GLEN RIDGE PUBLIC SCHOOLS
Curriculum Guide
Course Title: COLLEGE
PREP CALCULUS
Subject: Mathematics
Grade Level: 12
Department/School: Mathematics/High School
Duration: Full Year
Number of Credits: 5
Prerequisite: Math Analysis with a grade of “B” or
better
Or Honors Math Analysis
Elective or Required: Elective
Author: Catherine
McCarthy
Date Submitted: Summer 2007
Course Description
The College Prep Calculus course is designed for those students with a solid foundation in algebra, geometry, and math analysis. Students in this class should possess an interest in studying advanced mathematical topics as well as a desire to spend time solving problems that are of a challenging nature. The course is meant to serve as an introduction to differential and integral calculus. It is expected that the student will:
- Review the properties of functions and their graphs
- Evaluate both finite and infinite limits
- Calculate rates of change, including the definition of the derivative and derivative formulas
- Test for the continuity of functions
- Solve problems involving applications of the derivative, including tangents to curves, min/max problems, related rates, optimization problems, and velocity and acceleration problems
- Calculate antiderivatives, including both indefinite and definite integrals, Riemann sums, and the Fundamental Theorem of Calculus
- Solve problems involving applications of the integral, including area under a curve, average value, and distance and velocity from acceleration with initial conditions.
Students are required to explain problems using proper vocabulary and terms and are taught using the method of the rule of three; that is ideas can be investigated analytically, graphically, and numerically.
The graphing calculator is used throughout the course to help students develop an intuitive feeling for concepts before they are approached through typical algebraic techniques. Although it is emphasized as a tool to illustrated ideas and topics, it is expected that students also become proficient in using the calculator to:
· Find roots of an equation
· Sketch functions in a specified window
· Approximate the derivative at a point using numerical methods
·
Approximate the value of a definite integral
using numerical methods.
The students are therefore required to supply their own graphing calculator and the teacher will very often use a graphing calculator view screen.
Students are not required to take the Calculus AB Advanced
Placement exam, but may choose to do so with the permission of the teacher.
GLEN RIDGE PUBLIC SCHOOLS
MATHEMATICS
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 present 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 Standards:
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
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
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
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, iteration 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
Descriptive Statement: The mathematical processes described here highlight ways of
acquiring and using the content knowledge and skills delineated in the first
four mathematics standards.
Problem Solving Reasoning
Communication Representations
Connections Technology
Curriculum Description
UNIT I: FUNCTIONS
CCCS: 4.1, 4.3, 4.5
Objectives:
The students will be able to:
1.
Define and
develop the concept of a “function.”
2.
Analyze graphs of
functions with emphasis on the interplay between the geometric and analytic
information to explain the observed local and global behavior of a function.
3.
Generate the
graphs of equations and functions with a graphing calculator.
4.
Set and apply
appropriate viewing window when graphing functions on the calculator.
5.
Use a graphing
calculator to determine critical values of a function.
6.
Review
characteristics of linear functions such as point-slope form of the equation or
relationship between parallel and perpendicular lines.
7.
Perform
operations such addition, subtraction, multiplication, division on, and
composition of functions.
8.
Identify and
analyze families of functions such as linear, power, polynomial, exponential,
logarithmic, and trigonometric functions.
9.
Recognize the
characteristics of and graph special functions such as the absolute value
function and piecewise functions.
10.
Understand the
relationship between a function and its inverse.
11.
Determine a function’s
inverse if it exists.
12.
Solve equations
involving exponentials, logarithms, and trigonometric functions.
13.
Discover and
apply mathematical models to real-world situations.
Approximate duration:
25 days
UNIT II: LIMITS AND CONTINUITY
CCCS: 4.1, 4.3, 4.5
Objectives:
The students will be able to:
1. Define the
tangent line problem.
2. Develop and understand the concept of the
existence or non-existence of a limit.
3. Use a graphing calculator to analyze the
concept of a limit.
4. Calculate
limits using algebra and estimate limits from graphs or tables of data.
5. Define and apply basic limit theorems and
properties.
6. Understand
asymptotes in terms of graphical behavior and describe asymptotic behavior in
terms of limits involving infinity.
7. Compare
relative magnitudes of functions and their rates of change.
8. Understand
continuity in terms of limits.
9. Define and identify continuity at a point or
on an interval for a function.
10. Classify
discontinuities as either removable, jump, or infinite discontinuities.
11. Understand graphs
of continuous functions geometrically by applying Intermediate Value Theorem
and Extreme Value Theorem.
