//img.uscri.be/pth/10382c6fb3391712e888193df378b7d1b7697cb3
La lecture en ligne est gratuite
Le téléchargement nécessite un accès à la bibliothèque YouScribe
Tout savoir sur nos offres
Télécharger Lire

AP AUDIT

De
8 pages
AP Calculus AB Syllabus Pre-Requisites: Students intending to take AP Calculus AB are expected to have successfully completely 3 years of HS math and Pre-calculus either at their home school or as a summer class at a local community college. At their home school, Pre-Calculus students will study families of functions with emphasis on their behavior and modeling real world applications. Students will have used CBL’s in a lab setting as an introduction to certain functions and will have fit curves to the data collected. Written responses and explanation are required for many of the activities and assessments throughout the year. Course Overview: The students begin their study of Calculus before the end of their Accelerated Pre-Calculus course. They are initially exposed to the concepts of derivatives, integral, and limits from intuitive, graphical, and numerical perspectives. They continue on with Delta-epsilon definition of limit with applications, and limits algebraically and graphically. When they return in September, we spend approximately one and a half weeks reviewing these concepts, and learning the Intermediate and Extreme value theorems. The remainder of the year is devoted to exploring the concepts of AP Calculus AB in more depth. Students are encouraged to work in groups and to form study groups outside of the school day. Graphing Calculators: Each student either owns or is provided the use of a Graphing Calculator from ...
Voir plus Voir moins
AP Calculus AB
Syllabus
Pre-Requisites
:
Students intending to take AP Calculus AB are expected to have
successfully completely 3 years of HS math and Pre-calculus either at their
home school or as a summer class at a local community college.
At their
home school, Pre-Calculus students will study families of functions with
emphasis on their behavior and modeling real world applications.
Students
will have used CBL’s in a lab setting as an introduction to certain functions
and will have fit curves to the data collected.
Written responses and
explanation are required for many of the activities and assessments
throughout the year.
Course Overview
:
The students begin their study of Calculus before the end of their
Accelerated Pre-Calculus course.
They are initially exposed to the concepts
of derivatives, integral, and limits from intuitive, graphical, and numerical
perspectives.
They continue on with Delta-epsilon definition of limit with
applications, and limits algebraically and graphically.
When they return in
September, we spend approximately one and a half weeks reviewing these
concepts, and learning the Intermediate and Extreme value theorems.
The
remainder of the year is devoted to exploring the concepts of AP Calculus
AB in more depth.
Students are encouraged to work in groups and to form
study groups outside of the school day.
Graphing Calculators
:
Each student either owns or is provided the use of a Graphing Calculator
from the TI-83 or 84 families of calculators.
By the time they enroll in
calculus, the students have been using their calculators for at least a year.
Their Pre-Calculus course refines and expands on their algebraic technique
and builds their skills in using the graphing calculator for graphing functions
and solving equations using multiple methods (intersect, zeros, solver).
These skills are reviewed throughout the AP Calculus course.
In addition,
students will be able to find numerical derivatives at a specified point, the
definite integral over a specified domain, and the equation of a tangent line
with their calculators.
Course Outline:
May to June of Accelerated Pre-Calculus
Unit 1:
Introduction to Calculus
Average rate of change to approximate instantaneous rate of change –
numerically and graphically from functions, graphs, and tables.
Definite Integral as area – counting squares on a graph and Trapezoid
Rule on functions and tables.
Limits of a function from graphs and tables
Unit 2:
Limits
Limits numerically
Limits graphically
Limits algebraically
Delta/Epsilon definition of limits
Limits involving infinity.
September to May of AP Calculus AB
Unit 3:
Review of topics from units 1 and 2
Definition of continuity
( )
( )
c
f
x
f
c
x
=
lim
Intermediate and Extreme Value Theorems
Unit 4:
Difference quotients- forward, backward, and symmetric
Numerical and Algebraic definitions of a derivative (limit of a
difference quotient emphasis on derivative as a slope)
Analysis of functions and their derivatives graphically (preliminary
comparison of behaviors of one to predict behavior of the other)
Development of power rule using algebraic definition of derivative
(limit of difference quotient, formalization of notation such as
y
and
(
)
x
f
)
Introduction of velocity and acceleration as
1
st
and 2
nd
derivatives of a
position function
Unit 4 (cont.)
Equations of tangent and normal lines algebraically and
graphically
Unit 5:
Chain rule to find derivative of composition of functions
Derivatives of Sine and Cosine
Squeeze Theorem and proof of
( )
1
sin
lim
0
=
x
x
x
Applications for sine and cosine and derivatives
Introduction of anti-derivatives and u-substitution
Introduction of additional derivative notation (
2
2
,
dx
y
d
y
dx
dy
y
=
=
)
Unit 6:
Proof of product rule
Quotient rule (without proof)
Development of derivatives of secant, tangent, cosecant, and
cotangent
Unit 7:
Inverse trigonometric functions and their derivatives
Differentiability, continuity, and limits
Implicit differentiation
Unit 8:
Review of anti-derivatives
Linear approximation
Indefinite Integral-formal definition
Riemann sums-lower, upper, left, right, and midpoint
Mean Value Theorem
Rolle’s Theorem
Unit 9:
Proof of 1
st
Fundamental Theorem of Calculus
Definite integrals and their properties
Application of definite integral
(as accumulation function)
Integrals using u-substitution (including change of bounds)
Simpson’s Rule
Unit 10:
Review of exponential modeling
Derivatives and integrals of natural logarithms
2
nd
Fundamental Theorem of Calculus
Derivatives of exponential functions and logarithmic
differentiation
Two week review using selected MC and FRQ questions from released
exams followed by a cumulative midterm exam.
