Math Analysis
Math Analysis Syllabus:
Functions and Their Graphs, Polynomial and Rational Functions, Exponential and Logarithmic Functions, Trigonometric Functions, Analytic Trigonometry, Additional Topics in Trigonometry, Sequences, Series and Probability, Topics in Analytic Geometry, Limits and Introduction to Calculus
A.P. Calculus
Syllabus:
Mrs. Merchant
2008-2009
Teaching Strategies
Students taking AP Calculus are expected to have had a strong Algebra and PreCalculus foundation. As a group we will work extensively on study habits, algebra skills, appropriate use of the graphing calculators, and student communication—both oral and written. I will begin the year by setting high expectations for student performance, requiring clear work and providing time for collaborative learning. While many classes require some lecture, I will depend on it as little as possible. I spend some time presenting material to the class and then I will try to actively engage students in the learning process. This tends to ensure that they understand the material.
Homework
Homework assignments deal not only with current classroom topics, but with review as well. There is some time allowed at the beginning of class for students to ask questions regarding the previous night’s assignment. I limit the amount of time regarding assignments. Students are responsible for getting help from me regarding any questions that we do not have time to address in class. Each Wednesday morning before school we have a scheduled help session to correct any problems and address the assigned free response questions for that week.
Assessment
Some kind of graded experience, either a test or a quiz, will be given weekly in addition to two free-response questions that are assigned for that particular week. A multiple-choice nine weeks’ test may be given at the end of each quarter. As incentive to study and do well on the nine weeks’ test, I will allow students to use their nine weeks’ test score to replace a low test grade if time permits. Nine weeks’ averages are determined on a percentage system. The percentages are as follows: Tests 50%, Quizzes 30%, Free Response Questions (FRQs), Checked Homework 15% and Daily Homework 5%.
AP Problems
In addition to the regular assignments, students will have AP problems due each week beginning in September/October. The students will receive an assignment sheet giving assignment due dates and problems for each nine weeks. All of these problems are from previous AP Examinations. Students will write up these problems according to my guideline, provided in a separate handout. Students are allowed to talk with each other about the problems and may seek additional help from me only during scheduled help sessions. These assignments will be turned in each Friday and graded on a nine-point scale, (which is how the free response portion of the AP test is graded). We will strengthen communication skills, review topics, and discover weaknesses in understanding by working these problems.
Preparing for the AP Examination
To help develop confidence with the format of the AP exam, my multiple-choice and free-response questions are taken from previous AP examinations and they are incorporated into each test. Several practice examinations are given throughout the year. My mid-term examination consists of 45 multiple-choice questions. Students will also complete three free-response questions. Another practice examination is given in March. In April, all discussion of new topics is completed and we begin a concentrated review for the exam. Another practice examination is given one week before the actual AP examination. At that time we can focus on topics that appear to need reviewing the most. Students need to take practice tests under examination conditions.
The examination time for the AP Calculus Examinations will be 3 hours and 15 minutes. The multiple-choice section of the examinations will be 45 questions. In part A of the multiple-choice section (28 questions in 55 minutes), students are not allowed to use calculators. Part B of the multiple-choice section (17 questions in 50 minutes) contains some questions for which a graphing calculator is required. The free-response sections consists of two parts of 3 questions each. The first section is calculator active and the second section is calculator-free.
Technology in the classroom
Technology can be used in the classroom in a variety of ways. Students can use technology as a discovery tool to learn new material. Technology can reinforce concepts and visualize what they have already learned. Technology is also a problem solving tool that can assist in solving a problem with difficult mechanics.
In PreCalculus, students learn to find an appropriate viewing window to produce a complete graph of a function, find the zeros of a function, evaluate a function at a point, and determine symmetry. These topics are reinforced early in AP Calculus.
Students need to practice using their calculator to solve multiple-choice and free response questions. However, as much as I enjoy having calculator technology available, I also am unwilling to give up on the paper and pencil approach. There are times that my classes must put away their calculators away and think about the behavior of a graph of a function or find a volume of a solid of revolution by hand.
AP Calculus AB Course Outline
Advanced Placement Calculus AB is designed as college-level Calculus I. Students are required to take the College Board AB Examination in May to determine college credit awarded. Extensive use of a graphing calculator will be a major requirement of this course. Technology is used regularly by students and teachers to reinforce the relationship among the multiple representations of functions, to confirm written work, to implement experimentation, and to assist in interpreting results. Homework assignments will be enhanced by the use of the graphing calculator and many colleges require students to own a graphing calculator and be well trained in its use. Purchase of a graphing calculator is required; be advised however that colleges have their own preferences and requirements.
Goals:
• Students should be able to work with functions represented in a variety of ways: graphical, numerical, analytical, or verbal. They should understand the connections among these representations.
• Students should understand the meaning of the derivative in terms of a rate of change and local linear approximation and should be able to use derivatives to solve a variety of problems.
• Students should understand the meaning of the definite integral both as a limit of Riemann sums and as the net accumulation of a rate of change and should be able to use integrals to solve a variety of problems.
• Students should understand the relationship between the derivative and the definite integral as expressed in both parts of the Fundamental Theorem of Calculus.
• Students should be able to communicate mathematics both orally and in well-written sentences and should be able to explain solutions to problems.
• Students should be able to model a written description of a physical situation with a function, a differential equation, or an integral.
• Students should be able to use technology to help solve problems, experiment, interpret results, and verify conclusions.
• Students should be able to determine the reasonableness of solutions, including sign, size, relative accuracy, and units of measurement.
• Students should develop an appreciation of calculus as a coherent body of knowledge and as a human accomplishment.
