AP Calculus AB
Course Syllabus
Math Department
E-Mail:
Emily.lucas@pikeville.kyschools.us
Grade Level: 11-12
Contact Times:
1:15 P.M. – 2:05 P.M.
Credit: 1 hour
School Phone: 606-432-0185
Fees: None
Prerequisite: Algebra I, Algebra II, Geometry, Pre-Calculus
Course Description:
This 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.
Course Design and Philosophy:
Students do best when they have an understanding of the conceptual underpinnings of calculus.
Rather than making the course a long laundry list of skills that students have to memorize, the
“why’ behind the major ideas is stressed. If students can grasp the reasons for an idea or theorem,
they can usually figure out how to apply it to the problem at hand.
Teaching Strategies:
During the first few weeks, extra time is spent familiarizing students with graphing calculators.
Students are taught the rule of three: Ideas can be investigated analytically, graphically, and
numerically. Students are expected to relate the various representations to each other. The
graphing calculator is used to help students develop an intuitive feel for concepts before they are
approached through typical algebraic techniques. Finding a root, sketching a function in a
specified window, approximating the derivative at a point using numerical methods,
approximating the value of a definite integral using numerical methods, and other calculator
functions will ultimately be emphasized.
It is important for them to understand that graphs and tables are not sufficient to prove an idea.
Verification always requires an analytic argument.
It is important to maintain a high level of student expectation. Students will rise to the level that is
expected of them. A teacher needs to have more confidence in the students than they have in
themselves.
Communication is stressed as a major goal of the course. Students are expected to explain
problems using proper vocabulary and terms. Many times students take the role of teacher in
their small groups or upon occasion to the whole class. Communication between classmates is
encouraged and facilitated.
1
Resource Materials:
Primary Textbook Title: Calculus, 7th edition
Author: Larson/Hostetler/Edwards
Publisher: Houghton Mifflin Company
Copyright: 2002
ISBN: 0-618-14918-X
Resource Textbook Title: Calculus, 5th edition
Author: James Stewart
Publisher: Brooks/Cole
Copyright: 2003
ISBN: 0-534-39366-7
Resource Title: Multiple-Choice & Free-Response Questions in Preparation for the AP Calculus (AB)
Examination, 7th edition
Title: Accompanying Student Solutions Manual, 7th edition
Author: David Lederman & Lin McMullin
Publisher: D&S Marketing Systems, Inc.
Copyright: 1999
Resource Title: Fast Track to a 5: Preparing for the AP Calculus AB and BC Examinations
Author: Sharon Cade, Rhea Caldwell, & Jeff Lucia
Publisher: McDougal Little
Copyright: 2006
ISBN: 0-618-14944-9
Technology Resources: Geometer’s Sketchpad, Derive, TI-84 Plus Graphing Calculator
Online Help: http://archives.math.utk.edu/topics/calculus.html
http://www.math.com/homeworkhelp/Calculus.html
http://hotmath.com/
Grading Policy:
Nine Weeks Grade will comprise:
Test and Quizzes – 70% of grade
Homework and related assignments – 30% of grade
Grading Scale:
93-100
A
83-92
B
73-82
C
63-72
D
62-Below F
Performance Standards and Expectations:

Students will use graphing calculators to understand the relationships between the analytic and
graphical characteristics of functions, such as predicting and explaining the observed local and
global behavior of a function.
2
Performance Standards and Expectations: (Continued)


















