Syllabus for JFK High School, Sacramento, CA
David Van Natten, Principal
JOHN F. KENNEDY HIGH SCHOOL
COURSE SYLLABUS
DEPARTMENT OF MATHEMATICS
1. COURSE NUMBER, TITLE, UNITS AND PRINCIPAL/DEPARTMENT APPROVED
DESCRIPTION
Calculus AB 1-2 AP MCS 201-202 5 units + 5 units
Calculus AB curriculum follows the recommendations listed in the AP Course
Description for Calculus AB by the College Board. It is a two-semester course in the
study of the Calculus of functions of a single variable using the concepts and
techniques of limits, differentiation, and integration. Besides using algebraic
process to develop these concepts and skills, written, verbal, numerical, and
graphical representations/ methods will be applied whenever appropriate.
2. GENERAL INFORMATION
2015-2016 Jennifer Manzano-Tackett C308 (916) 433-5200 EXT 1308
jennifer-manzano@scusd.edu
3. TEXTBOOKS AND/OR RECOMMENDED OR REQUIRED READINGS
Calculus of a Single Variable by Larson, Hostetler & Edwards ,8th Edition,
Published by Houghton Mifflin Company, copy right 2006
Barron’s AP Calculus Test Preparation (no earlier than the 11th edition)
John F. Kennedy High School, Sacramento City USD
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Syllabus for JFK High School, Sacramento, CA
David Van Natten, Principal
4. GENERAL OVERVIEW
The syllabus of Calculus AB submitted to AP Central of the College Board for
review has been approved and is now used to guide the teaching of this course.
The Calculus AB syllabus begins with a review of the various concepts on function
learned in pre-Calculus course which are essential in the study of Calculus. The
first new concept introduced is the limit of a function. An informal definition of this
new concept is used to replace the formal definition which is too abstract for most
high school students. After that, the concept of continuity of a function at a point
and on an interval will be introduced to make way for the next two main concepts of
Calculus-Differentiation and Integration of function of a single variable. Students will
be required to memorize the limit definitions of the derivatives and be able to use
them to prove the rules of differentiation and later to apply these rules/short cuts to
solve word problems. After they have mastered the techniques of differentiation,
they will be introduced to the concept of Antidifferentiation, Riemann Sums and
finally the definition of a definite integral. They will also learn five Existence
Theorems which are inter-related to one another. Eventually these theorems will be
used to prove the Fundamental Theorem of Calculus part I and part II. The
Fundamental Theorem is the most important part of our syllabus. Many
mathematicians would like to call this theorem the most important
theorem/breakthrough in Mathematics. In the past, many students got very excited
when this Theorem was introduced to them. The rest of the syllabus contains the
applications of these two important techniques of Calculus-differentiation and
integration to various types of functions and their applications.
5. COURSE OBJECTIVES
1. To provide the students the opportunities to acquire the concepts and
techniques in solving application problems of differential and integral calculus.
2. To provide the students the opportunities to use their graphing calculators to
expand and explore their understanding of calculus.
3. To provide the students the opportunities to incorporate different ways
(analytical, numerical, and graphical approaches) to solve real world problems, as
well as the opportunity to verbalize explanations of their solutions and thought
process.
4. To provide a strong Mathematics background for students who intend to major
in sciences, social sciences, or in any field in which the understanding of higher
level Mathematics is beneficial.
5. To prepare students for the AP test in Calculus AB so that they can be better
prepared for college and earn college credits.
6. COURSE REQUIREMENTS, ATTENDANCE AND SPECIFIC GRADING POLICY
John F. Kennedy High School, Sacramento City USD
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Syllabus for JFK High School, Sacramento, CA
David Van Natten, Principal
AP Calculus is a pretty rigorous course and the problems in their homework
assignments or in the chapter tests are generally more difficult than regular Math
classes. In order to encourage students to stay in the class, the grading scale is a
little bit lower than that in regular Math classes. But that doesn’t imply that students
could get an easy ‘A’ or ‘B’ in this class. They have to work hard to earn good
grades.
