AP+Calculus


 * Resources **

Text: Each student has a copy of Calculus, Larson 7th Edition (available in different volumes, including Calculus of a Single Variable and Calculus with Analytic Geometry - the material we cover is identical across these volumes). This text is used as a reference and for homework.

The coursework also draws from other sources, most extensively from Stu Schwartz’s Calculus materials from MasterMathMentor.com, for notes and homework. These materials constitute the equivalent of a textbook: guided notes, class problems, and homework problems. The Schwartz "text" explicitly covers all AP Calculus AB materials.

All students are provided a TI-84+. While most class tests are calculator-free, graphing calculators are used regularly to check work (particularly with graphing and evaluating definite integrals) and to help investigate new concepts, most notably a numerical understanding of the limit and derivative and a graphical understanding of the derivative and antiderivative.


 * Coursework **

Homework is assigned daily. Answers are provided the following day, and students work in small groups to discuss and explain problems to one another at the beginning of each class period. I’ve found that peer explanations are both easier for students to grasp and can deepen the understanding of concepts for both the struggling student and the one who can explain, but I monitor this time to ensure correctness. If problems persist in the groups, the class reconvenes, a student is selected to work out the problem for everyone, and the class watches while I pause regularly, asking questions of the student and the class to check for understanding.

Each week, two AP-style free response questions are assigned for homework during the week. Quizzes are assigned about once or twice a week, and emphasize interpreting answers not only analytically but graphically, numerically, and in written sentences. These check understanding of recent content, or draw from the bank of free response questions that have been assigned.

Tests are given about once every two weeks. They are a combination of multiple-choice and free response questions integrating numerical, graphical, analytical, and verbal (written sentence) solutions to problems. Tests are cumulative for all material covered up to that point. Students may correct tests for partial credit. A key aspect of test correction is a written explanation of the student’s error and how they will adjust their approach to similar tasks in the future.


 * Course Schedule **

Timing is, of course, approximate. Some “wiggle room” is included for each semester. The approximate chapter equivalent from the Larson text is included with each unit.

If time allows, the antiderivative is introduced at the close of the Fall semester.


 * Semester 1 **

Precalculus Review (~Larson Chapter P) - 1 week
 * Review key graphing, algebraic, and trig concepts from Algebra II and PreCal
 * Review of graphing calculator functions and investigating graphs with various windows, tables, etc.
 * Preview of Calculus

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">Limits (~Larson Chapter 1) - 2 weeks > (3.5) Limits at Infinity and Horizontal Asymptotes
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">Understanding Limits Graphically and Numerically
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">Evaluating Limits Analytically
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">Continuity and One-Sided Limits
 * Intemediate Value Theorem
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">Infinite (Unbounded) Limits and Vertical Asymptotes
 * Comparing infinities and different kinds of growth (exponential, linear, etc)

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">Introduction of the Derivative (~Larson Chapter 2.1) - 1 week > Introduction to the relationship of f and f’, especially graphically.
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">The Derivative as a Limit of the Slope of a Secant Line
 * Graphical approximations of tangents
 * By hand
 * Using calculators and GeoGebra applet
 * Numerical limit of secant line at a point
 * By hand
 * Using tables on graphing calculators
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">The derivative as the limit of the rate of change of a function at all x
 * Limit of the difference quotient = instantaneous rate of change
 * Alternate Form (xc)
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">Continuity and derivatives
 * Where derivatives fail to exist (cusps, discontinuities)
 * Vertical tangent lines
 * Differentiability implies continuity

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">Differentiation (~Larson Chapter 2) - 4 weeks > Product and Quotient Rules
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">Basic Differentiation Rules
 * Introduction to rates of change and applications of the derivative, including linear motion (particle on a line, free-fall, etc)
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">Higher-Order Derivatives
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">The Chain Rule
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">Implicit Differentiation
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">Related Rates

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">Applications of Differentiation (~Larson Chapter 3) - 3 weeks
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">Extrema on an interval
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">Rolle’s Theorem and the Mean Value Theorem
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">Increasing and Decreasing Functions and the First Derivative Test
 * Local extrema where increasing/decreasing behavior changes
 * Monotonic and strictly monotonic functions
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">Concavity and the Second Derivative Test
 * Points of Inflection where concavity changes
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">Curve Sketching
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">Optimization
 * Using f, f’, and f’’ to predict and describe the behavior of a function, using numerical, verbal, written, and graphical demonstrations


 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">Semester 2 **

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">The Antiderivative (~Larson Chapter 4) - 3 weeks > The Second Fundamental Theorem of Calculus
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">Slope Fields
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">Antiderivatives and Indefinite Integration
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">Area, Riemann Sums, and Definite Integrals
 * Numerical Integration (Left-, Right-, Mid-point and Trapezoidal methods)
 * Definite Integral as Limit of Riemann Sums
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">The Fundamental Theorem of Calculus
 * Definite Integral as Accumulation of a Rate of Change
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">Integration by Substitution
 * Change of variable and change of limits of integration

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">Applications of Integration (~Larson Chapter 6) - 3 weeks
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">Linear Motion
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">Area between Curves
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">Volumes of Rotation
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">Volumes of Known Cross Section

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">Transcendental and Inverse Functions (~Larson Chapter 5) - 4 weeks > Inverse Trigonometric Functions
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">Natural Logarithm: Differentiation and Integration
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">e: Differentiation and Integration
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">Exponential functions with bases other than e
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">Inverse Functions and Calculus
 * Relationship between rates of change of inverse functions
 * Implicit differentiation of inverse functions
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">Differential Equations: Separation of Variables
 * Exponential growth
 * L'Hopital's Rule

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">Intensive Review for AP Test - 3 weeks
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">AP Test: May 7

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">Post-Test Concepts - 2 weeks
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">Physics and Calculus: Force, Work, etc