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AP Calculus AB
Course Description:
Advanced Placement Calculus AB is primarily concerned with developing the students’ understanding
of the concepts of calculus and providing experiences with its methods and applications. The four major
topics of this course are limits, differential calculus, integral calculus, and their applications. All
students will be required to analyze problems graphically, numerically, analytically and verbally in this
course. The course content will follow the outlines set forth by the College Board and the state. All
students will be required to use a TI-89 series graphing calculator for this course.
Course Prerequisites:
All students enrolling in AP Calculus AB should successfully complete Pre-Calculus. By successfully
completing Pre-Calculus all students should have a working understanding of linear, polynomial,
rational, exponential, logarithmic, trigonometric, inverse trigonometric, and piecewise defined functions.
Students must also be familiar with the properties of functions, the algebra of functions, the graphs of
functions, the language of functions (domain, range, odd, even, periodic, symmetry, zeros, intercepts,
etc.) and understand the concept of the unit circle.
Textbook:
Larson, Hostetler, Edwards. Calculus, 6th Edition, Houghton Mifflin, © 1998
AP Review Supplemental Workbook:
Lederman, David. Multiple-Choice & Free-Response Questions in Preparation for the AP Calculus
(AB) Examination, 8th Edition, D&S Marketing Systems, © 2003
Student Evaluation:
1st, 2nd & 3rd Nine Weeks: Exams 65%
Quizzes 15%
Daily Warm-ups 10%
Assignments 10%
4th Nine Weeks Exam 40%
AP Free-Response 25%
AP Multiple Choice 25%
Assignments 10%
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Student Assessments:
Exams will be given at the completion of a unit (typically at the end of a chapter). Most exams will span
over a 57 minute class period. The majority of the exam will be without the use of a calculator and the
remainder of the exam will require the use of a TI-89 series graphing calculator.
Quizzes will be given during the coverage of the chapter, usually covering 2 – 4 sections of textbook
material. These quizzes will be completed during one class period and some will be with the use of a
graphing calculator and others will be without.
Daily warm-ups are calculus problems given at the beginning of each class to assess the students’
understanding of concepts taught a few days earlier. Daily warm-up problems are collected and graded
each day.
Assignments are graded works which assess the students understanding of the basic calculus concepts.
Assignments are given daily.
AP Review Problems will be assigned daily beginning with the fourth nine weeks. These problems are
past AP Free Response and AP Multiple Choice questions. All AP free response questions will be
assigned to students to complete from the past five examination cycles. AP Multiple Choice will be
review assignments from the D&S Marketing review workbook. Students will practice the multiple
choice questions in these workbooks to help them prepare for the AP examination.
Old AP questions will be given throughout the year as appropriate on quizzes and exams, as well as
homework assignments.
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AP Calculus AB: Course Outline
The 2007-2008 school year begins on August 20, 2007 in Orange County Florida. The AP Calculus
Exam will be administered on May 7, 2008. This gives OrangeCounty students 158 in session school
days to learn calculus and prepare for the AP Calculus exam. All of the days listed below are an
approximation. This is a typical schedule, but it does change slightly from year to year with the number
of days spent on each topic.
First Semester:
Chapter 1: Limits and Their Properties 15 days
Section 1.1: A Preview of Calculus 1 day
Section 1.2: Finding Limits Graphically and Numerically 1 day
Section 1.3: Evaluating Limits Analytically 3 days
Section 1.4: Continuity and One-Sided Limits 3 days
Review (Sections 1.1 – 1.4) 1 day
Quiz (Sections 1.1 – 1.4) 1 day
Section 1.5: Infinite Limits 2 days
Section 3.5: Limits at Infinity 1 day
Review (Chapter 1) 1 day
Exam (Chapter 1) 1 day
Chapter 2: Differentiation 29 days
