AP CALCULUS AB SYLLABUS

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AP CALCULUS AB SYLLABUS Powered By Docstoc
					                                   AP CALCULUS AB SYLLABUS

TEXT:
Larson, R., Hostetler, R., Edwards, B., and Heyd, D. (2002) Calculus of a Single Variable. Boston:
Houghton Mifflin.

MAIN REFERENCE MATERIALS:

Albert, B. & P. Hillis. Calculus Calculator Labs. Andover: Skylight Publishing.

College Board Professional Development Workshop Materials

Finney, Demana, Waits, Kennedy. Calculus, Graphical, Numerical, Algebraic. (2003) Prentice Hall,
ISBN: 0-13-063131-0

Foerster, P. Calculus Concepts and Applications. Key Curriculum Press. ISBN: 0-471-48481-4

Kamischke, E.A Watched Cup Never Cools. Key Curriculum Press. ISBN: 1-55953-318-8

Lederman, D. Multiple Choice & Free Response Questions in Preparation for the AP Calculus (AB) or
(BC) Examination. 8th Edition. D & S Marketing Systems, Inc.

Lipp, Alan. Little Books of Big Ideas. The Peoples Publishing Group, Inc. ISBN:1-4138-1322-4

Course Description:
 3113A AP Calculus           Pre-requisites and/or Requirements: Calculus Honors
This course is concerned with developing the students’ understanding of the concepts of differential and
integral single variable calculus providing experience with its methods and applications. The course
emphasizes a multipresentational approach to calculus, with concepts, results, and problems being
expressed graphically, numerically, analytically, and verbally. The course represents a college-level
mathematics. The students will be expected to participate in the corresponding advanced placement exam.



COURSE PLANNER:

Unit 1: Prerequisites for Calculus (3 weeks)

       Review solving equations and using the graphing calculator
            o Intercepts, symmetry, domain, range
       Linear models and rates of change
       Functions and graphs
            o Linear, power, inverses, exponential, logarithmic, trigonometric, inverse trigonometric,
                piecewise and composite functions
            o Transformations of functions and corresponding graphs
            o Analysis of graphs


Unit 2: Limits and Continuity (3 weeks)

       Finding limits graphically and numerically
       Evaluating limits analytically
       Continuity and one-sided limits
       Infinite limits
       AP - Limits of functions (including one-sided limits).
           o An intuitive understanding of the limiting process
           o Calculating limits using algebra
           o Estimating limits from graphs or tables of data

       AP - Asymptotic and unbounded behavior
            o Understanding asymptotes in terms of graphical behavior
            o Describing asymptotic behavior in terms of limits involving infinity
            o Comparing rates of change of exponential, polynomial and logarithmic functions

       AP - Continuity as a property of functions
           o An intuitive understanding of continuity
           o Understanding continuity in terms of limits
           o Geometric understanding of graphs of continuous functions
           o Intermediate Value Theorem and Extreme Value Theorem


Unit 3: Differentiation (6 weeks)

       The Derivative and the Tangent Line Problem
       Discover the Derivative Function from a Table of Values and by Exploration
       Basic Differentiation Rules and Rates of Change
       The Product and Quotient Rules and Higher-Order Derivatives
       The Chain Rule
       Implicit Differentiation
       Related Rates
       Derivatives of Exponential and Logarithmic Functions

       AP – Concept of the Derivative
            o Derivative presented graphically, numerically, and analytically
            o Derivative interpreted as an instantaneous rate of change
            o Derivative defined as the limit of the difference quotient
            o Relationship between differentiability and continuity

       AP - Derivative at a point
           o Slope of a curve at a point
           o Tangent line to a curve at a point
           o Instantaneous rate of change as the limit of average rate of change
           o Approximate rate of change from graphs and tables of values

       AP - Computation of derivatives
            o Knowledge of derivatives of basic functions, including power, exponential, logarithmic,
               trigonometric and inverse trigonometric functions
            o Basic rules for the derivatives of sums, products, and quotients of functions
            o Chain rule and implicit differentiation


Unit 4: Applications of Derivatives (4 weeks)

       Extrema on an Interval
       Rolle’s Theorem and the Mean Value Theorem
       Increasing and Decreasing Functions and the First Derivative Test
       Concavity and the Second Derivative Test
       Limits at Infinity
       A Summary of Curve Sketching
       Optimization Problems
       Newton’s Method
       Differentials
            o Linear Approximation

       AP – Derivative of a function
           o Corresponding characteristics of f, f’, and f’’
           o Relationship between the increasing and decreasing behavior of f and the sign of f’
           o The Mean Value Theorem
           o Equations involving derivatives. Verbal descriptions translated in equations involving
               derivatives and vice versa.

