### Calculus Content

AP® Calculus AB

Course Outline

Chapter P – Preparation for Calculus (5 days)

·                     Graphs and Models

·                     Linear Models and Rates of Change

·                     Functions and Their Graphs

·                     Fitting Models to Data

Chapter 1 – Limits and Their Properties (12 days)

·            Preview of Calculus
·       Limits Graphically and Numerically
·       Evaluating Limits Analytically

·       Continuity and One-sided Limits

·       Infinite Limits

Chapter 2 – Differentiation (12 days)

·       The Derivative and Tangent Line
·       Basic Differentiation and Rates of Change

·       Product and Quotient Rules

·       Higher-Order Derivatives

·       The Chain Rule

·       Implicit Differentiation

·       Related Rates

Chapter 3 – Applications of Differentiation (15 days)

·      Extrema
·       Rolle’s Theorem and the Mean Value Theorem

·       The First Derivative Test

·       Concavity and the Second Derivative Test

·       Infinite Limits

·       Curve Sketching

·       Optimization Problems

·       Differentials

Chapter 4 – Integration (15 days)

·      Antiderivatives and the Indefinite Integral
·       Area

·       Reimann Sums and Definite Integrals

·       The Fundamental Theorem of Calculus

·       Integration by Substitution

·       Numerical Integration

Chapter 5 – Logarithmic, Exponential, and Other Transcendental Functions (18 days)

·      Differentiation of the Natural Logarithmic Function
·       Integration of the Natural Logarithmic Function

·       Inverse Functions

·       Exponential Functions

·       Other Bases and Applications

·       Differential Equations: Growth and Decay

·       Differential Equations: Separation of Variables

·       Differentiation of Inverse Trigonometric Functions

·       Integration of Inverse Trigonometric Functions

Chapter 6 – Applications of Integration (15 days)

·      Area Between Two Curves
·       Volume: Disk Method

·       Volume: Shell Method

·       Volume: Slices

·       Surfaces of Revolution

·       Work Problems

Chapter 7 – Integration Techniques, L’Hopital’s Rule, and Improper Integrals (6 days)

·      Basic Integration
·       Integration by Parts

·       Trigonometric Integrals

·       Trigonometric Substitution

Multiple Representations

Most functions and problems in this course are presented analytically (or symbolically), but many of them can also be viewed graphically, verbally, and numerically.  The integration of these different representations is one of your major goals of the course.  Any function or problem given in one representation should always be analyzed for its representation in other mediums.  You should be able to correlate symbolic analysis with graphical analysis and explain this correlation in a logical and easily understandable manner.  At least once every three weeks, each student will be assigned one problem to present and explain to the class.  This presentation should include all applicable representations.

Graphing Calculators

Graphing calculators are an extensive part of AP® Calculus AB.  I personally prefer the Casio graphing calculators, but you may use any other brand as well.  Graphing calculators are particularly helpful in numerical and graphical limits, numerical integration, and slope fields, but are an absolute necessity to find approximations for functions that cannot be integrated and for approximating roots of higher-order functions.  You are expected to use your graphing calculator, as appropriate, for all homework and tests.  It is also your responsibility to determine when a function can be integrated analytically or when it cannot be integrated analytically and requires an approximation from the definite integral function of your calculator.

Justification of Techniques

Problem solving in this course requires not only correct solutions, but an explanation or justification for the techniques you employed.  This is the “what, where, why, and how?” of this course.  Verbal and/or written explanations are required for any points to be awarded for any item presented in a “problem solving” format or for any problem that states “justify.”

AP® Exam Review

The review for the AP® Exam will cover the five weeks prior to the exam.  The format for the review will be different than the format for the normal curriculum. We review the entire curriculum in five weeks. The pace is very fast.  Three weeks we cover many multiple choice calculus questions from old AP® exams, assigning small tests throughout the review. Along with the multiple-choice, we go over twelve to fifteen applications problems. The additional two weeks we spend working out of the study guide “Multiple-Choice & Free-Response Questions in Preparation for the AP Calculus (AB) Examination,” Eighth edition. Then we wrap everything up by taking the 1997 AP Calculus AB examination. Specific lectures are by advanced request only.

After the AP® Exam

After the AP® exam, students are assigned as a group of three a topic to present to the class. They are randomly assigned the most challenging topics of the curriculum throughout the year.  Students must prepare teach that lesson to the class, assign homework, and grade the homework. After all the groups have presented their topics, the course final will be given.

Textbook

Larson, Ron, Robert P. Hostetler, and Bruce H. Edwards.  Calculus with Analytic Geometry.  7th ed.  Boston: Houghton Mifflin Company, 2002.