Section 2.1: The Derivative and the Tangent Line Problem 2 days
Section 2.2: Basic Differentiation Rules and Rates of Change 2 days
Section 2.3: The Product and Quotient Rules and Higher-Order Derivatives 2 days
Section 2.4: The Chain Rule 3 days
Review (Sections 2.1 – 2.4) 1 day
Quiz (Sections 2.1 – 2.4) 1 day
Section 2.5: Implicit Differentiation 3 days
Section 2.A: Supplemental Unit: Particle Motion 5 days
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Review (Section 2.5 & Particle Motion) 1 day
Quiz (Section 2.5 & Particle Motion) 1 day
Section 2.6: Related Rates 5 days
Supplemental Worksheet on Related Rates 1 day
Review (Chapter 2) 1 day
Exam (Chapter 2) 1 day
Chapter 3: Applications of Differentiation 27 days
Section 3.1: Extrema on an Interval 2 days
Section 3.2: Rolle’s Theorem and the Mean Value Theorem 2 days
Section 3.3: Increasing and Decreasing Functions and the First Derivative Test 2 days
Section 3.4: Concavity and the Second Derivative Test 2 days
Section 3.6: A Summary of Curve Sketching 2 days
Supplemental worksheets on Sketching Functions 2 days
Review (Sections 3.1 – 3.6) 1 day
Test (Sections 3.1 – 3.6) 1 day
Section 3.7: Optimization Problems 3 days
Supplemental Worksheet on Optimization Problems 1 day
Section 3.8: Newton’s Method 2 days
Section 3.9: Differentials 2 days
Review (Sections 3.7 – 3.9) 1 day
Quiz (Sections 3.7 – 3.9) 1 day
Section 3.10: Business and Economic Applications 1 day
Review (Chapter 3) 1 day
Exam (Chapter 3) 1 day
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Second Semester:
Chapter 4: Integration 23 days
Section 4.1: Antiderivatives and Indefinite Integration 2 days
Section 4.2: Area 4 days
Section 4.3: Riemann Sums and Definite Integrals 2 days
Review (Sections 4.1 – 4.3) 1 day
Quiz (Sections 4.1 – 4.3) 1 day
Section 4.4: The Fundamental Theorem of Calculus 3 days
Section 4.5: Integration by Substitution 3 days
Review (Sections 4.4 – 4.5) 1 day
Quiz (Sections 4.4 – 4.5) 1 day
Section 4.6: Numerical Integration (Trapezoidal Rule and
Simpson’s Rule) 2 days
Review (Chapter 4) 2 days
Exam (Chapter 4) 1 days
Chapter 5: Logarithmic, Exponential, and Other Transcendental Functions 27 days
Section 5.1: The Natural Logarithmic Function: Differentiation 2 days
Section 5.2: The Natural Logarithmic Function: Integration 2 days
Section 5.3: Inverse Functions 2 day
Section 5.4: Exponential Functions: Differentiation and Integration 2 days
Section 5.5: Bases Other than e and Applications 3 days
Review (Sections 5.1 – 5.5) 1 day
Quiz (Sections 5.1 – 5.5) 1 day
Section 5.6: Differential Equations: Growth and Decay 1 days
Section 5.7: Differential Equations: Separation of Variables 2 days
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Section 5.A Supplemental Unit on Slope Fields 1 day
Supplemental worksheet on Slope Fields & Differential Equations 1 day
Section 5.8: Inverse Trigonometric Functions: Differentiation 2 days
Section 5.9: Inverse Trigonometric Functions: Integration 2 days
Review (Sections 5.6 – 5.9, 5.A) 1 day
Quiz (Sections 5.6 – 5.9, 5.A) 1 day
Section 7.1: Basic Integration Rules 1 day
Review (Chapter 5) 1 day
Exam (Chapter 5 and Section 7.1) 1 day
Chapter 6: Applications of Integration 9 days
Section 6.1: Area of a Region Between Two Curves 1 day
Section 6.2: Volume: The Disk Method 3 days
Section 6.3: Volume: Shell Method 2 days
Section 7.7: Indeterminate Forms and L’Hôpital’s Rule 1 day
Review (Sections 6.1 6.3, 7.7) 1 day
Exam (Sections 6.1 6.3, 7.7) 1 day
Review: Review for AP Examination 20 days
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AP Calculus AB: College Board’s Topic Outline & Correlation to Textbook
The sections listed next to each topic, lists a section (or supplemental unit) of the textbook which
corresponds to each topic. Although several of these topics appear throughout a calculus course, the
section where it is first introduced is included for reference.
TOPIC TEXTBOOK SECTION
I. Functions, Graphs, and Limits
Analysis of Graphs
With the aid of technology, graphs of functions are often easy to produce. (1.2, 1.3)
The emphasis is on the interplay between the geometric and analytic
information and on the use of Calculus both to predict and to
explain the observed local and global behavior of a function.