       AP – Derivative at a point
           o Local linear approximation

       AP – Second derivatives
           o Corresponding characteristics of the graphs f, f’, and f’’
           o Relationship between concavity of f and the sign of f’’
           o Points of inflection as places where concavity changes

       AP – Applications of Derivatives
           o Analysis of curves, including the notions of monotonicity and concavity
           o Optimization, both absolute (global) and relative (local) extrema
           o Modeling rates of change, including related rates problems
           o Use of implicit differentiation to find the derivative of an inverse function
           o Interpretation of the derivative as a rate of change in varied applied contexts, including
               velocity, speed, and acceleration


Unit 5: Integration (4 weeks)

       Antiderivatives and Indefinite Integration
       Area
       Riemann Sums and Definite Integrals
       The Fundamental Theorem of Calculus
       Integration by Substitution

       AP - Interpretations and properties of definite integrals
            o Computation of Riemann Sums using left, right, and midpoint evaluation points
            o Definite integral as a limit of Riemann Sums over equal subdivisions
            o Definite integral of the rate of change of a quantity over an interval interpreted as the
                 change of the quantity over the interval:
                 b

                  f ( x)dx  f (b)  f (a)
                 a
            o    Basic properties of definite integrals

       AP – Applications of Integrals

       AP – Fundamental Theorem of Calculus
           o Use of the Fundamental Theorem to evaluate indefinite integrals
           o Use of the Fundamental Theorem to represent a particular antiderivative, and the
               analytical and graphical analysis of functions so defined

       AP – Techniques of antidfferentiation
             o   Antiderivatives following directly from derivatives of basic functions


    Unit 6: Differential Equations and Mathematical Models (4 weeks)

       Numerical Integration; Trapezoidal Rule
       Antiderivatives and Slope Fields
       Integration of Logarithmic and Exponential Functions
       Exponential Growth and Decay
       Population Growth and Numerical Methods

       AP – Numerical Approximations to definite integrals
           o Use of Riemann and trapezoidal sums to approximate definite integrals of functions
               represented algebraically, graphically, and by tables of values

       AP – Geometric interpretation of differential equations via slope fields and the relationship
        between slope fields and solution curves for differential equations

       AP – Applications of Integrals

       AP - Techniques of antidifferentiation
            o Antiderivatives by substitution of variables (including change of limits for definite
               integrals)

       AP – Applications of antidifferentiation
           o Solving separable differential equations and using them in modeling (i.e. studying the
               equation and exponential growth)


    Unit 7: Applications of Integration (3 weeks)

       Area of a Region Between Two Curves
       Volume: The Disk Method
       Volume: The Shell Method
       Arc Length and Surfaces of Revolution

       AP – Applications of integrals
           o Finding specific antiderivatives using initial conditions, including applications to motion
               along a line
           o Solving separable differential equations and using them in modeling (i.e. studying the
               equation and exponential growth)


REVIEW FOR THE AP EXAM

Beginning in January, I assign an AP free-response question for students to work on every week. Students
may work on these questions with one other person and come to me for extra help. I randomly select from
these questions an AP free-response question on every test. Students are graded as they would be graded
on an AP Exam. Every test beginning in January will consist of a multiple-choice section of AP Calculus
Released Exams. Times allotments should allow ample weeks of review for the AP exam, which is
scheduled for the 2nd week of May. The review will consist of sessions with parts of previous exams. This
will vary between calculator and non-calculator problems from the multiple choice and free response
sections. The review period will also include a mock scenario using the 2003 exam in order to simulate the
actual conditions of the exam.
STUDENT EVALUATION

Grades for this course will be based on two categories:

     1.   Quizzes, homework, classwork, and projects (40% of each grading period average). Daily
grades will be taken from homework, classwork, and bell ringers. Short quizzes (25 points) cover the
previous day’s homework; longer quizzes (50 points) cover 1 or 2 concepts or book sections. Labs
(examples follow) usually count 50 points; projects (examples follow) count 100 points.

      2. Tests and 9-week examinations (60% of each grading period average). Chapter tests count 100
points and are given at the end of each chapter. The 9-week exam counts the same as a chapter test. The
exams given at the end of the first and second 9-week grading periods will test the material covered during
that grading period. The exam given at the end of the third grading period will be comprehensive and use
AP style questions. The fourth grading period exam will be given after the AP exam and will cover
any additional topics covered that grading period. Quizzes, tests and exams will feature both calculator and
non-calculator sections.


LABS/PROJECTS
The following is a brief description of several of the homework/projects assigned:


             1.   To review functional transformations a group project in the form of bingo is used. A list
                  of functions is provided and the class chooses 25 at random to add to their bingo sheet.
                  The corresponding graphs are then shown using the projector. The class is expected to
                  match the graphs appropriately based on their knowledge of functions, graphs, and
                  transformations.

             2.   Activity #2 is repeated using only graphs of functions and their corresponding graphs of
                  the first derivative and second derivative.

             3.   Class activity involving rules of differentiation. Again, bingo is used, but in this case, the
                  boxes are already completed with the derivatives and the students are provided a list of
                  functions.

