My main objective in this course is to give the students a strong foundation in Calculus that will enable them to do well in the AP exam and subsequent mathematics courses.
The students will master the concepts by using various tools where relevant (examples: technology, tables of numeric data…) and learn how to apply the concepts by solving well chosen and designed exercises. They will later on develop an expertise in problem solving and communication using the appropriate mathematics language and technology.
Quarter 1
Unit 1: Pre-Calculus Review (2-3 weeks)
A.
Lines
1.
Slope as rate of change
2.
Parallel and perpendicular lines
3.
Equations of lines
B.
Functions and graphs
1.
Functions
2.
Domain and range
3.
Families of function
4.
Piecewise function
5.
Composition of functions
C.
Exponential and Logarithmic Functions
1.
Exponential growth and decay
2.
Inverse functions
3.
Logarithmic functions
4.
Properties of logarithms
D.
Trigonometric Functions
1.
Graphs of basic trigonometric functions i.
Domain and range ii.
Transformations iii.
Inverse trigonometric functions
2. Applications
Unit 2: Limits and Continuity(3weeks)
A.
Rates of change

B.
Limits at a point
2.
Properties of limits
3.
Two-sided
4.
One-sided
C.
Limits involving infinity
1. Asymptotic behavior
2. End behavior
3. Properties of limits
4. Visualizing limits
D. Continuity
1. Continuous functions
2. Discontinuous functions i. Removable discontinuity ii. Jump discontinuity iii. Infinite discontinuity
E. Rates of Change and Tangent Lines
Unit 3: Derivatives (4 weeks)
A.
Derivative of a function
B.
Differentiability i. Local linearity ii. Numeric derivatives using the calculator iii. Differentiability and continuity
C.
Rules for differentiation
D.
Velocity and other rates of change
E.
Derivatives of Trigonometric Functions
F.
Chain Rule
G.
Implicit Differentiation i. Differential method
ii. y’ method
Unit 3: Derivatives continued (1 week)
Quarter 2
H.
Derivatives of inverse trigonometric functions
I.
Derivatives of logarithmic and exponential function
Unit 4: Applications of Derivatives (4 weeks)
A.
Extreme Values of Functions i. Local (relative) extrema ii. Global (absolute) extrema

B.
Using the derivative i. Mean value theorem ii. Rolle’s Theorem ii. Increasing and decreasing functions
C.
Analysis of graphs using the first and second derivatives i. Critical values ii. First derivative test for extrema iii. Concavity and points of inflection iv. Second derivative test for extrema
D.
Optimization problems
E.
Linearization models
F.
Related Rate Problems
Unit 5: The Definite Integral (3 weeks)
A.
Approximating areas i. Riemann sums ii.Trapezoidal rule iii. Definite integrals
B.
The Fundamental Theorem of Calculus
C.
Definite Integrals and antiderivatives i. The Average Value Theorem
Quarter 3
Unit 6: Differential Equations and Mathematical Modeling (4 weeks)
A.
Antiderivatives
B.
Integration by Parts
C.
Integration by Substitution
D.
Separable differential equations i. Growth and decay ii. Slope fields iii. General differential equations
E.
Numerical Methods
Unit 7: Applications of Definite Integrals (3 weeks)
A.
Integrals as Net Change
B.
Areas in the Plane
C.
Volumes i. Volumes of solids with known cross-sections ii. Volumes of solids of revolution a. Disk method b. Shell method