Approximate duration: 20 days
UNIT III: DERIVATIVES
CCCS: 4.1, 4.3, 4.5
Objectives:
The students will be able to:
1.
Develop the
concept of the derivative.
2. Understand the concept of the derivative
geometrically, numerically, and analytically.
3. Interpret the derivative as an instantaneous rate of
change.
4. Define the derivative as the limit of the difference
quotient.
5. Understand the relationship between differentiability
and continuity.
6. Define the derivative of a function at a point.
7. Find the slope of a curve at a point.
8. Find the equation of the tangent line to a curve at a
point.
9. Find the instantaneous rate of change as the limit of
average rate of change.
10. Approximate rate of change from graphs and tables of
values.
11. Develop and apply techniques of differentiation.
12. Find derivatives of basic functions, including
constant, power, exponential, logarithmic, trigonometric, and inverse
trigonometric functions.
13. Use basic rules for the derivative of sums, products,
and quotients of functions.
14. Find derivatives using the Chain Rule and Implicit
differentiation.
15. Use Implicit differentiation to find derivatives of
inverse functions.
16. Find higher order derivatives.
17. Use graphing calculator to find numerical derivatives.
Approximate duration: 45 days
UNIT IV: APPLICATIONS OF THE DERIVATIVE
CCCS: 4.1, 4.3, 4.5
Objectives:
The students will be able
to:
1. Identify corresponding characteristics of the graphs
of f(x) and f’(x).
2. Understand the relationship between increasing and
decreasing behavior of f(x) and the sign of f’(x).
3. Know and apply the Mean Value Theorem and its
geometric consequences.
4. Know and apply Rolle’s Theorem and its geometric
consequences.
5. Find the linearization of a function.
6. Use the linearization of a function to estimate the
value of a function.
7. Translate verbal descriptions into equations involving
derivatives and vice versa.
8. Identify corresponding characteristics of the graphs
of f(x), f(x)’, and f”(x).
9. Understand the relationship between the concavity of f
and the sign of f”.
10. Identify points of inflection as places where
concavity changes.
11. Analyze curves, identifying minimum, maximum, and
inflections points.
12. Develop and apply first and second derivative tests
for critical points.
13. Solve optimization problems.
14. Model rates of change, including related rates
problems.
15. Interpret the derivative as a rate of change in varied
applied contexts, including velocity, speed, and acceleration.
16. Use
Approximate duration: 30 days
UNIT V: INTEGRATION
CCCS: 4.1, 4.2, 4.3, 4.5
Objectives:
The students will be able to:
1.
Understand the
concept of a Riemann sum over equal subdivisions.
2. Calculate Riemann sums using left, right, and midpoint
evaluation points and a graphing calculator.
3. Use the method of setting up a Riemann sum and
representing its limit as a definite integral
4. Evaluate definite integrals using areas.
5. Find antiderivatives of basic functions.
6. Find antiderivatives by substitution of variables.
7. Use methods of finding antiderivatives to simplify
indefinite integrals.
8. Define and use the Fundamental Theorem of Calculus to
represent a particular antiderivative.
9. Evaluate definite integrals by using the Fundamental
Theorem of Calculus.
10. Use the graphing calculator to evaluate definite
integrals.
Approximate duration:
20 days
UNIT VI: APPLICATIONS OF THE DEFINITE INTEGRAL
CCCS: 4.1, 4.2,
4.3, 4.5
Objectives:
The students will be able
to:
1. Use appropriate integrals to model physical, social,
or economic situations.
2. Use the integral as a rate of change to give
accumulated change.
3. Use integrals to find the area of a region.
4. Use integrals to find the average value of a function.
5. Use integrals to find the distance traveled by a
particle along a line.
6. Find specific antiderivatives using initial
conditions, including applications to motion along a line.
7. Solve separable differential equations and use them in
modeling exponential growth and decay.
8. Use the Trapezoidal rule to approximate definite
integrals of functions represented algebraically, geometrically, and by tables
of values.
Approximate duration: 20 days
UNIT VII: COLLEGE PLACEMENT TEST
REVIEW
CCCS: 4.1, 4.2, 4.3, 4.5
Objectives:
The
students will be able to:
1.
Review algebraic,
geometric, and trigonometric concepts required on college placement tests.
2.
Practice taking
sample college math placement tests as found on the internet.
Approximate duration: 5 days
List of texts,
resources, and/or literature:
Primary Text: Anton, Howard, Iri Bivens, Stephen Davis,
Calculus: Early Transcendentals Sinle
and Multivariable, 8th Edition,
Supplementary
Materials: Anton, Howard, Calculus: Early Transcendentals Single and
Multivariable Solutions Manual,