             4.   Discover the derivative by exploration.
                                             ln( x  h)  ln( x)
                  Exploration 1. Let Y1 =                        . Set up the Table: Tbl Start 0; ΔTbl = 1
                                                      h
                  Looking at the Table, can you guess the derivative of y = ln(x)?
                                                                          x      e x  0.001  e x
                  Exploration 2. To discover the derivative of y =e , graph Y1 =                   . Use
                                                                                      0.001
                  ZOOM4. Can you determine the derivative function?

                  Exploration 3. We want to investigate the derivative of y = sin x, using one of the
                                                          f ( x  h)  f ( x )              sin( x  L1 )  sin x
                  definitions of the derivative,   lim                         . Enter Y1 =                       .
                                                   h 0            h                                 L1
                  Press STAT, EDIT, and enter 5, 1, 0.1,and 0.001 in L1. Use ZOOMTRIG to graph Y1.
                  Each value in L1 is the “h” value in the derivative definition, so there will be four graphs
                  displayed. As the “h” values approach 0, what does the graph of Y 1 resemble? You can
                  graph Y2 = cos x simultaneously with Y1. Does this confirm your guess?
             5.   Students explore the concept of average rate of change and discover the concept of
                  instantaneous rate of change in a ball toss lab. We use the Vernier computer interface,
                  Logger Pro, Vernier Motion Detector, or the Calculator-Based Laboratory. The students
                  examine the distance-verses-time graph. They fit the data with a quadratic equation for
                  the position function to determine how high the ball traveled and time traveled. The
                  students calculate the average velocity of the ball over a period of time. Students are then
                  asked to investigate the exact velocity at different points in time. Depending on the use
                  of either the Vernier computer interface and Logger Pro or CBL the students either trace
                  or zooms in on the position functions graph to find the slope of the line and compare it to
                  their estimation of the instantaneous velocity.

             6.   During the study of related rates, students suck on Tootsie Roll Pops to determine the rate
                  of change of their radius; they then calculate the rate of change of the Pops’ volume.
                  Students measure the initial radius of a Pop with dental floss. They then suck on the Pop
                  for 30 seconds, record its radius, suck for another 30 seconds, etc. They model the rate of
                  change of the radius with some function of time. Students then use this rate of change to
                  estimate the rate of change of the volume of the Pop when its radius is three-fourths of its
                  original radius. This lab, “How Many Licks?,” can be found in Ellen Kamischke’s book,
                  A Watched Cup Never Cools.

             7.   Optimization Lab - Working with a variety of materials, some students will form prisms
                  and pyramids of maximum volume. First, the maximum volume possible will be
                  calculated, then the solid will be formed. Other students will calculate and form solids of
                  minimum surface area which would contain a given volume. A short paper will be
                  written describing the process.

             8.   As an exploration problem and introduction to optimization problems, the students are
                  asked to form a 5-sided box from a standard sheet of poster board with a maximum
                  volume.

             9.   Volume Lab -- Working with a partner, the student will build a model of a solid formed
                  by rotating a function around an axis. The model will be formed by shells or disks. The
                  volume of the model will be calculated. A short paper will be written discussing the
                  project and findings.


Technology

Each of my students must have a graphing calculator of his or her own. We use graphing calculators daily
in class to introduce, explore, discover, and reinforce the concepts of calculus; therefore, the students are
expected to have it in class each day. The calculator is also used to analyze data, analyze graphs, and
compare derivatives. The majority of students either use a TI-83 Plus, TI-84, or TI-89. We begin using the
graphing calculators with the review of many functions.

We investigate roots, domain, range, minimums, maximums, and continuity using the different features of
the graphing calculator. We continue the use of the calculator to investigate limits. We use the table and
trace features to view the graphs and make the connection of the limit graphically, numerically,
analytically, and verbally.

The students use the calculator when guessing the derivative function from a table of values. Before the
students learn the rules for differentiating the elementary functions, the students take some simple functions
and create a small table of values for f(x), say at x = 0, 1, 2, 3, then uses the difference quotient to
approximate f’(x). The students look at the values in the table, try to see the pattern, and discover the
equation for f’(x).
The students use the graphing calculator to explore and discover for themselves certain derivatives.
(lab/project #4)

Students explore the concept of average rate of change and discover the concept of instantaneous rate of
change as described in the ball toss lab (#5 lab/project).

We explore the position function verses velocity function verses acceleration function in a simple
pendulum lab similar to the ball toss lab in the investigation of the sine and cosine functions.

We continuously use the graphing calculator in making connections between f, f’, and f’’ in tables and in
the graphs. Investigating functions differentiable at x = a as well as not differentiable at x = a will be
included algebraically and graphically. Algebraically and graphically the students will investigate and
confirm increasing/decreasing behavior using f’.

Students use graphing technology exploring Riemann sums. Although the investigation of area is the basic
context we include regions below the axis as well as those above. Students discover that Riemann sums are
not always good predictors of the answer if area is the question.