Advanced Placement Calculus AB
Course Syllabus
Course Overview:
Students in this course will develop a strong foundation in Calculus. I want students to appreciate the necessity for Calculus
and provide them the experiences to be successful in future mathematics courses. To guide the students towards a more
comprehensive knowledge, we will examine all main ideas graphically, numerically and analytically. Students will be
expected to communicate their mathematical knowledge both verbally and in a written form. In addition, students will
develop an understanding of the connectedness of the multiple representations. Students will discus individual problems in
small groups and present responses using graphing calculators, SMART board© technologies and verbal explanations. This
course will enhance the students’ ability to use graphing calculators to help solve problems, explore concepts, interpret
results and support their conclusions. This course will utilize a wide range of assessment tools including free response,
multiple choice, oral response, and graphical demonstrations. Students will be required to put in a great deal of work in
preparation for the AP Exam including completing, scoring and presenting solutions to previous AP exams (FreeResponse).
Course Outline:
Unit 1: Pre-Calculus Review and Preparation for Calculus: Chapter P and Appendix (2 weeks)
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Properties of Real Numbers and the Cartesian plane.
Interval Notation as a method to express solutions.
Identification of basic graphs: families of functions, piece-wise functions, composition of functions,
domain, range, intercepts, symmetry, asymptotes, zeros, etc.
Trigonometry review.
Transformation of basic graphs.
Use function notation to represent and evaluate a function.
Interpret mathematical models for real–life data
Fit a mathematical model to a set of real-life data using technology (graphing calculators)
Use of the Graphing Calculator.
a.
Zeros
b.
Points of intersection
c.
Modeling data, predicting future events
d.
Graphing a function in an arbitrary viewing rectangle
Unit 2: Limits and Their Properties: Chapter 1
(2 weeks)
1.
Understand that the tangent line problem and area problem are basic to calculus.
2.
Find limits numerically and geometrically including one-sided limits. (Includes the use of tables and
graphs.)
3.
Use the properties of limits to calculate the sum, difference, product or quotient of functions.
4.
Determine whether a limit exists using numerical, geometrical and analytical approaches.
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Applying the definition of continuity at a point.
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Finding points of discontinuity for a function.
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Understand the concept of indefinite limits and how to locate vertical asymptotes and describe asymptotic
behavior.
8.
The Intermediate Value Theorem and its applications.
Unit 3: Differentiation
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Chapter 2
Explore the tangent line problem using graphing calculators.
Find the slope of a tangent line and write the equation of the tangent line using the limit process.
Define and compute the derivative of a function using the limit process.
Apply the concepts of Instantaneous and average rate of change of a function.
Explore the concept of differentiability and Local Linearity using a geometrical approach (Graphing
calculators).
Understand the relationship between differentiability and continuity of a function.
Use the differentiation rules to compute derivatives algebraically.
Use the chain rule to differentiate composite functions.
Use Implicit differentiation to find the derivative of implicitly defined functions.
Compute successive derivatives of a function.
Use the derivative to find the slope of a curve and to write the equations of tangent and normal lines to
the curve.
Use the graphing calculator to compute the numerical derivative of a function.
Solve problems involving related rates of change.
Unit 4: Applications of the Derivative
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Chapter 3
(4 weeks)
Apply the Extreme Value Theorem to locate the Global (Absolute) Extrema on a closed interval.
Apply Rolle’s Theorem and the Mean Value Theorem.
Use the 1st derivative test to determine where a function is increasing or decreasing and to locate the
Local (relative) Extrema.
Use the 2nd derivative test to determine where a function is concave up or concave down and to locate the
inflection points of a function.
Use the 2nd derivative test to analyze the relative extrema of a function.
Apply the information from the 1st and 2nd derivative tests to determine a sketch of the function.
Explore the relationships between f , f ' and f '' .
Find limits at infinity to determine horizontal asymptotes.
Apply derivatives to solve optimization problems. .
Apply the derivative to position, velocity and acceleration applications.
Use Newton’s method to approximate zeros of a function.
Unit 5: Integration
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(4 weeks)
Chapter 4
(4 weeks)
Compute anti-derivatives.
Find the general and particular solutions to a differential equation with separable variables.
Understand the concept of area under a curve using a Riemann’s sum.
Compute a Riemann’s sum using left endpoints, right endpoints, and midpoints as evaluation points.
Use the limit of a Riemann’s sum to calculate a definite integral.
Use the Fundamental Theorem of Calculus to evaluate a definite integral.
