and solved.      Very few or no steps are
correctly written and solved.     correctly written and solved.      Some steps are shown, but          shown making the process
All steps are neatly shown        Most, but not all, steps are       some are skipped making the        difficult to follow.
until a conclusion is found.      neatly shown until a               process difficult to follow.       More than four mistakes.
No mistakes.                      conclusion is found.               Three or four mistakes.
Only one or two mistakes.
Numerical Representation of    A very clear data table shows     A mostly clear data table          A data table shows the             No data table included
your process (Data Table of    the behavior of all key           shows the behavior of all key      behavior of some, but not all,
functions and derivatives)     functions and derivatives that    functions and derivatives that     key functions and derivatives.
support your design strategy.     support your design strategy.      Few or none of the most
All of the most important         Some of the most important         important data values are
data values are highlighted       data values are highlighted        highlighted. Incorrect or no
and their significance to your    and their significance to your     use of labels and units.
solution is accurately            solution is described with few
described. Columns are            mistakes. Columns are
labeled with correct units.       labeled with correct units.
4                                3                                2                                1
Graphical Representation of   Very clear graphs show the       Graphs show the behavior of      Graphs show the behavior of      Graphs are not included or
your process                  behavior of all key functions    all or most key functions and    some functions and               have numerous major
and derivatives that support     derivatives that support your    derivatives. Few of the          mistakes.
your design strategy. All of     design strategy. Most of the     important points on the
the important points on the      important points on the          curves are highlighted and
curves are highlighted and       curves are highlighted and       their significance to your
solution is clearly described.   solution is described with few   Axis are not properly labeled.
All axis are properly labeled    mistakes. Most axis are
and have correct units.          properly labeled and have
correct units.
Use of calculus concepts      Appropriate and accurate         Appropriate and accurate         Appropriate and accurate         Appropriate and accurate
application and interpretation   application and interpretation   application and interpretation   application and interpretation
of derivatives is an important   of derivatives is part of your   of derivatives is part of your   of derivatives is not part of
part of your design solution.    design solution. These           design solution. These           your design solution or
These concepts are clearly       concepts are incorporated in     concepts are incorporated in     contains many mistakes.
algebraic, numeric, and          numeric, and graphical           algebraic, numeric, and
graphical representations.       representations with some        graphical representations
mistakes.                        with many mistakes.
Actual Cereal Box             You have constructed the full    You have constructed full size   You have constructed the         You have not constructed the
size cereal box that is          cereal box that is mostly        cereal box that is somewhat      cereal box or you have more
All key dimensions of the box    with one or two mistakes.        with three or four mistakes.
are clearly labeled with units   Most key dimensions of the       Some key dimensions of the
(length, width, height, volume   box are labeled with units       box are labeled with units
and surface area)                (length, width, height, volume   (length, width, height, volume
and surface area)                and surface area)
On Time                       The completed project is         The completed project is         The completed project is         The completed project is
delivered on or before the       delivered within one day after   delivered within three days      delivered more than three
due date:                        the due date:                    after the due date:              days after the due date:
Start of class on Tuesday        Start of class on Tuesday        Start of class on Tuesday        Start of class on Tuesday
December 9.                      December 9.                      December 9.                      December 9
Calculus
Project – Optimize our Zoo Using Functions (Revised)
Introduction

You have been hired by the local community to advise them on how to best design their new zoo. They have
limited materials to use and they want to ensure that they get the most out of the materials. You will need to
examine their requirements, determine the best way to meet the requirements, and make a scale model or map

Objectives:

   You will be able to model real situations with functions.
   You will be able to represent the functions algebraically, numerically (data table), and graphically.
   You will be able to use proportions to produce a map to scale.

Zoo Requirements

1. Gorilla
a. We have _________ feet of fence to use for the gorilla cage.
b. We want the gorilla to live in a rectangular space with the maximum area possible to swing around.

c.

2. Sheep and Pigs
a. We have _____________ feet of fence to use for the sheep and pigs.
b. We want the sheep and pigs to live in two adjacent rectangular pens.
c. One side of the sheep and pigs area will be bordered by a river for drinking
d. We want to give the sheep and pigs the maximum area possible.
e. The pig area should be the same size as the sheep area.

f.
3. Lions, Tigers, and Bears
a. We have _____________ feet of fence to use for the lions tigers and bears.
b. The animals will live in one rectangular area divided into three equal parts as shown below.
c. We want the lions, tigers, and bears to live in the maximum area possible.

d.

4. Shark Aquarium
a. We want a rectangular aquarium with the left, right, front, back, and bottom is made of clear glass
so people can see the sharks
b. The top is open (no glass used for top)
c. The length should be ______________ times the width.
d. The tank should hold a volume of _____________.
e. We want to minimize the amount of glass that we use because the material is very expensive.

