Advanced Placement Calculus AB by HC120704005715


									                 Advanced Placement Calculus AB

Course Description

Calculus AB is primarily concerned with developing the student's
understanding of the concepts of calculus and providing experience
with its methods and applications. The course will consist of a four
pronged approach to calculus. Concepts, results, and problems will be
expressed graphically, numerically, analytically, and using writing in
context. The connections among these representations also are
important are constantly related. Broad concepts and applicable
methods are emphasized. The focus of the course is neither
manipulation nor memorization of a predetermined list of functions,
curves, theorems, or problem types. Although students must deal with
manipulation and computational competence, they are not the sole
purpose of this course. Technology, in the form of a TI-84 calculator,
will be used throughout the course to reinforce the relationships
among the various representations of functions; f(x) to f ’(x) to f ’’(x),
to confirm written work, to encourage experimentation, and to assist
in interpreting results relating to slope, maxima/minima, and area
using tables, rectangular and trapezoidal approximations, and graphs.
Through the use of the calculator, students will become aware of
connections between derivatives, integrals, limits, approximations, and
applications with modeling. As a result, the course becomes a
connected whole rather than a collection of unrelated topics.


Calculus: Graphical, Numerical, Algebraic by Finney, Demana, Waits,
Kennedy; Scott Foresman,Addison Wesley 1999

Multiple-Choice & Free-Response Questions in Preparation for the AP
Calculus AB Examination by David Lederman and Lin McMullin, 2004,
D&S Marketing


All students either have or are supplied with TI-84 calculators. The
students have been using these calculators in Algebra II Honors, Pre-
calculus Honors, and in AP Calculus AB to interprete graphs, determine
limits, identify asymptotes, find maxima and minima points, and to
determine where a function is increasing, decreasing concave up or
down using graphs and tables.

Calculation of Grades

Students will be tested approximately every 2 to 4 weeks. Each test
will consist of questions using a format that is similar to the AP Exam.
These questions will contain multiple choice questions that include
methodology questions, interpretation of graphs using calculators and
derivatives, numerical tables and their relationship to slope of a curve,
and essay questions from past exams that will include justification
using written statements to interpret what is happening
mathematically. Each test will consist of 4 components. Component 1
will be a timed multiple choice test of approximately 9-10 questions.
This component is non-calculator active. Component 2 will be a timed
multiple choice test of 5-6 questions. This component is calculator
active. Component 3 will consist of 1 essay question (show your work)
consisting of multiple parts that must be completed in 15 minutes.
This question will be calculator active. Component 4 will be 1 essay
question to be completed in 15 minutes. Calculators may not be used
on this question. Grading of the test will be completed in the following
manner: Components 1 and 2 will be combined. Students will earn
credit based upon right, wrong, and blank answers using the formula
(# correct - 1/4 # wrong) x 1.2 = points earned. Blank answers will
not count against the student. 2. Components 3 and 4 will be graded
based upon the student's work. Answers only will only earn the answer
point. Components 1 and 2 are worth 18 points and Components 3 and
4 are worth 9 points each for a total of 36 possible points on each test.

Grades are assigned in the following manner:

36 points----------100
31-35 points--------95
25-30 points--------93
19-24 points--------90
13-18 points--------83
7-12 points--------74
1-6 points--------65

Students may earn bonus points weekly on 10 minutes quizzes given
on Tuesday and Thursday. These points will be totaled at the end of
the nine weeks and divided into 5 zones. Zone 1 will earn 0 points.
Zone 2 will earn 1 point. Zone 3 will earn 2 points. Zone 4 will earn 3
points. Zone 5 will earn 4 points which will be added to the nine weeks
average. These quizzes cannot hurt the student’s grade, but they do
consist of questions that are AP formatted with regards to the multiple
choice form.

Each nine weeks average will be computed as a simple average. Each
semester average will be computed as 1st nine weeks times .4 plus
2nd nine weeks times .4 plus the exam grade. This total will then be
divided by 5 to obtain the semester average.

