AP® Calculus AB - Syllabus

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AP
®
Calculus AB
-

Syllabus

Course Overview

The main objective in teaching AP
®
Calculus is to enable students to appreciate the
strength of calculus and receive a thorough foundation that will give them the tools to
succeed in future mathematics courses.
Calculus with Analytic Geometry I, includes the
study of real numbers, limits, continuity, differential and integral calculus of functions of
one variable.


Technology:




All students have a TI
-
83/TI
-
83+/TI84 graphing calculator for use in class, at
home,
and on the AP exam. Students will use their graphing calculator
extensively throughout the course.



Various applets on the Internet



APCD software


Textbook:
Calculus, 2
nd

Edition, Robert T. Smith and Roland B. Minton,
McGraw Hill


Course Planner
[C2]

A L
ibrary of Functions (Chapter 0)

1
st

Quarter

Lab: Exploring function transformations
f
(
x
+
h
),
f
(
x
) +
k
,
a
*
f
(
x
),
f
(
b
*
x
),
f
(|
x
|),|
f
(
x
)|



Multiple representations of functions



Absolute value and interval notation



Domain and range



Categories of fun
ctions, including linear, polynomial, rational, power,
exponential, logarithmic, and trigonometric



Even and odd functions



Function arithmetic and composition



Inverse functions



Parametric relations


Key Concept: Limits and Continuity (Chapter 1
)

1
st

Quarter


1. Concept of Limits


2. Computation of Limits


3. Continuity and its consequences


4. Limits involving infinity


5. Formal Definition of a limit





Differentiation (Chapter 2)

2
nd

Quarter


1. Tangent Lines and Velocity


2. The Derivative


3. The Derivative at a point


4. Powers and Polynomials


5. The Exponential/logarithmic Function


6. The Product and Quotient Rules


7. The Chain Rule


8. The Trigonometric Functions


9. Applications of the Chain Rule and Related Rates


10. Implicit Functio
ns


11. Mean Value Theorem


Activity: CBL Prime Number Lab
[C3] [C5]

Activity: Graphing the Derivative of a Function (See example)
[C4]


Application of the Derivative (Chapter 3)

2
nd

Quarter


1. Linear Approximation and the Derivative


2. L’Hopital’s Ru
le


3. Newton’s Method


4. Using Local Linearity to Find Limits


5. Using First and Second Derivatives


6. Families of Curves


7. Optimization and Modeling


8. Theorems About Continuous and Differentiable Functions



Activity: Optimization Project
[C3] [C4]


Key Concept: Integration (Chapter 4)

2
nd

Quarter


1. How Do We Measure Distance Traveled?


2. Antiderivatives Graphically and Numerically

3. Constructing Antiderivatives Analytically

4. Sums and Sigma Notation

5. Reimann Sums

Activity: Who’s winning th
e race?
[C3][C5]






Midterm Exam




Constructing Antiderivatives (Chapter 4)

3
rd

Quarter

1. Fundamental Theorem of Calculus


2. The Definite Integral


3. Interpretations of the Definite Integral

4. Theorems About Definite Integrals


5. Integration by Su
bstitution


6. Differential Equations


7. Second Fundamental Theorem of Calculus


8. The Equations of Motion


9. Numerical Integration

Activity: Jimmy Stewart’s Car


Applications of the Definite Integral (Chapter 5)

3
rd

Quarter


1.

Area and Volumes

2.

Arc Leng
ths and Surface Areas

3.

Projectile Motion

4.

Work, Moments and Hydrostatic Force

5.

Probability

Activity: Three
-
dimensional foam project.
[C3][C4]


Exponentials, Logarithms and other Transcendental Functions
(Chapter 6)

4
th

Quarter

1.

Natural Logarithm

2.

Inverse Funct
ions

3.

The Exponential Function Revisited

4.

Growth and Decay Problems

5.

Separable Differential Equations

6.

Euler’s Method

7.

Slope Fields

8.

Inverse Trig

Drawing Slope Fields
[C3] [C4] [C5]

C2

The course teaches all topics associated with Functions, Graphs, and Limits;

Derivatives; Integrals; and Polynomial
Approximations and Series as delineated in the Calculus AB Topic Outline in the
AP Calculus Course Description
.


C5

The course teaches students how to use graphing calculators to help solve problems, experiment, int
erpret results, and support
conclusions.


C4

The course teaches students how to communicate mathematics and explain solutions to problems both verbally and in written
sentences.


C3

Evidence of Curricular Requirement: The course provides students with th
e opportunity to work with functions represented in a
variety of ways

graphically, numerically, analytically, and verbally

and emphasizes the connections among these representations.

Teaching Strategies

During the first few weeks of school I start with a
transformation packet. This packet
requires students to use their prior knowledge about all types of functions to discuss
transformations. The students work alone for a time and then I have them work together
in groups to discuss their findings. This is

how I structure the class for the rest of the
year. I incorporate the use of technology throughout the year so that students have
multiple means of interpreting a problem. I encourage students to explore and discover
throughout the course.


I like to
incorporate whiteboard activities while learning new topics. Sometimes I will
give the entire class the same problem but have them work on it in groups of two. They
then have to write up their findings on the whiteboard. Once all are done I call for a

board meeting”. All the students hold up their board and we then discuss the different
findings. The students enjoy this atmosphere. I use technology throughout the year.
These include TI series calculators (this varies depending on which one the studen
t
owns), TI InterActive!, APCD, CBL’s, and LabPro. I use four simple rules when
presenting topics. They are graphically, numerically, algebraically, and verbally. This
allows the students to gain an in
-
depth understanding of the material.


Student Evaluat
ion

In order to give students practice with AP type questions, I start with free
-
response
questions after the students have mastered derivatives. I give them a question a week and
we go over them on Friday’s. This allows the students to come in for help
and work
together on solving them. I then incorporate free
-
response questions into the exam for
each chapter. After the December holiday break, we start working on multiple
-
choice
questions. I use them as warm
-
ups and we go over them after 15 minutes of

work. The
number of problems I give as warm
-
ups depends on the difficulty of the question.
Usually there are 2 to 3 questions per warm
-
up. We then go over the questions as a class.
Oftentimes, I will select students to display their work on the board.


Example of Student Activity

Graphing with the Derivative

Work done on this assignment will be assessed against all criteria. The use of technology is required and
will be assessed.

Consider the function f(x) =

1.

Where are the roo
ts of the function?

2.

What are the intervals for which f is increasing and decreasing. Show all work.

3.

Where are the local maxima and minima of f?

4.

What are the intervals for which f is concave up and concave down?

5.

Where are the inflection points?

6.

Name all ve
rtical and horizontal asymptotes.

7.

Sketch the graph of f.

8.

On the same grid sketch the graph of f, f’, and f’’.

9.

Summarize the graph and make links to the information that you found in 1
-
7.