Grading System in Calculus AB
Assessments-----------------------------------------70%
Daily Assignments --------------------------------10%
Final Exam--------------------------------------------20%
85% -- 100%
75% -- 84.9%
60% -- 74.9%
50% -- 59.9%
49.9% and below
A
B
C
D
F
Homework will be checked and stamped for “reasonable completion” (at least 75 %
complete with work shown) for each assignment. An assignment that does not
receive a stamp will only be work half credit when the homework packet is collected
at the end of each unit. Late homework stamps will only be given for excused
absences and are the responsibility of the student to seek out upon returning from
the absence. Unit tests are given at the end of each unit and there are quizzes in
between. A correct answer to a non-calculator problem without showing work will at
most get partial credit, but often receive no credit.
7. DESCRIPTION OF MAJOR ACTIVITIES/EXERCISES/PROJECTS
Unit tests and a final exam at the end of each semester will be given through the
school year. Students who have signed up to take the AP test will be given a final
exam before the AP Calculus AB test in May. Keep in mind, given that all
material required for the AP test must be taught before the beginning of May,
the concepts taught in the first 3 quarters will be delivered at a very quick
pace. There will be time for review (and thus the potential for raising the class
grade) in the second semester, but not the first. Therefore, 1st semester
grades in this course are often lower than second semester grades on
average. However, given that the purpose of this course is to gain a deep
understanding of the introductory concepts in Calculus, steps will be taken to
reteach and reassess whenever necessary for the general population of the
class.
John F. Kennedy High School, Sacramento City USD
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Syllabus for JFK High School, Sacramento, CA
David Van Natten, Principal
8. OUTLINE OF CLASS SESSIONS
Unit
Unit
Unit
Unit
Unit
Unit
Unit
Unit
9.
1:
2:
3:
4:
5:
6:
7:
8:
Functions & Trigonometry Review
Limits and Continuity of Functions
Differentiation
Applications of Derivatives
Antidifferentiation and Definite Integrals
Differentiation and Integration of Transcendental Functions
Differential Equations
Applications of Definite Integrals
9 days
15 days
29 days
25 days
18 days
14 days
11 days
10 days
GENERAL STATEMENTS
Students are expected to adhere to and be familiar with all school and district
policies pertaining to behavior, attendance, testing, cheating and plagiarism, and
final exams.
10. CROSS INDEXING KEY OF COURSE OBJECTIVES TO REQUIRED
STANDARDS
Unit 1: Functions & Trigonometry Review
1. Trigonometric Functions-definitions, evaluation, and graphs
2. Multiple representations of functions—verbal, graphical, numerical, and
algebraic.
3. Classifications of functions—algebraic and non-algebraic.
4. Function transformations—horizontal and vertical shifting, reflecting,
compressing, and expanding.
5. Graphs of basic functions-linear and polynomial
6. Special functions—absolute value, rational, exponential, and logarithmic.
7. Use of graphing calculators to graph functions, to evaluate a function, to
John F. Kennedy High School, Sacramento City USD
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Syllabus for JFK High School, Sacramento, CA
David Van Natten, Principal
find the zeros of a function, and to find the point of intersection of the
graphs of two functions,to find the extrema, in addition to being able to
interpret the relevance of these points and use them to justify arguments
made about the function.
8. Applications—functions as mathematical models.
Unit 2.
Limits and Continuity of Functions
1. Introduction to the concept of the limit of a function using graphical
and numerical approach.
2. Informal definition of the limit of a function, including written/verbal
explanations of cases in which limits do not exist, supported by evidence
found in tables/graphs of a graphing utility.
3. Evaluation of limits using algebraic methods—substitution,
cancellation, and rationalization.
4. Use the Squeeze Theorem to evaluate a special trigonometric limit.
5. Properties (or theorems) of limits of functions.
6. One-side limits.
7. Intuitive understanding of continuity of a function at a point.
8. Definition of the continuity of a function at a point and on a closed interval.
9. Properties (or theorems) of continuity of a function.
10. The Intermediate Value Theorem.
11. Infinite limit and vertical asymptotes.
12. Limits at infinity.
13. Horizontal and oblique asymptotes.
Unit 3.
Differentiation
1. A graphical and numerical introduction of the concept of the derivative
using the classical tangent line problem and velocity of a falling object
problem.
2. Derivative interpreted as the slope of a secant line as it approaches
to the position of the tangent line.
3. Derivative as defined by the instantaneous rate of change of the
outputs in relation to the inputs.