The grading scale for our District is as follows:

            A     93-100
            B     85-92
            C     77-84
            D     70-76
            F     69 and below

Our students do receive additional grade point value for each grade
due to AP calculus being considered an honors course.

Topics and Timeline for First Semester

Section     Topic                                             Timeline
1.1 – 1.3   Equations of lines; domain, range, and            1day
            zeroes of functions; exponential growth
            and decay functions; parametric forms
1.4-1.6     Continuity; vertical and horizontal               1day
            Asymptotes (calculator activity)
2.1         Average/instantaneous rate of change;             3 days
            Definition of limit; Properties of limits;
            Left/Right-hand limits; Sandwich Theorem
            (calculator activity-analysis of functions)
2.2         Limits as x approaches form of infinity;          2 days
            Limits of infinity; End behaviors
            (calculator activity-table analysis of function
2.3         Continuity at a point; Continuous function;       2 days
            Combinations/Composition of continuous
            Functions; Intermediate Value Theorem
2.4         Average rate of change and slope; Equations       2 days
            of tangent lines; Slope of a curve; Normal
            lines to a curve; Speed
3.1         Definition of derivative; Notation; Connections   4 days
           between graphs of f and f ’; Graph derivatives
           from data and tables

           Test Review                                             2 days

Test # 1   Covers all topics listed above                          2 days

3.2        Failure of f ’(x) to exist; Differentiability implies    2 days
           linearity and continuity; Intermediate Value
           Theorem for derivatives; Use calculator to find
           derivatives and graph same
3.3        Rules for differentiation: positive integer              3 days
           powers; Sums and differences; Products and
           quotients; Negative integer powers; Higher
           order derivatives
3.4        Instantaneous rate of change; velocity; motion           3 days
           along a line; sensitivity to change; Applications
3.5        Derivatives of sine/cosine; Simple harmonic              2 days
           motion; Jerk; Derivatives of tangent/cotangent
           and Secant/Cosecant

           Test Review                                              2 days

Test # 2   Derivatives of above, definition of derivative,     2 days
           multiple derivatives, domain of a function,
           velocity/acceleration, limits, reflections, zeroes,
           equations of tangent/normal lines, parallel/
           perpendicular lines, position, speed, jerk

3.6        Derivatives of composition; Repeated use of        5 days
           chain rule; Slopes of parametric curves; Chain
           rule with powers of functions
3.7        Implicit functions and their derivatives; Tangents 2 days
           and normals; implicit derivatives of higher order;
           rational powers and their derivatives
3.8        Derivatives of inverse trigonometric functions      2 days
3.9        Derivatives of all forms of exponential functions; 3 days
           derivatives of all forms of logarithmic functions

           Test Review                                               2 days

Test # 3   Derivatives of any type of function mentioned             2 days
           above, zeroes of functions, multiple derivatives,
           laws of logarithms, continuity, asymptotes,
           equations of tangent/normal lines, derivatives
           of composition, definition of derivative, odd/even
           functions, vertical/horizontal tangents, slope of
           a curve,

4.1        Absolute extreme values; Local extreme values; 2 days
           finding extreme functional values
4.2        Mean Value Theorem; interpretation; increasing/ 2 days
           decreasing functions
4.3        First derivative test for local extrema; concavity; 3 days
           points of inflection; second derivative test for
           local extrema; functions from derivatives

           Test Review                                          2 days

Test # 4   Asymptotes, increasing/decreasing functions,          2 days
           Concavity, maxima/minima, point of inflection,
           Odd/even functions, behavior of tangent lines,
           Equations of tangent lines, implicit differentiation,
           Derivatives, limits, maxima/minima with justification,
           Concavity/point of inflection with justification,
           Sketch graph using first derivative information,
           Use graph of first derivative to find maxima/minima
           with written justification, use graph of first derivative to
           find inflection points with written justification

4.4        Modeling and optimization Methodology; Business 5 days
           and industry; Mathematics; Economics
4.5        Linearization; Newton’s method; Differentials;  5 days
           L”Hopital’s Rule
           Estimating change with derivatives; Absolute,
           relative, and percent of change
4.6        Related rates; solution strategy                5 days

           Test Review                                          2 days

Test # 5   Normal/tangent lines, related rates, derivatives, 2 days
           Definition of derivative, limits, maxima/minima,
           Evaluate derivatives, odd/even and symmetry,
           Differentiability, zeroes of a function, Mean Value
           Theorem for derivatives
First Semester Exam—covers all topics listed above; Split format: 28
questions-multiple choice without calculator and 17 questions-multiple
choice with calculator.

