GLEN RIDGE PUBLIC SCHOOLS
Curriculum Guide
Course Title: ADVANCED PLACEMENT CALCULUS (AB)
Subject: Mathematics
Grade Level: 12
Department/School: Mathematics/Glen Ridge High School
Duration: Full Year
Number of Credits: 5
Prerequisite: Math Analysis Honors, Grade of “B” or higher and teacher recommendation
Elective or Required: Elective
Author:
Date Submitted: Summer 2007
Course Description
Advanced Placement Calculus is appropriate for a student with a strong foundation in algebra, geometry and trigonometry who also possesses a drive to learn mathematical concepts at a higher level. The AP Calculus course is designed to give the students a solid understanding of broad sweeping concepts then apply them to specific and concrete problem solving situations. The course prepares students for the AP Calculus Exam that is administered in May. Students may then use their results from the exam to receive college credits or placement in accelerated courses from their college or university.
GLEN RIDGE PUBLIC SCHOOLS
MATHEMATICS MISSI ON STATEMENT
Mathematics
and Computer Science are an integral part of our lives. Students must be actively involved in their
mathematics education with problem solving being an essential part of the
curriculum. The mathematics and computer
science curricula should emphasize thinking skills through a balance of
computation, intuition, common sense, logic, analysis and technology. Students will be engaged and challenged in a
student-centered learning environment that is developmentally appropriate. Students will communicate mathematical ideas
effectively by applying hands-on manipulatives, basic computational skills,
mathematical models, and technology in order to solve practical problems.
The Mathematics Standards consist of five statements, which
describe what is essential to excellent mathematics education, and presents a
view of mathematics teaching and learning that integrates the processes of
mathematical activity, the content of mathematics, and the learning environment
in the classroom. The following
standards were adopted by the New Jersey State Board of Education.
This course will cover the following Core Curriculum Content
Standards:
STANDARD 4.1 (NUMBER AND NUMERICAL OPERATIONS) ALL
STUDENTS WILL DEVELOP NUMBER SENSE AND WILL PERFORM STANDARD NUMERICAL
OPERATIONS AND ESTIMATIONS ON ALL TYPES OF NUMBERS IN A VARIETY OF WAYS.
Descriptive Statement: Numbers and
arithmetic operations are what most of the general public think about when they
think of mathematics; and, even though other areas like geometry, algebra, and
data analysis have become increasingly important in recent years, numbers and
operations remain at the heart of mathematical teaching and learning. Facility with numbers, the ability to choose
the appropriate types of numbers and the appropriate operations for a given
situation, and the ability to perform those operations as well as to estimate
their results, are all skills that are essential for modern day life.
Number Sense
Numerical Operations
Estimation
STANDARD 4.2 (GEOMETRY AND MEASUREMENT) ALL STUDENTS
WILL DEVELOP SPATIAL SENSE AND THE ABILITY TO USE GEOMETRIC PROPERTIES,
RELATIONSHIPS, AND MEASUREMENT TO MODEL, DESCRIBE AND ANALYZE PHENOMENA.
Descriptive Statement: Spatial sense
is an intuitive feel for shape and space.
Geometry and measurement both involve describing the shapes we see all
around us in art, nature and the things we make. Spatial sense, geometric modeling, and
measurement can help us to describe and interpret our physical environment and
to solve problems.
Geometric Properties
Transforming Shapes
Coordinate Geometry
Units of Measurement
Measuring Geometric Objects
STANDARD 4.3 (PATTERNS AND ALGEBRA) ALL STUDENTS WILL
REPRESENT AND ANALYZE RELATIONSHIPS AMONG VARIABLE QUANTITIES AND SOLVE
PROBLEMS INVOLVING PATTERNS, FUNCTIONS, AND ALGEBRAIC CONCEPTS AND PROCESSES.
Descriptive Statement: Algebra is a
symbolic language used to express mathematical relationships. Students need to understand how quantities
are related to one another, and how algebra can be used to concisely express
and analyze those relationships. Modern
technology provides tools for supplementing the traditional focus on algebraic
procedures, such as solving equations, with a more visual perspective, with
graphs of equations displayed on a screen.
Students can then focus on understanding the relationship between the equation
and the graph, and on what the graph represents in a real-life situation.
Patterns
Functions and Relationships
Modeling
Procedures
STANDARD 4.4 (DATE ANALYSIS, PROBABILITY, AND DISCRETE
MATHEMATICS) ALL STUDENTS WILL DEVELOP AN UNDERSTANDING OF THE CONCEPTS AND
TECHNIQUES OF DATA ANALYSIS, PROBABILITY, AND DISCRETE MATHEMATICS, AND WILL
USE THEM TO MODEL SITUATIONS, SOLVE PROBLEMS AND ANALYZE AND DRAW APPROPRIATE
INFERENCES FROM DATA.