Unit 8 :Review of the Key Concepts of Calculus (2 weeks)
Quarter 4
Unit 9 :Review for the Advance Placement Examination
A.
L’ Hospital’s Rule.
B.
Multiple Choice Questions
C.
Free Response Questions
D.
Practice Drills
E.
Mock AP Examination
F.
Formula quizzes
A.
Integration by Parts
B.
Partial Fractions
C.
Improper Integrals
The first few weeks are like a training-camp or workshop. During this time, while reviewing Pre-Calculus, I set the tone and establish working habits that I expect students to follow throughout the course. The students are taught how the same idea can be investigated analytically, graphically, and numerically. Students learn how to relate those three representations to each other.
Also, the students are taught how to communicate their results both verbally and in written sentences by using the appropriate mathematic language.
For each lesson, there is a roadmap that I hand to the students, where the objective is clearly stated and key Ideas pointed out. The exploration, group-work, independent activities, homework and some exercises specifically designed for the AP Exam are clearly indicated.
I see myself as a coach, a facilitator, a mentor and a teacher.
As a teacher, I explain the objectives and motives of all the new concepts involved and how they are related to prior knowledge. At a designated time, I will cover all the theoretical aspects of the lesson.
The answer to each exercise is presented on the board by the teacher or a student.
Whoever does the work on the board follows the same protocol which consists of writing

the answer using the appropriate mathematics notation, explaining verbally which concept was used, how and where and finally illustrating the result by using a graphic calculator connected to a TV screen.
As a coach, I supervise the exploration of new concepts. During this activity, graphic calculators are extensively used. Also, I make myself available during groups and independent activities.
When a concept is covered, we complete a concept worksheet provided by the authors, which provides a feedback of our strengths and weaknesses. I re-emphasize on our weak areas.
Another aspect of my strategy is to organize oral presentation. Each student is assigned a particular problem to work on for a few days; then on a designated date is allotted an average of 20 minutes to give a talk. This is where I play the role of a facilitator and a mentor.
To achieve this goal, I pick exercises from textbooks, especially well designed ones such as [3] where the exercises labeled “Focus on Concepts” were chosen.
Here are some samples of questions and student responses.
Sample1:
Determine whether the statements are true or false.
If the statement is false, find functions for which the statement fails to hold. a.
If f and g are increasing on an interval, then so is f+g. b.
If f and g are increasing on an interval, then so is fg. c.
If f and g are concave up on an interval, then so is f+g. d.
If f and g are concave up on an interval, then so is fg.
To solve b. and d. the student presented two counter-examples.
In b. the student took f(x) = g(x) = x, therefore (fg)(x) = x
2
, the student show the work analytically by using the derivatives and illustrate the result by using a graphic calculator.
In d. She use the same protocol by taking f(x) = x
2
, and g(x) = e
-x
.
Sample2 a.
Let f(x) =


 x
2 x
2
, x

1 ,
 x
0

0
Show that lim x
 f
0

' ( x )
 lim x f

0
'

( x )
But that f’(0) does not exist.

b.
Let f(x) =


 x x
3
2
, x
, x


0
0
Show that f’(0) exists but f”(0) does not.
In a., the student used the definition of the derivative and established the result, then illustrated by graphing the function. She used the zoom to show that the function has a cusp at 0.
Last but not the least, we thoroughly cover the 2003 Exam release and all the2006 free responses and also some exam-type problems provided by the authors.
We use TI-83 graphic calculators which are provided by the school on a daily basis.
However, some students use TI-84 and TI-89.
United Streaming provides a different insight on some concepts. We use it for a better understanding of the Application of derivative and the concept of integrals. Winplot is also very good for exploration.
The grading factors include class-work, quizzes, tests, quarter finals, projects and oral presentations.
The quizzes deal with definitions and are non-cumulative. We take at least one every week.
The tests are cumulative; one every two to three weeks and each of them include past AP exam questions, multiple choices and free responses.
The quarter final models the AP test.
Oral presentations as described previously, gives me the best feedback of how well the students are grasping the concepts.
Projects come after the AP examination.
Textbook:[1] Calculus Graphical, Numerical, Algebraic
By Finney, Demana, Waits, and Kennedy
Published by Scott Foresman, Addison Wesley, 1999
ISBN 0-201-32445-8

Other Materials: [2] Calculus (Eight Edition) By Larson, Hostetler, and
Edwards
[3] Calculus (Eight Edition, Early Transcendentals) By Anton,
Bivens, and Davis.
[4] AP Calculus (CollegeBoard Advanced placement
Program 2005-2006 Professional Development Workshop Materials)