f.
Element                     4 Mastered                               3 Proficient                           2 Developing                            1 Emerging
Labeled picture of      You always do these with no             You do most of these with few (1        You do most of these with some (3       You do few or none of these or
description             errors:                                 2) minor errors:                        6) errors                               have major errors.
draw a clear picture of each           Draw a clear picture of each           Draw a clear picture of each           Draw a clear picture of each
application clearly labeled with        application clearly labeled with        application clearly labeled with       application clearly labeled with
all known and unknown                   all known and unknown                   all known and unknown                  all known and unknown
variables.                              variables.                              variables.                             variables.
Use Algebra to create   You always do these no errors:          You do most of these with few (1        You do most of these with some (3       You do few or none of these or
the Appropriate          write key equations/ functions        2) minor errors:                        6) errors                               have major errors.
Function(s)                that show how each of the             write key equations/ functions         write key equations/ functions         write key equations/ functions
variables are related to one            that show how each of the               that show how each of the              that show how each of the
another . (i.e. V=lwh).                 variables are related to one            variables are related to one           variables are related to one
arrange these equations to write         another . (i.e. V=lwh).                 another . (i.e. V=lwh).                another . (i.e. V=lwh).
an appropriate function(s) that       arrange these equations to write       arrange these equations to write       arrange these equations to write
can be used to describe each            an appropriate function(s) that         an appropriate function(s) that        an appropriate function(s) that
problem (i.e. Surface Area              can be used to describe each            can be used to describe each           can be used to describe each
function)                               problem (i.e. Surface Area              problem (i.e. Surface Area             problem (i.e. Surface Area
function)                               function)                              function)
Numerical Behavior of   You always do these no errors:          You do most of these with few (1        You do most of these with some (3       You do few or none of these or
the function             produce a clear table(s) of           2) minor errors:                        6) errors                               have major errors.
values that show the behavior of       produce a clear table(s) of            produce a clear table(s) of            produce a clear table(s) of
your function(s)                         values that show the behavior of        values that show the behavior of       values that show the behavior of
on your table(s) so it’s very easy     Use labels, variables, and units       Use labels, variables, and units       Use labels, variables, and units
to know what the data means.             on your table(s) so it’s very easy      on your table(s) so it’s very easy     on your table(s) so it’s very easy
Highlight any data on the                to know what the data means.            to know what the data means.           to know what the data means.
table(s) important to your             Highlight any data on the              Highlight any data on the              Highlight any data on the
solution and clearly explain why         table(s) important to your              table(s) important to your             table(s) important to your
it helps you solve the problem.          solution and clearly explain why        solution and clearly explain why       solution and clearly explain why
it helps you solve the problem.         it helps you solve the problem.        it helps you solve the problem.
Graphical Behavior of   You always do these with no             You do most of these with few (1        You do most of these with some (3       You do few or none of these or
the Function            errors:                                 2) minor errors:                        6) errors                               have major errors.
produce a clear graph(s) that          produce a clear graph(s) that          produce a clear graph(s) that          produce a clear graph(s) that
show the behavior of your               show the behavior of your               show the behavior of your              show the behavior of your
function(s)                             function(s)                             function(s)                            function(s)
Use labels, variables, and units       Use labels, variables, and units       Use labels, variables, and units       Use labels, variables, and units
on your graph(s) so it’s very easy      on your graph(s) so it’s very easy      on your graph(s) so it’s very easy     on your graph(s) so it’s very easy
to know what the curve means            to know what the curve means            to know what the curve means           to know what the curve means
Highlight any point(s) on the          Highlight any point(s) on the          Highlight any point(s) on the          Highlight any point(s) on the
curve that is important to your         curve that is important to your         curve that is important to your        curve that is important to your
solution and clearly explain why        solution and clearly explain why        solution and clearly explain why       solution and clearly explain why
it helps you solve the problem.         it helps you solve the problem.         it helps you solve the problem.        it helps you solve the problem.
Element                  4 Mastered                             3 Proficient                          2 Developing                            1 Emerging
Solve the problem   You always do these with no            You do most of these with few (1       You do most of these with some (3      You do few or none of these or
errors:                                2) minor errors:                       6) errors                              have major errors.
Clearly interpret all the             Clearly interpret all the             Clearly interpret all the             Clearly interpret all the
tables, and algebra to solve the       tables, and algebra to solve the       tables, and algebra to solve the       tables, and algebra to solve the
problem.                               problem.                               problem.                               problem.
The solution could be one             The solution could be one             The solution could be one             The solution could be one
number or a set of numbers so          number or a set of numbers so          number or a set of numbers so          number or a set of numbers so
labels and units are used to           labels and units are used to           labels and units are used to           labels and units are used to
clarify.                               clarify.                               clarify.                               clarify.
If possible, a picture is used to     If possible, a picture is used to     If possible, a picture is used to     If possible, a picture is used to
that you understand how the            that you understand how the            that you understand how the            that you understand how the
numbers solve the original             numbers solve the original             numbers solve the original             numbers solve the original
problem.                               problem.                               problem.                               problem.
Check to see that your solution       Check to see that your solution       Check to see that your solution       Check to see that your solution
makes sense.                           makes sense.                           makes sense.                           makes sense.
Scale Map           You always do these with no            You do most of these with few (1       You do most of these with some (3      You do few or none of these or
errors:                                2) minor errors:                       6) errors                              have major errors.
Make a creative, accurately           Make a creative, accurately           Make a creative, accurately           Make a creative, accurately
scaled map that represents your        scaled map that represents your        scaled map that represents your        scaled map that represents your
proposal for the configuration of      proposal for the configuration of      proposal for the configuration of      proposal for the configuration of
the zoo.                               the zoo.                               the zoo.                               the zoo.
Show a legend that indicates the      Show a legend that indicates the      Show a legend that indicates the      Show a legend that indicates the
scale of actual lengths to map         scale of actual lengths to map         scale of actual lengths to map         scale of actual lengths to map
lengths.                               lengths.                               lengths.                               lengths.
Label all required areas and          Label all required areas and          Label all required areas and          Label all required areas and
clearly indicate any key               clearly indicate any key               clearly indicate any key               clearly indicate any key
measurements                           measurements                           measurements                           measurements
On Time             You’re project is completed and        You’re project is completed and        You’re project is completed and        You’re project is completed and
handed in before or at the             handed in within 2 days past the       handed in within 4 days past the       handed in more than 4 days past
beginning of class on the due date:    due date:                              due date:                              the due date:
January 14, 2009                       January 14, 2009                       January 14, 2009                       January 14, 2009

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