Exam is modeled on the
format of the AP Calculus AB exam.
Unit 11:
Derivatives and Integrals involving
x
e
L’Hospital’s rule
Unit 12:
Separable differential equations
Applications involving differential equations
Slope fields
Euler’s method for solving non-separable differential equations
Predator/Prey models
Unit 13:
Analysis of functions and 1
st
and 2
nd
derivatives in order to graph a
function including review of behavior near vertical asymptotes,
behavior of functions as x approaches
±
, and monotonicity.
1
st
derivative test and critical points.
2
nd
derivative test, points of inflection, and concavity
Applications involving optimization
Unit 14:
Area of plane regions
Volume of solids by disk/washer method
Volume of solids by cylindrical shells method
Volume of solids
of known cross sections
Unit 15:
Displacement, distance, velocity, and acceleration
Average value function
Related rates problems
Remaining time is used to review for the AP Calculus AB exam.
Student Activities:
Exploration
s:
Supplementary resource with primary textbook.
Students
complete selected calculator-active explorations introducing topics and
develop an intuitive sense of the topic prior to formal presentation.
Writing
:
In addition to writing explanations supporting their work, students are
required to keep a journal during the year.
Entries include responses to
prompts in the textbook allowing them to explain their understanding of the
current topics, reaction to math articles both print and internet based, and
responses to readings about calculus and the people involved in its
development.
Review activity
:
PBS Mathline “Bottles and Divers” activity
(
http://www.pbs.org/mathline
)
Students work in small groups on the Bungee Jumper Problem and the Dive
Problem.
Students review rates of change and use average rates of change to
approximate the instantaneous rate of change.
Students review the
techniques used to find the rates of change from tables, graphs and
functions.
They review methods to simplify calculations using their
graphing calculator.
Groups share their results and generalize their findings
as a class.
Trig Derivatives Activity
:
Students work in small groups to derive a formula for the derivative of
secant, tangent, cosecant, and cotangent using their knowledge of derivatives
(power rule, product and quotient rules, derivatives of sine and cosine) and
trig identities.
Each group finds one of the derivatives and the groups
present their results to the class on the whiteboard or chart paper.
Slope Field Activity
:
Students, working in small groups, are assigned a differential equation and
selected sample points.
They calculate the value of
dx
dy
at each point and plot
the slope field on grid chart paper.
After all groups have completed their
graph, the class makes generalizations about the slope fields.
The class will
then make predictions about the solution curves for the differential equations
based on the slope fields.
Volumes of Solids of Revolution
:
Classic Coke Bottle Activity
(disk method):
Students, working in groups of two or three, measure the circumference of a
classic coke bottle at evenly spaced intervals.
These values are entered into
the statistics editor of their calculators.
Other lists are used to generate the
corresponding radii and areas of the circles.
The students use their
knowledge of integrals to calculate the volume of the soda in the bottles.
The groups compare their volumes and the actual volume of the soda given
on the bottle.
They discuss the reasons for the differences between their
answers and the actual volume.
The students also create scatter plots of the
data and find a regression model for the bottle.
Calculus Cake Day
(washer and cylindrical shells methods):
Students calculate the volume of a bundt cake and an angel food cake using
equations modeling the cross section of the cake rotated about the y and x-
axes, respectively.
(The equations were found using regression techniques
on the pans the cakes were baked in.)
After the calculations are complete,
the students eat the models.
Student Assessment
:
Students grades are determined using a combination of quizzes, tests, daily
homework, spiral review assignments (Do These Quickly exercises in the
primary textbook), Graded Homework (prior to each test), and Journals.
Quizzes and tests are drawn from or model released AP exam questions.
MC questions are modified to short answer questions in the early part of the
year.
Graded homework in the latter part of the course consist of released
FRQ questions related to the current topic.
Primary Textbook:
Foerster, Paul A.,
CALCULUS: Concepts and Applications,
Key
Curriculum Press, Berkley, CA, 1998
Supplementary Textbooks:
Hughes-Hallett, et al, CALCULUS, John Wiley & Sons, Inc., New York,
1994
Anton, Howard, Calculus with Analytic Geometry, 5
th
Edition (Brief), John
Wiley & Sons, New York, 1995
Additional resources:
Print:
Explorations by Paul Foerster
Internet:
PBS Mathline: Bottles and Divers Activity
Hadley Math Page
AP Central
The Calculus Page
Surfin’ Sinefeld
Calculus on the Web
Calculus-Help.com
Whiteboard movies
Powerpoint presentations by Greg Kelly
The MacTutor History of Mathematics archive
PLUS magazine