Students will find limits algebraically, estimate limits graphically and numerically, and use limit
notation correctly.
Students will know the definition and properties of a continuous function as they relate to limits
and graphical interpretations; explain continuity at a point, and be able to identify functions
which are continuous.
Students will understand asymptotes in terms of graphical behavior and in terms of limits
involving infinity.
Students will find horizontal and vertical asymptotes using limits.
Students will be able to compare relative magnitudes of functions and their rates of change (for
example, contrasting exponential growth, polynomial growth, and logarithmic growth).
Students will be able to explain and derive the limit definition of the derivative with the
understanding of derivative as an instantaneous rate of change, and explain the relationship
between differentiability and continuity.
Students will be able to estimate derivatives of functions graphically, numerically, and using a
graphing calculator.
Students will be able to explain the difference between average and instantaneous rate of
change, approximate rates of change from graphs and tables, and sketch a graph of f´(x) given
the graph of f(x).
Students will be able to use implicit differentiation when applicable, such as the derivative of an
inverse function.
Students will be able to explain how to find critical points and extreme values, state and apply
the Mean Value Theorem.
Students will be able to sketch a graph of f(x), given characteristics of f, f´, f´´ (increasing and
decreasing behavior of f and the sign of f´; relationship between the concavity of f and the sign
of f´´; points of inflection as places where concavity changes). In addition, curve sketching will
draw upon the student’s knowledge of domain and range, vertical and horizontal asymptotes,
symmetry, continuity and differentiability.
Students will be able to use Calculus to solve application problems such as related rates,
optimization (absolute and relative extrema), velocity, speed, acceleration, and position.
Students will be able to differentiate and integrate basic functions (power, exponential,
logarithmic, trigonometric, and inverse trigonometric functions) using rules for the derivative,
such as sums, products, quotients, chain, and implicit.
Students will understand that rules for differentiation are derived from the limit definition of the
derivative.
Students will be able to define a definite integral as a limit of Riemann sum using left, right, and
midpoint summation. In addition, students will use trapezoidal sums to approximate definite
integrals of functions represented algebraically, graphically, and by tables of values.
Students will be able to solve separable differential equations and use them in modeling (in
particular, studying the equation y´ = ky and exponential growth).
Students will be able to interpret differential equations using slope fields and understand the
relationship between slope fields and solution curves to differential equations.
Students will be able to use the Fundamental Theorem of Calculus to solve Definite Integrals
and understand its relationship to rate of change of a quantity over an interval. In addition,
students will use the Fundamental Theorem to represent a particular anti-derivative, and the
analytical and graphical analysis of functions so defined.
3
Performance Standards and Expectations: (Continued)



Students will solve anti-derivatives using substitution of variables (including change of limits
for definite integrals) and recognize anti-derivatives that follow directly from derivatives of
basic functions.
Students will be able to find specific anti-derivatives using initial conditions, including
applications to motion along a line.
Students will be able to use integrals to find accumulated change, the area of a region, volume
of a solid of revolution (Disk, Washer, and Shell method), volume of a solid by known cross
section, average value of a function, and the distance of a particle along a line.
Topics to be Covered:
Limits and Their Properties (≈4 weeks)
1.) Finding Limits Graphically and Numerically
2.) Evaluating Limits Analytically
3.) Continuity and One-Sided Limits
(Intermediate Value Theorem)
4.) Infinite Limits
5.) Limits at Infinity(Horizontal Asymptotes)
Differentiation (≈6 weeks)
1.) Derivative and the Tangent Line Problem
2.) Basic Differentiation Rules and Rates of Change
3.) Product and Quotient Rules and Higher-Order Derivatives
4.) Chain Rule
5.) Implicit Differentiation
Differentiation and Applications (≈6 weeks)
1.) Related Rates
2.) Extrema on an Interval
(Extreme Value Theorem)
3.) Rolle’s Theorem and the Mean Value Theorem
4.) Increasing and Decreasing Functions,1st Derivative Test,
Concavity and the Second Derivative Test.
5.) Curve Sketching
6.) Optimization
`
Integration (≈5 weeks)
1.) Antiderivatives/Indefinite Integration
6.) Riemann Sum and Trapezoidal Rule
7.) Definite Integral
8.) Fundamental Theorem of Calculus
9.) Integration by Substitution
Logarithmic, Exponential, and other Transcendental Functions (≈6 weeks)
1.) Natural Logarithm (Differentiation and Integration of Inverse
Functions
2.) Exponential (Differentiation and Integration)
Logarithmic, Exponential, and other Transcendental Functions (cont.)
4
3.)
4.)
5.)
6.)
7.)
Bases other than e
Inverse Trig. Functions (Differentiation and Integration)
Differential Equations (Growth and Decay)
Slope Fields
Differential Equations (Separation of Variables)
Applications of Definite Integrals (≈5 weeks)
1.) Areas of a region between two curves
2.) Volumes of Solids of Revolution
3.) Overview and Test Preparation
4.) ArcLength and Surfaces of Revolution
5.) Indeterminate Forms and L’Hopital’s Rule
Activities:
The following sample activities demonstrate ways to help students gain an increased understanding
of calculus.
Example 1
The “table” feature of the TI graphing calculator can be used to zoom in on a
limit numerically. For example, to find
 x2 
lim  2