This outline of topics in intended to indicate the scope of the course, but it is not necessarily the order in which the topics are to be taught. The time spent is only an estimate of the average number of days allotted to the topic. Actual time varies from year to year depending on the students’ abilities and interests. Although the examination is based on the topics listed in the topical outline, enrichment topics are included in this course.
Analysis of graphs (Covered in summer assignment)
With the aid of technology, graphs of functions are often easy to produce. The emphasis is on the interplay between the geometric and analytic information and on the use of calculus both to predict and to explain the observed local and global behavior of a function.
• Slopes and equations for lines
• Relations, functions, and their graphs
• Identify graphs of the following functions: constant, linear, quadratic, cubic, square and cube root, absolute value, greatest integer, exponential, logarithmic, and trigonometric
• Domain, range, intercepts, symmetry, asymptotes, zeros, and odd and even function
• Shifts, reflections, stretches, and shrinks
• Solving equations and inequalities
• Relations, functions, and their inverses
• A review of trigonometric functions
Limits of a function (15 days or 7.5 blocks)
• An intuitive understanding of the limiting process
• Instantaneous Velocity
•
• Finding limits algebraically
• Continuous functions
• Determine when a limit does not exist
• Evaluate one-sided limits
• Limits involving infinity
• Asymptotic and unbounded behavior
• Understanding asymptotes in terms of graphical behavior
• Describing asymptotic behavior in terms of limits involving infinity
• Comparing relative magnitudes of functions and their rates of change.
(For example, contrasting exponential growth, polynomial growth, and logarithmic
growth.)
• Geometric understanding of graphs of continuous functions (Intermediate Value Theorem and Extreme Value Theorem).
Derivatives (25 days or 12.5 blocks)
Concept of the derivative.
• The concept of the derivative presented geometrically, numerically, and analytically
• Derivative defined as the limit of the difference quotient
• Slopes, tangent lines, and derivatives
• Differentiation Rules
• Velocity, Speed, and Other Rates of Change
• Determine the derivative of a function in a variety of ways including slope of the tangent line, rate of change of the function, and instantaneous velocity
• Derivatives of Trigonometric Functions
• Derivatives of Logarithmic and Exponential Functions
• Relationship between differentiability and continuity
• Slope of a curve at a point
• Chain Rule, Product Rule, Quotient Rule
• Implicit Differentiation and Fractional Powers
• Logarithmic Differentiation
• Instantaneous rate of change as the limit of average rate of change.
Applications of derivatives (30 days or 15 blocks)
• Curve sketching involving derivatives and sign lines
• Use the First Derivative Test to find the intervals on which a function is increasing and decreasing and to determine relative extrema of a function
• Use the Second Derivative Test to determine intervals of concavity of a function and locate inflection points
• Corresponding characteristics of graphs of f and f′
• Relationship between the increasing and decreasing behavior of f and the sign of’
• Corresponding characteristics of the graphs of f, f′, and f′′
• Relationship between the concavity of f and the sign of f′′
• Points of inflection as places where concavity changes
• Optimization, both absolute (global) and relative (local) extrema
• Modeling rates of change, including related rates problems.
• Graphing summary including roots, domain and range, asymptotes, symmetry, extrema, and concavity
Integrals (30 days or 15 blocks)
• Understand the concept of area under a curve using a Riemann sum over equal subdivisions
• Use the limit of a Riemann sum to calculate a definite integral
• Definite integrals and Antiderivatives
• Fundamental Theorem of Calculus
• Use the graphing calculator to compute definite integrals numerically
• Techniques of antidifferentiation including integration by parts and simple partial fractions
• Integration of trigonometric functions
• Differentiation and integration of inverse trigonometric functions
• Numerical Integration: Trapezoidal Rule and Simpson’s Rule
• Find general and particular solutions to differential equations with separable variables
• Slope fields
Applications of antidifferentiation (20 days or 10 blocks)
• Use definite integrals to find the area under a curve
• Use definite integrals to find the area between two curves
• Volumes of Solids of Revolution—Disks and Washers
• Cylindrical Shells
• Average Value of a Function
• Volumes of solids with known cross sections
References and Materials
Major Textbook
Larson, Hostetler, and Edwards Calculus 8th edition--2006, Houghton—Mifflin
Supplementary Materials
2006, 2007 AP Calculus AB and BC Course Description
AP Calculus Teacher’s Guide
AP Calculus Free-Response Questions and Solutions 1989-1997
AP Calculus Free-Response Questions and Solutions 1979-1988
AP Calculus Free-Response Questions and Solutions 1969-1978
AP Calculus Multiple-Choice Question Collection 1969-1998
Stewart, James. Calculus. 5th edition. Belmont, CA: Brooks/Cole—Thomson Learning. 2003
Bradley, Gerald L., Karl Smith, Single Variable Calculus, Prentice-Hall, 1995.
Dick, Thomas P., and Charles Patton. Calculus of a Single Variable. Boston: PWS Publishing Company, 1994.
Edwards, Penney, Calculus and Analytic Geometry, 6th edition (2002) Prentice-Hall
Hughes-Hallet, Deborah, and Andrew M. Gleason, et. al. Calculus. New York: John Wiley & Sons, 1994
Swokowski, Earl W., Michael Olinick, Dennis Pence, and Jeffery A. Cole. Calculus. 6th edition. Boston: PWS Publishing Company, 1994.
The Calculus Problem Solver, Staff of Research and Education Association
Hockett, Shirley O., How to Prepare for Advanced Placement Examinations in Mathematics, Barron’s Educational Series
Lederman, David, Multiple Choice Questions in Preparation for the AP Calculus Examination
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