Topics and Timeline for Second Semester

Section                  Topic                             Timeline

5.1        Distance traveled; Rectangular Approximation       2   days
           Left, right, and midpoint rectangular approx-
5.2        Summation notation; Reimann sums;                  2   days
           Notation of integration; Definite integrals and
           area; Constant functions; Integrals on calculator;
           Discontinuous integrable functions
5.3        Properties of definite integrals; Average value    3   days
           of a function; Mean Value Theorem for Definite
           Integrals; Connecting Differential and Integral
5.4        Forms of Fundamental Theorem of Integral           3   days
           Calculus; Area connection; Applications
5.5        Trapezoid and Simpson’s approximations             2   days
6.1        Initial value, anti-derivatives and indefinite     2   days
           integrals, properties of indefinite integrals,
6.2        Integration by substitution, change of limit,      2   days
           separation of variables for differential equations

           Test review                                       2 days

Test # 6   Symbolic differentiation, First Fundamental Th. 2 days
           of Integral Calculus, definite integrals, properties of
           integrals, definition of derivative, integrals with
           absolute value, Fundamental Theorem of Integral
           Calculus (alternative version), integration with unknown
           limits, max/min, symmetry, inverses, area between
           curves, equations of tangent lines, points of inflection
           with justification

6.3        Integration by parts, repeated use, solve unknown 2 days
6.4        Solve exponential change using separation of        3 days
6.5        Exponential and logistics growth models; Partial    2 days
6.6        Euler’s method                                      2 days

           Test review                                         2 days

Test # 7   Definite integrals, derivatives, integrals with      2 days
           unknown limits, derivatives and distance using
           graphs, tangent or normal lines, functions from
           graphs, increasing or decreasing functions,
           concavity, Fundamental Theorem of Integral
           Calculus, area under curve, definition of derivative
           and continuity, Mean Value Theorem for
           derivatives, Maxima and minima with analysis,
           roots and average value of a function, initial
           conditions problem, velocity

7.1        Linear motion, net change from data, work          3 days
7.2        Area between curves, boundaries with               3 days
           changing functions, integration with respect
           to y
7.3        Volumes of solids: disc, washer, and shell;        6 days
           known cross-sections and volume
7.4        Length of smooth curves, vertical tangents,        3 days
           corners, and cusps

           Test Review                                        2 days

Test # 8   Concavity, indefinite integrals, tangent lines,    2 days
           approximation of zeroes, limits from graphs,
           definite integrals, relative extrema, derivatives,
           initial conditions problems, area using properties
           of integrals, area between curves, volume of
           known cross-sectional area, integration by
           parts, Intermediate Value Theorem, instantaneous
           and average velocity, total distance traveled,
           area between curves, length of a curve

Review for AP exam Practice problems in exam format. 20-25 days
          Students will initially work in groups of 2-3.
          Practice problems will be worked in class in an
             exam format. Students will consult with each
             other or the teacher to address specific difficulties.
             After 10 class days, the students will work
             individually on practice tests under a time format
             that conforms to AP guidelines. After each
             session, questions may be asked about
             specific problems.

Practice AP exam Previous AP multiple-choice exam and last 1 day
          year’s essay exam will be given following exam
          time and form requirements

Final Review Students will discuss the practice exam or any     5 days
           specific topics they feel are necessary to help them
           achieve their desired score on the exam

Total days                                                       162 days

AP Exam

Last days of class Review of essay exam and preparation 14 days
                   for BC calculus

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