AP Calculus AB Syllabus
Course Overview
Advanced Placement Calculus is a course that requires a student to learn the fundamental concepts and mathematics of calculus and to recognize and formulate connections between topics. It is expected from this course that students will gain mathematical skill, understanding and use of technology to help them be successful in further mathematics classes and in their future careers. Students are expected to think hard, try different approaches to problems, and enjoy seeing their understanding of mathematics grow.
Course Outline
Below is an outline of the topics covered in AP Calculus AB.
First Semester
Section
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
Topics
Functions and Change
Exponential Functions
Functions
• Composite
• Stretches
• Symmetry
• Inverses
Quiz
Logarithmic Functions
Trigonometric Functions
Power Functions
• Polynomials
• Rational Functions
Quiz
Continuity: Intermediate Value Theorem
Limits and Continuity
Timeline (in days)
1.5 1.5
2
1.5
1.5
2
1
3
Review
Test Chapter 1
2.1
2.2
2.3
2.4
2.5
2.6
3.1
3.2
3.3 3.4
3.5
3.6
3.7
3.9
3.10
1
Average and Instantaneous Velocity
Derivative at a Point
• Average Rate of Change
• Definition of Derivative
Derivative Function
Quiz
Interpretations of the Derivative
• Alternative Notation
Second Derivative
Differentiability
• Relationship with Continuity
Review
Test Chapter 2
1
1.5
Derivatives • Powers • Polynomials
Derivatives • y = ax
• y = ex
Product and Quotient Rules
Quiz
Chain Rule
Trigonometry
• Review of Unit Circle
• General and Restricted Circles
• Inverses
• Derivatives
Chain Rule and Inverse Functions
Quiz
Implicit Differentiation
Linear Approximation
Theorems about Differentiable Functions
• Mean Value Theorem
• Increasing Function Theorem
2
2.5
0.5
1
1
1
2
2
3
4
1.5
2
1
1
• Constant Function Theorem
• Racetrack Principle
Review
Test Chapter 3
4.1
4.2
4.3
4.5
4.6
4.7
Using First and Second Derivatives
• Critical Points
• Inflections Points
• Local Extrema
• First and Second Derivative Tests
Families of Functions
• The Effect of Changing Parameters
3
Optimization
• Global Maximum/Minimum
• Extreme Value Theorem
• Upper/Lower Bounds
Quiz
Optimization and Modeling
Related Rates
L’Hopital’s Rule
2.5
1
3
2.5
1
Review
Midterm Exam Also included in the first semester are several days for review, multiple choice and free response practice, group work and presentation, quizzes and tests for assessment.
Second Semester
Section Topics
5.1
Measuring Distance Traveled
• Velocity vs. Time • Area under a Curve
• Left and Right Sums
5.2
Definite Integral
• Area under a Curve
• Riemann Sums
Timeline (in days)
1
2.5
5.3
5.4
Fundamental Theorem of Calculus
Definite Integral Theorems
• Sums and Constants
• Area between Curves
• Even and Odd Integrals
Review
Test Chapter 5
6.1
Antiderivatives
1
• Graphs
• Calculate using the Fundamental Theorem
Finding Antiderivatives
2
Differential Equations and Initial Value Problems 1.5
Second Fundamental Theorem of Calculus
• Construction Theorem
Review
Test Chapter 6
6.2
6.3
6.4
7.1
7.5
8.1
8.2
8.4
1.5
2.5
Methods of Integration
• Guess and Check
• Substitution
Quiz
Approximation of Definite Integrals
• Left/Right Rules
• Midpoint Rule
• Trapezoid Rule
Review
Test Chapter 7
3
Finding Areas and Volumes
• Horizontal and Vertical Slicing Quiz
Volumes of Revolution
Density and Center of Mass
Review
Test Chapter 8
3
2
2
2
11.1
11.2
11.4
11.5
11.6
Differential Equations
• First and Second Order
Slope Fields
Quiz
Separation of Variables
Growth and Decay
Applications and Modeling
Review
Test Chapter 11
2
2
3
2
1
Review for AP Exam
AP Exam
11.7
7.2
8.3
9.1
9.2
Models of Population Growth
Integration by Parts
Area and Arc Length using Polar Coordinates
Sequences
Geometric Series
2
2
2
2
2
Final Second Semester Exam
Also included in the second semester are several days for review, multiple choice and free response practice, group work and presentation, quizzes and tests for assessment.
Primary Textbook
Hughes­Hallet, Deborah, et al. Calculus: Single Variable. 4th ed. New York: Wiley, 2005.
In addition to the textbook, additional materials include multiple choice and free response questions from previous AP Calculus AB exams and Hughes­Hallet supplemental materials. Curricular Requirements
The following is a list of the curricular requirements for AP Calculus AB along with evidence of how these requirements are fulfilled within this course.
Curricular Requirement 1
The course teaches all topics associated with Functions, Graphs, and Limits; Derivatives; and Integrals as delineated in the Calculus AB Topic Outline in the AP Calculus Course Description. •
The topics listed above in the course outline are aligned with the topics outlined in the AP Calculus Course Description. Curricular Requirement 2
The course provides students with the opportunity to work with functions represented in a variety of ways ­­ graphically, numerically, analytically, and verbally ­­ and emphasizes the connections among these representations. •
In each class, a graphical representation of functions that follow from a table is shown. It is important to be able to explain what is happening and also to be able to physically solve the problems (with integration for example).
•
In a specific example, students examine the effects of changing parameters on the functions. The students are given a function like y = A sin(Bx) from which they must decide what happens to the graph of the function when the parameters of A and B are changed. Each group of students will have a different function. They must analyze the graphs of the functions and how they transform based on the parameters. Each group then presents their conclusions to the class. Curricular Requirement 3
The course teaches students how to communicate mathematics and explain solutions to problems both verbally and in written sentences. •
The students solve problems in small groups and give presentations to the class.
•
Some homework assignments require the students to give a written explanation to their solution.
•
Students often come to the whiteboard or overhead projector to explain their solutions to the class. They must learn how to speak in a way that communicates the mathematics using the correct vocabulary and showing each step. By having they students teach each other, they begin to learn what is important to say and show in their work in order to clearly express the mathematics so that others understand.
Curricular Requirement 4
The course teaches students how to use graphing calculators to help solve problems, experiment, interpret results, and support conclusions. •
Graphing calculators are used daily to look at the behavior of functions based on both graphs and tables. Students who are somewhat unfamiliar with their graphing calculator become proficient at using their calculator for many different purposes.
•
Calculator technology is used to find numerical approximations of integrals with Riemann sum approximations. In one lesson, students enter a program on their calculator which allows them to approximate integrals based on Left, Right, Midpoint, and Trapezoidal sums. From this, they can see the effect of the number of subintervals and make a best estimate of the area under a curve.
•
Calculators are used to find solutions to equations using numerical methods. For example, if an equation cannot be solved using any known method (factoring, quadratic formula, logarithms), the students become adept at using the graph or table on their calculator to find the solution(s) to the equation.
•
In general, TI­83+, TI­84, and TI­89 calculators are utilized by the students. For presentation and demonstration, the TI­84 is used most often.
Student Evaluation
Students are assessed on the basis of tests (80 points), quizzes (30 points), and homework (variable 1­4 points per graded assignment). The total points per semester ranges from 500­600 points. The students are required sometimes to complete homework or assessments with and without the use of calculators. The two quarter grades per semester are worth 40 percent of the final grade. The midterm or final exam completes the final 20 percent of the grade.