x2 x  4


we view the values of the function from x-values of 1.5 to 2.5 with an increment step of 0.1. At
x = 2 the table records “error” or “not defined.” Students should see that the y-values seem to
follow a pattern. Redo the process beginning at 1.9 with a step size of 0.01, and observe that
the y-values are converging to 0.25. The process can be repeated with smaller and smaller
steps.
The limit can also be shown visually by graphing the function in a window that has a pixel step
of 0.1. Trace the function beginning at x = 1. Each step shows the corresponding x- and ycoordinates, but at x = 2, the y-coordinate disappears. It “reappears” when the tracing
continues at x = 2.1. Students can see graphically that the y-coordinates cluster at about 0.25
as x is near 2.
For comparison, do the same exploration with
 x2  4 

lim 
x 2
 x2 
This function is also undefined at x = 2, but the y-values do not converge as x approaches 2.
Instead, the values “explode”, giving students a numerical look at asymptotic behavior.
Example 2
The “Round Barn” assignment that follows is assigned at the beginning of the year. Students
are not aware of ways in which calculus could be applied to the problem. Students are
5
reassigned the problem at the end of the school year, but with the charge that they must use
calculus in significant ways to justify their arguments. Their final task is to present their
justifications to the class and the following scoring criteria sheet is used to assess. Classmates
and the teacher fill out the scoring criteria form, complete with comments. Comments are to
be constructive criticism. Presenters are given these critiques later and asked to read them
looking for themes representing areas of improvement.
6
Agriculture has always played a large
part in the history of West Virginia.
Round barns, though never quite as
common as the standard square or
rectangular shaped barns, were seen
as a time and money saver. The
selling point for the round barn
design was the large loft space, which
allowed farmers to store hay in the
barn, eliminating the cost of building
and maintaining outbuildings for hay
storage. This also reduced the
farmer’s work since they did not need
to haul the hay to the barn to feed the
livestock. The Hamilton Round
Barn, built in 1912 by Amos C.
Hamilton as a dairy barn, also
incorporates a technology more
commonly associated with the
Pennsylvania bank barns. It has two
main entrances, one on the lower
level for livestock and another,
accessible by ramp, to the second
level where the farmer would store
equipment and silage.
The above picture and description
was taken from the Frontiers to
Mountaineers Heritage Tourism brochure on Mountaineer Architecture. There are two statements I
would like for you to justify.
Statement 1: “Round barns, though never quite as common as the standard square or rectangular
shaped barns, were seen as a time and money saver.”
Statement 2: “The selling point for the round barn design was the large loft space, which allowed
farmers to store hay in the barn.”
Task 1:
Mathematically support or contradict the claim made by statements 1 and 2 in the
brochure. (Note: Hay could have been stored in square and rectangular barn lofts as
well. So, statement 2 implies that the loft space of the round barn is more efficient.)
Follow-up question: Would the way hay is stored today, as compared to 1912, make the round barn
more attractive or less attractive? Justify your answer mathematically.
Task 2:
What other shapes might barns be built in? Compare at least two other shapes to a
square, rectangle, and circle supporting claims of comparison mathematically.
7
The Round Barn
Scoring Criteria
1=strongly agree
2=agree
3=average performance
4=disagree
5=strongly disagree
1.) Addressed all questions.
1
2
3
4
5
Comments
_________________________________________________________
_________________________________________________________
_________________________________________________________
2.) Neat and well organized.
1
2
3
4
5
Comments
_________________________________________________________
_________________________________________________________
_________________________________________________________
3.) All propositions are
mathematically justified.
1
2
3
4
5
Comments
_________________________________________________________
_________________________________________________________
_________________________________________________________
4.) Significant use of Calculus
1
2
3
4
5
Comments
_________________________________________________________
_________________________________________________________
_________________________________________________________
Additional comments or suggestions for improvement
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
8