Limits of Functions (Including one-sided limits)
An intuitive understanding of the limiting process (1.2)
Calculating limits using algebra (1.3)
Estimating limits from graphs or tables of data (1.2)
Asymptotic and Unbounded Behavior
Understanding asymptotes in terms of graphical behavior (1,4, 1.5, 3.5)
Describing asymptotic behavior in terms of limits involving infinity (3.5)
Comparing relative magnitudes of functions and their rates of change (5.6)
(for example, contrasting exponential growth, polynomial growth, and
logarithmic growth)
Continuity as a Property of Functions
An intuitive understanding of continuity. (The function values can be made (1.4)
as close as desired by taking sufficiently close values of the domain.)
Understanding continuity in terms of limits (1.4)
Geometric understanding of graphs of continuous functions (Intermediate (1.4)
Value Theorem and Extreme Value Theorem)
II. Derivatives
Concept of the Derivative
Derivative presented graphically, numerically, and analytically (2.1, 2.2)
Derivative interpreted as an instantaneous rate of change (2.2, 2.A)
Derivative defined as the limit of the difference quotient (2.1)
Relationship between differentiability and continuity (2.1)
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Derivative at a Point
Slope of a curve at a point. Examples are emphasized, including points (2.1)
at which there are vertical tangents and points at which there are
no tangents.
Tangent line to a curve at a point and local linear approximation (2.1)
Instantaneous rate of change as the limit of average rate of change (2.2)
Approximate rate of change from graphs and tables of values (2.2)
Derivative as a Function
Corresponding characteristics of graphs of f and f (3.3)
Relationship between the increasing and decreasing behavior of f and the (3.3)
sign of f
The Mean Value Theorem and its geometric consequences (3.2)
Equations involving derivatives. Verbal descriptions are translated into (5.6)
equations involving derivatives or vice versa.
Second Derivatives
Corresponding characteristics of the graphs of f, f , f (3.3, 3.4, 3.6)
Relationship between the concavity of f and the sign of f (3.4, 3.6)
Points of inflection as places where concavity changes (3.4, 3.6)
Applications of Derivatives
Analysis of curves, including the notations of monotonicity and concavity (3.6)
Optimization, both absolute (global) and relative (local) extrema (3.1, 3.7)
Modeling rates of change, including related rates problems (2.6)
Use of implicit differentiation to find the derivative of an inverse function (5.3)
Interpretation of the derivate as a rate of change in varied applied contexts, (2.2, 4.4, 2.A)
including velocity, speed, and acceleration
Geometric interpretation of differential equations via slope fields and the (5.A)
Relationship between slope fields and solution curves for differential
equations
Computation of Derivatives
Knowledge of derivatives of basic functions, including power, exponential (2.2, 2.3, 5.1
logarithmic, trigonometric, and inverse trigonometric functions 5.3, 5.4, 5.8, 5.9)
Basic rules for the derivative of sums, products, and quotients of functions (2.2, 2.3)
Chain rule and implicit differentiation (2.4, 2.5)
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III. Integrals
Interpretation and Properties of Definite Integrals
Definite integral as a limit of Riemann sums (4.2)
Definite integral of the rate of change of a quantity over an interval (4.4)
interpreted as the change of the quantity over the interval:
b
a
f x dx f b f a
Basic properties of definite integrals (examples include additivity and (4.4)
linearity)
Applications of Integrals
Appropriate integrals are used in a variety of application to model physical, (3.10, 4.1, 4.2,
biological, or economic situations. Although only a sampling of applications 4.3, 4.4, 6.1,
can be included in any specific course, students should be able to adapt 6.2, 6.3)
their knowledge and techniques to solve other similar application problems.
Whatever applications are chosen, the emphasis is on using the method of
setting up an approximating Riemann sum and representing its limit as a
definite integral. To provide a common foundation, specific applications
should include using the integral of a rate of change to give accumulated
change, finding the area of a region, the volume of solid with known
cross sections, the average value of a function, and the distance traveled
by a particle along a line.
Fundamental Theorem of Calculus
Use of the Fundamental Theorem to evaluate definite intergrals (4.4)
Use of the Fundamental Theorem to represent a particular antiderivative, (4.4)
and the analytical and graphical analysis of functions so defined
Techniques of Antidifferentiation
Antiderivatives following directly from derivatives of basic functions (4.1)
Antiderivatives by substitution of variables (including change of (4.5)
limits for definite integrals)
Applications of Antidifferentiation
Finding specific antiderivatives using initial conditions, including (4.5)
applications to motion along a line
Solving separable differential equations and using them in modeling (5.6, 5.7)
(in particular, studying the equation y ky and exponential growth)
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Numerical Approximations to Definite Integrals
Use of Riemann sums (using left, right, and midpoint evaluation points) (4.2, 4.6)
and trapezoidal sums to approximate definite integrals of functions
represented algebraically, graphically, and by tables of values.