4. Definition of the derivative of a function at a point—the limit definition.
5. Derivative as a number and as a derived function.
6. Relationship between continuity and differentiability.
7. Derivative from the negative side and from the positive side.
8. Approximation of the derivative of a function at a point
using graphical and numerical data.
9. Rules of differentiation of algebraic and trigonometric functions.
10. Implicit differentiation for implicitly defined functions.
11. Solving related rates problems using implicit differentiation.
Unit 4.
Applications of Derivatives
John F. Kennedy High School, Sacramento City USD
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Syllabus for JFK High School, Sacramento, CA
David Van Natten, Principal
1. Use of derivatives to find the intervals of increasing or decreasing
for a given functions, including written/verbal explanations of the
significance of these intervals on both the original function and
derivative
function graphs.
2. The critical value of a function, including written/verbal explanations of
the significance of this value on the both the original function and
derivative function graphs.
3. Use the First Derivative Test to find the relative (local) extrema of a
function and provide written/verbal explanations of the significance of
these points for both the original function and the derivative function
graphs.
4. The Extreme Value Theorem.
5. Find the absolute extrema of a function defined on a closed interval,
both algebraically and by using the graphing utility, in addition to
providing written/verbal explanations of the significance of these points
for
both the original function and the derivative function graphs.
6. The Second Derivative Test.
7. Use the second derivative of a function to determine the concavity and
point of inflection of the graph of a function.
8. Relationships of graphs of the function, its first derivative, and
its second derivatives, with a focus on the ability to provide written/
verbal explanations for these relationships using appropriate calculus
vocabulary.
9. Applications of derivatives—optimization and linear approximation.
10. Differential—definition and applications
11. Analysis of motion of a particle---position, velocity, and acceleration
12. The Rolle’s Theorem and the Mean Value Theorem.
13. Use of graphing utilities to verify and interpret the significance of all
maxima/minima, zeros, points of inflection, intervals of
decreasing/increasing, etc. of functions and their derivatives.
Unit 5.
Antidifferentiation and the Definite Integrals.
1. Antiderivative of a given function, found both algebraically and by using
a graphing utility.
2. A particular antiderivative that satisfies a given initial value condition.
3. Numerical Integrations--Approximation of the area under a curve
using n rectangles. Functions could be given in algebraic, numeric, or
graphical forms and various points of evaluation could be used, i.e., the
left-endpoint, the right-endpoints, or the midpoint in the ith subinterval.
4. The Riemann Sum of a function over a closed interval using a regular
partition.
5. Definition of a Definite Integral—defined as the limit of a Riemann Sum.
7. Properties of Definite Integrals.
John F. Kennedy High School, Sacramento City USD
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Syllabus for JFK High School, Sacramento, CA
David Van Natten, Principal
8. Applications and proofs of The Fundamental Theorems of
Calculus-Part I and Part II.
9. The Mean Value Theorem for Integrals and the average value of a
function.
10. The Trapezoidal Rule, in addition to the general concept of finding
areas with trapezoids.
11. Integration by Substitution.
12. Written/verbal explanations of the relationships between functions,
their derivatives, and their antiderivatives, supported by evidence
found by using a graphing utility to find key characteristics of their
graphs and interpreting the meaning of those characteristics.
Unit 6
Differentiation and Integration of Transcendental Functions
1. The natural logarithmic function defined by a definite integral.
2. The derivative of y=ln x and the completion of the Power Rule for
Integration.
3. The behavior of the graph of y=ln x.
4. The derivative of an inverse function.
5. The natural exponential Function defined as the inverse of the
natural logarithmic function.
6. The behavior of the graph of y=e^x.
7. The derivative of the inverse trigonometric functions.
8. Applications--exponential growth or decay models.
Unit 7
Differential Equations
1. Solving simple differentiations including initial values problems.
2. Solving separable differential equations.
3. Slope Fields—a graphical approach to solving problems
involving differential equations.
4. Exponential growth and decay models.
Unit 8
Applications of Definite Integrals
1. Area of a region between two curves.
2. Volume of a sold generated by revolving a bounded region
about a line—the Disc Method and the Washer Method.
3. Volume of solid with known cross sections.
4. Problems that require setting up a Riemann Sum and then taking its
limit.
John F. Kennedy High School, Sacramento City USD
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Syllabus for JFK High School, Sacramento, CA
John F. Kennedy High School, Sacramento City USD
David Van Natten, Principal
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