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AP Calculus AB Course Outline/Syllabus
Text: Stewart, James (2008). Calculus (sixth edition). Belmont, California: Thomson-Brooks/Cole Learning.
Additional Resources
- College Board AP Released Questions 2000-2007
- Worksheets will supplement text where needed to ensure a substantial representation of analytical, verbal, graphical, and numerical problems.
Technology
All students are required to have a graphing calculator. A TI-83 or 84 is preferred. If a student cannot afford a graphing calculator, a TI-83 will be issued. Calculator demonstrations are made using a TI-84 on the overhead projector. Some visual demonstrations will be made using the TI-Nspire calculator. Class time is allocated to teaching students how to use the calculator effectively. Calculator usage includes but is not limited to:
- Finding zeros
- Verifying and computing numerical derivatives
- Verifying and computing definite integrals
- Adjusting viewing window to include critical points, inflection points and asymptotes to verify graphs of functions
- Using the “table” capability to explore limits, rates of change, etc.
General Instructional Guidelines
Various instructional techniques (lecture, group work, guided discovery, and projects) and approaches (graphical, numerical, and analytical) will be used. Every effort will be made to show the connections between these various representations. Homework assignments and test questions reflect this philosophy. Students are often asked to explore and expand on the topics by offering different scenarios (such as what if the function was not continuous, etc.) to deepen their understanding.
The objective of the course is for the students to master all topics listed in the AP Syllabus and for the student to be able to think and write mathematically. Students are always required to support their answers regardless of the approach. Students will often be asked to verify their results by using a second method. Homework assignments and test questions reflect this philosophy of supporting their response and demonstrating the connections. In addition, homework often contains problems from released test questions from previous AP Calculus AB exams to allow students more practice to the wording and expectations on the exam.
Student Evaluation
Tests and quizzes are given to monitor student progress and there is at least one test per chapter. Each test will consist of a calculator part and a no calculator allowed part. For each question, work must be shown to justify answers and proper symbolism is required at all times. When appropriate, students must justify a response with a written sentence/paragraph. When grading tests, wrong answers with correct work receive partial credit in order to encourage the students to show all steps.
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Course Content
Unit 1 – Functions and Models
Key Concepts:
- Review of basic ideas concerning functions, their graphs, and ways of transforming and combining them
- Stress four ways to represent a function: by an equation, in a table, by a graph, or in words
- Investigation of the main types of functions that occur in calculus and using these functions as mathematical models of real-world phenomena
Unit 2 – Limits and Continuity
Key Concepts:
- Investigation of limits and their properties
- Finding limits using calculator “table” and graphs
- Finding limits analytically
- Infinite limits
- Limits at infinity
- Continuity at a point and graphic interpretations
- Intermediate Value Theorem and continuity on an interval
- Squeeze Theorem and limits of trig functions
Unit 3 – The Derivative and Differentiation
Key Concepts:
- Interpreting derivatives as slopes and rates of change & the tangent line problem: approximating the slope of the tangent line
- Estimating derivatives of functions given by tables of values
- Graphing derivatives of functions defined graphically
- Limit definitions of the derivative
- Differentiation and continuity
- Rules for differentiations
- Derivatives of Trig functions
- The Chain Rule
- Implicit Differentiation
- Higher order derivatives and acceleration
- Graphical interpretation: velocity and acceleration curves
- Related rates
- Linear Approximations and differentials
Unit 4 – Applications of Differentiation
Key Concepts:
- Relative extrema and critical points
- Absolute extrema
- Rolle’s Theorem and the Mean Value Theorem
- First derivative test: analysis of the sign with sentence justifications
- Graphical interpretations of the relationship between f and f ‘
- Concavity and inflection points: analysis of the sign with sentence justifications
- Graphical interpretations of the relationships between f, f ‘ and f “
- Second derivative test
- Limits at infinity and horizontal asymptotes
- Curve sketching: Analysis of the graphs of functions, guidelines for sketching a curve by hand and with a graphing calculator
- Optimization word problems and justifying absolute extrema
- Antiderivatives
- Rectilinear motion (position and velocity)
Unit 5 – Definite Integral and Integration
Key Concepts:
- The Area Problem: Using area and distance problems to help formulate the idea of the definite integral
- Sigma Notation
- Approximating area under a curve by Riemann Sums
- The Definite Integral
- Properties of the Definite Integral and the Mean Value Theorem for Integrals
- The Fundamental Theorem of Calculus
- Indefinite Integrals
- Technique of integration: u-substitution
- Differential equations and rectilinear motion
- Differential equations from word problems
Unit 6 – Applications of Integration
Key Concepts:
- Areas between curves
- Trapezoid Rule
- The definite integral as an accumulator and graphical and numerical interpretations
- Volume of a solid object: cross section, disk, washer, shell (solids used to show shapes and illustrate methods)
Unit 7 – Inverse Functions: Logarithmic & Exponential
Key Concepts:
- Inverse Functions
- Exponential functions and their derivatives
- Logarithm Review
- Derivatives of Logarithms
- Natural log from a calculus point of view, differentiation and integration
- Integrals that yield natural log
- Natural base, differentiation and integration
- Power function (ax), differentiation and integration
- Logbx, differentiation and integration
- Word problems involving differentiation that model exponential growth and decay
Unit 8 – Inverse Trig Functions and Slope Fields
Key Concepts:
- Inverse trig function
- Derivatives of inverse trig funtions
- Integrals that yield inverse trig functions
- Creating slope fields, drawing solution curves and matching given slope fields with given differential equations
Unit 9 – Practice Problems for the AP Calculus AB Exam
Key Concepts:
- Review Course
- Practice timed tests for Multiple Choice and Free Response formats
AFTER THE AP EXAM
Topics covered after the exam, by formal lectures or projects, include (as time allows):
- Solving problems involving integration by parts
- Evaluating trigonometric integrals
- Using trigonometric substitution to evaluate integrals
- Integrating rational functions by partial fractions
- Evaluating improper integrals
- Finding arc length
- Graphing curves defined by parametric equations
- Determining derivatives and integrals involving parametric equations
- Graphing curves in polar form
- Finding areas of regions bounded by curves in polar form
- Finding lengths of curves in polar form
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