Calculate anti-derivatives using substitution of variables and change of limits.
Use the graphing calculator to compute definite integrals numerically.
Apply the 2nd Fundamental Theorem of Calculus.
Apply the Average Value Theorem.
Use the trapezoidal rule to approximate the area under a curve.
Unit 6: Logarithmic, Exponential and other Transcendental Functions
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Chapter 5
(4 weeks)
Explore properties of logarithmic and exponential functions.
Differentiate logarithmic and exponential functions.
Find the derivative of an inverse of a function.
Use logarithmic differentiation to find the derivatives of complicated derivatives.
Differentiate and integrate general exponential and logarithmic functions.
Apply the anti-derivative to solve problems.
Find the derivatives of inverse trigonometric functions.
Evaluate integrals that yield inverse trigonometric functions.
Unit 7: Differential Equations. Chapter 6
(3 weeks)
1.
What are differential equations?
2.
Use separation of variables to solve simple differential equations
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Use initial conditions to find the particular solutions of a differential equation.
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Sketching a slope
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Using a graphing calculator to explore slope fields and approximate solutions to differential equations.
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Use Euler’s method to approximate solutions to differential equations.
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Use exponential functions to model growth and decay in applied problems.
Unit 8: Applications of Integration.
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Chapter 7
(2 weeks)
Use definite integrals to find the area under a curve.
Use definite integrals to find the area between two curves.
Find the volume of a solid of revolution using the methods of Discs, Rings and Shells.
Find the volume of a solid with a known cross section.
Unit 9: AP Exam Preparation
Supplementary Materials
(2 weeks)
Highlighted Teaching Strategies and Student Activities:
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All students will be using graphing calculators on a regular basis. Most students choose a TI-84. The
majority of my demonstrations are on the TI 84 calculator. Students also have access to Geometers’
Sketchpad to aid in their explorations of the calculus concepts. Any student that has difficulty affording a
calculator will be given a graphing calculator to be used throughout the course.
The beginning of the course has students reviewing many Pre-Calculus concepts. However, the students
will be forced to examine their own understanding by looking problems from numerical, geometrical and
analytical perspectives. Students will be working in small groups and individually. Students are also
required to verbally explain their solutions in small group and/or whole class discussions.
One of our school goals is to improve student’s written and verbal communication skills. Students are
regularly asked to provide written explanations to support their work. In addition, our textbook has
writing prompts in most sections that we often use.
My classes are a combination of a lectures and “guided explorations” activities. Many of the exploration
activities incorporate the use of technology. We explore the concept of local linearity to support
differentiability. In groups, students will create slope fields from a given differential equation and then
explore possible solutions to the differential equation. By using their graphing calculators, students will
then verify their solution curves. Another activity has small groups of students bring in a bottle (usually a
small soda) and the students will calculate the volume of the bottle by measuring the diameter (or
circumference) in various locations. Students can input the data, find a regression model for the bottle and
use a Riemann’s Sum to calculate the volume of the bottle. Students will present their results and discuss
the “accuracy” with the class. Several other exploration activities are done during the year.
During the AP review unit, students prepare for the exam using a variety of activities including taking old
AP exams and “scoring” their own, and other student’s free-response tests using the “scoring guidelines”.
Students find this activity very beneficial because they really see how they will be graded on the real
exam.
Student Evaluation:
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Grades are determined by weekly problem sets, quizzes, unit test and final exam. Depending of the
material, the assessments may be with or without a calculator. Most assessments, including the final, have
a section where the students may not use a calculator and a section where a calculator is required to solve
the problem.
Practice problem sets are given on a daily basis, but are not formally evaluated. Practice problems sets are
discussed in class.
Students are expected to work collaboratively, outside of class, on the weekly problem sets. These
problem sets are mostly application based or lead to the discovery of future calculus concepts. Often,
students will present solutions to the class.
Primary Textbook:
Larson, Hostetler and Edwards. Calculus of a Single Variable. 8th edition. Boston, MA.
Houghton Mifflin Company, 2006
Supplementary Resources:
The College Board / AP Central website. http://apcentral.collegeboard.com/apc/public/exam/index.html
1.
Exam Information
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Old Exam Questions
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Scoring guides and commentary
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Exam tips
National Council of Teachers of Mathematics website: http://nctm.org/
Clements, Pantozzi and Steketee. Exploring Calculus with Geometer’s Sketchpad. Emeryville, CA.
Key Curriculum Press, 2002.