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AP Calculus AB: Evidence of Curricular Requirements
Curricular Requirement #1:
The course teaches all topics associated with Functions, Graphs, and Limits; Derivatives; and Integrals
as delineated in the Calculus AB Topic Outline in the AP Calculus Course Description.
This curricular requirement is met and on the four previous pages the entire Calculus AB topic outline
was reproduced and the section or supplemental unit where it is covered is denoted. In addition, some
topics that are not in the Calculus AB Topic Outline are also covered in this course. Examples of these
topics include Simpson’s Rule, L’Hôpital’s Rule, and Volume: Shell Method.
Curricular Requirement #2
This course provides students with the opportunity to work with functions represented in a variety of
ways – graphically, numerically, analytically, and verbally – and emphasizes the connections among
these representations.
Example #1:
Students are first introduced to limits numerically, then graphically, then analytically. Students will
construct tables of values, then graph the function to see if the two answers agree. During the next class
the analytical method is introduced and the three methods are compared. Students are encouraged to
verbalize their explanations and answers in class on a daily basis.
Example #2:
To find the area under the curve the students are exposed to several approximation techniques prior to
learning how to evaluate the definite integral algebraically.
Examples of this would include problems of the following nature:
Problem #1: (Numerical)
Suppose a volcano is erupting and readings of the rate r t at which solid materials are spewed into the
atmosphere are given in the table. The time t is measured in seconds and the units for r t are tones
(metric tons) per second.
t 0 1 2 3 4 5 6
r t 2 10 24 36 46 54 60
Use the Trapezoidal Rule with 3 subdivisions of equal length to approximate
6
0
r t dt .
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0.5 1 1.5 2 2.5
x
2
4
6
8
y
Problem #2: (Graphical)
Use the midpoint rule to approximate the area of the region bounded by the graph of the function and the
x-axis over the indicated interval. Use n 4 subintervals.
2 f x x 3 ; 0,2
Problem #3: (Analytical)
A particle moves along the x-axis so that its acceleration at any time t is given by a t 6t 18. At
time t 0 the velocity of the particle v 0 24 , and at time t 1 its position is x 1 20.
(A) Write an expression for the velocity v t of the particle at any time t.
(B) For what value(s) of t is the particle at rest?
(C) For what value(s) of t is the particle moving to the right on the time interval 0,8 ?
(D) Write an expression for the position x t of the particle at any time t.
At the conclusion of this students can present their finding to class by writing their solutions on the
board and verbally explaining the steps to their classmates.
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Curricular Requirement #3
The course teaches students how to communicate mathematics and explain solutions to problems both
verbally and in written sentences.
Students are encouraged to verbally express their suggestions on solving problems in class on a daily
basis. Students will also come to the board and demonstrate their solutions to the class.
Examples of questions that have been used on examinations for students to express the mathematics in
written sentences appear below.
Example #1:
If oil leaks from a tank at a rate of r t gallons per minute at time t, what does
120
0
r t dt represent?
Example #2:
During the first 40 seconds of a flight, the rocket is propelled straight up so that in t seconds it reaches a
height of 3 s 5t feet.
It can be shown that 2 15
ds
t
dt
.
Interpret
ds
dt
in the context of the problem.
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Curricular Requirement #4
The course teaches students how to use graphing calculators to help solve problems, experiment,
interpret results, and support conclusions.
Example for Solving Problems:
Each student enrolled in this course will have a TI-89 series graphing calculator to use. The students are
expected to bring their graphing calculators to class with them each day. The calculators are used
frequently in class to explore, discover and reinforce the concepts that we are learning.
Example for Experiment:
When introducing the limit definition of a derivative, the students will sketch a function using their
graphing calculator and graph various secant lines to the curve which pass through a specific point
x, f x on the curve. Students will calculate the slope of their secant lines. To graph the other secant
lines students will choose a second point on the curve closer to fixed point of x, f x . They will
continue this pattern until they pick a point very close to the fixed point x, f x .
Example of Interpretation of Results:
Using the example above, students will notice that our secant line is very close to slope of the curve at
the fixed point x, f x . They will explore this further by “zooming in” on their original function
close to the fixed point and noticing that it appears almost linear over a small fixed interval.
Example of Supporting Conclusions:
The first function students experiment with is usually a quadratic. I continue this experimentation with
polynomial, exponential and logarithmic functions.