Descriptive Statement: Data analysis,
probability, and discrete mathematics are important interrelated areas of
applied mathematics. Each provides
students with powerful mathematical perspectives on everyday phenomena and with
important examples of how mathematics is used in the modern world. Two important areas of discrete mathematics
are addressed in this standard: a third area, iterations and recursion, is
addressed in Standard 4.3 (Patterns and Algebra).
Data Analysis
Probability
Discrete Mathematics – Systematic Listing and Counting
Discrete Mathematics – Vertex-Edge Graphs and
Algorithms
STANDARD 4.5 (MATHEMATICAL PROCESSES) ALL STUDENTS
WILL USE MATHEMATICAL PROCESSES OF PROBLEM SOLVING, COMMUNICATION, CONNECTIONS,
REASONING, REPRESENTATIONS, AND TECHNOLOGY TO SOLVE PROBLEMS AND COMMUNICATE
MATHEMATICAL IDEAS.
Descriptive Statement: The mathematical
processes described here highlight ways of acquiring and using the content
knowledge and skills delineated in the first four mathematical standards.
Problem Solving Reasoning
Communication Representations
Connections Technology
Course Description
The students have learned the topics in Unit I in PreCalculus Honors. It is recommended that the AP Calculus teacher give an appropriate summer assignment for students to review these necessary topics.
UNIT I: PREREQUISITES FOR CALCULUS
CCCS: 4.1, 4.2, 4.3, 4.5
Objectives:
Students will be able to:
1. Apply properties of lines.
2. Understand and apply the properties of functions.
3. Interpret and apply exponential functions.
4. Use, graph, and apply properties of logarithmic functions to problem solving situations.
5. Apply trigonometric functions to problem solving and solving equations involving trig functions and graphs.
Approximate duration: 7- 10 days (Review and Assessment)
UNIT II: LIMITS AND CONTINUITY
CCCS: 4.1, 4.2, 4.3, 4.5
Objectives:
Students will be able to:
1. Calculate rates of change and evaluate limits.
2. Evaluate limits involving infinity.
3. Determine if a function is continuous or not and apply continuity properties to functions.
4. Calculate instantaneous rates of change by applying limit properties.
5. Calculate the slope of a tangent line by applying limit properties.
Approximate duration: 20-22 days
UNIT III: DERIVATIVES
CCCS: 4.1, 4.2, 4.3, 4.5
Objectives:
Students will be able to:
1. Find the derivative of a function using the definition.
2. Understand and graph a function’s derivative.
3. Identify when f’(x) fails to exist.
4. Relate continuity to differentiability.
5. Apply rules to differentiating more complex functions.
6. Relate derivatives to velocity and other rates of change.
7. Find the derivatives of the trigonometric functions and understand their connections to each other.
8. Apply the Chain Rule to finding the derivatives of composite functions.
9. Apply implicit differentiation.
10. Find the derivative of the inverse trig functions.
11. Find the derivatives of exponential and logarithmic functions.
Approximate duration: 30- 35 days
UNIT IV APPLICATIONS OF DERIVATIVES
CCCS: 4.1, 4.2, 4.3, 4.5
Objectives:
Students will be able to:
1. Find the extreme values of functions using derivatives and apply to curve sketching.
2. Apply the Mean Value Theorem to functions.
3. Connect the graphs of f, f’ and f’’.
4. Apply derivatives to modeling and optimization problems.
5. Use derivatives to solve related rates problems.
Approximate duration: 35 – 40 days
UNIT V: THE DEFINITE INTEGRAL
CCCS: 4.1, 4.2, 4.3, 4.5
Objectives:
Students will be able to:
1. Estimate areas with finite sums.
2. Define and apply the definition of the definite integral.
3. Define integrals and antiderivatives.
4. Define and apply the Fundamental Theorem of Calculus.
5. Calculate areas by applying the Trapezoidal Rule.
Approximate duration: 30 – 35 days
UNIT VI: DIFFERENTIAL EQUATIONS AND MATHEMATICAL MODELING
CCCS: 4.2, 4.3, 4.5
Objectives:
Students will be able to:
1. Graph a slope field and use it to overlay a solution given an initial condition.
2. Antidifferentiate by substitution.
3. Solve exponential growth and decay problems by applying antidifferentiating rules.
Approximate duration: 30 – 35 days
UNIT VII: APPLICATIONS OF DEFINITE INTEGRALS
CCCS: 4.1, 4.2, 4.3, 4.5
Objective:
Students will be able to:
1. Interpret the integral as net change.
2. Calculate the areas in the plane by using integrals.
3. Calculate the volumes of solids formed by areas.
4. Calculate the volumes of solids formed by revolving areas around axes.
Approximate duration: 30 – 35 days
Lists of texts,
resources and/or literature
TEXTBOOK: Finney, Demana, Waits, Kennedy, Calculus Graphical, Numerical, Algebraic,
Pearson Education Inc., publishing as Pearson Prentice Hall,