AP CALCULUS AB



AP CALCULUS AB

2012/2013

School: South Gibson County High School

Instructor: Beth Davis

Brief Description of Course: This course is designed to cover one semester of college calculus material. It covers an extensive study of function, graphs, limits, derivatives, definite integrals and applications of all of the above. Each of these topics is approached via the “Rule of Four”, with activities that emphasize expressing mathematics from graphical, numerical, analytical and verbal representations. Students must become familiar with functions written not only as equations, but also shown as a graph, table or in words. Students must become adept with the concepts of limits, derivatives and integrals given these various forms of function representation. Students must learn that many calculus concepts can be written with symbols or words that they must become familiar with.

Some examples of verbal representations:

Function is always increasing = first derivative is always positive

Function is increasing at a decreasing rate =

increasing and concave down (first derivative positive and second derivative negative)

Find where function is increasing fastest = maximize the first derivative

Find where average rate of change is equal to the instantaneous rate of change =

apply the Mean Value Theorem ( or find where algebra slope = calculus slope )

These are just some of the examples of what students will learn in this class about the language of calculus. This will help students become familiar with the language of calculus and how this language applies to the real world.

Course Overview: My main objective in teaching AP Calculus AB is to provide students with an opportunity to explore higher levels of mathematics. Through this exploration and interaction with mathematics I hope to enable students to appreciate the higher intricacies of problems and develop a solid foundation in the Calculus AB topic outline as it appears in the AP Calculus Course Description, which they can take with them to their higher level classes. I expect a lot from my students, whether it is in class in discussion and group time, or at home working on homework and AP sample problems.

In order to reach all students, I strive to present all topics in many different ways. Among these are graphical, numerical, analytical and verbal approaches to almost all topics.

I use a TI-84 calculator to help students use the table feature, or math menu (zeros, derivative at a point or numerical integral) in graphical mode, to get a more numerical approach to problems. All students are required to have a graphing calculator, with about half of the class using a TI-83+ or TI-84+ and half purchasing a TI-89. Graphing calculators will be used daily to explore, discover, and reinforce the concepts of calculus. Graphing calculators will be used to solve problems, complete experiments, interpret results, and support conclusions throughout the course.

Java applets, PowerPoint presentations, Geometer’s Sketchpad sketches, WinPlot, Graphmatica, and other technology-based visual aids will be used in the delivery of instruction in this AP Calculus course.

Assessment: Students are assessed in my class daily, weekly and once a unit. Each day students will have an assignment addressing the topics we covered in class. Each week students will work on problems from released AP exams, both multiple choice and free response designed to target the subject from the week before. On both of these assignments (homework and class assignments) students are encouraged to work together to discuss and complete the problems, but are required to write their own solutions to the problems. At the end of each unit students are given an exam covering all material in the unit. These exams are generally written in two parts, one calculator active, and one without calculators. Problems are generally open ended, and students are required to show work to get full credit. Often there is at least one released Free Response question from previous AP exams, which is scored in the same manner as they will be on the AP exam. We will also have some multiple-choice questions from released AP exams for practice.

It is important to me to convey to students the importance of knowing how to do a problem and also for students to know what their solutions actually mean in the context of the problem. Students will be asked to explain work and justify answers whenever they are working with open-ended questions. They will be shown that it is equally if not more important to know how they are finding the answer they are finding and what their answer means than to actually get the correct answer. They will be asked to explain their work and answers using complete sentences using correct calculus language and symbols and to use no calculator syntax. This is a great way to show students how calculus relates to the real world.

Course Format: Students enrolled in AP Calculus AB have successfully completed Honors Precalculus. Our class will meet for 90 minutes a day, 5 days a week all year long. Each student receives 5 points added to each grading period’s final grade. If the student receives a 3, 4 or 5 on the Calculus AB exam, our school district will pay the students their AP exam fee back.

Textbooks: Larson, Ron, Hostetler, Robert P., and Edwards, Bruce H. Calculus of a Single Variable 7TH edition. Boston, Houghton Mifflin Company. 2002

Rogawski, Jon. Single Variable Calculus: Early Transcendentals. New York, W. H. Freeman and Company. 2008

Other Resources Used:

These are some of the many other resources that I use for Calculus AB.

- AP Calculus AB Teacher’s Manual by Duke University Talent Identification Program 2nd Edition

-Be Prepared for the AP Calculus Exam by Skylight Publishing 2005

-Multiple Choice and Free Response Questions in Preparation for the AP Calculus Exam by Lin McMullin. D and S Marketing.

-Preparing for the AP Calculus Examination by George Best. Venture Publishing.

-AP Calculus Teachers Guide by The College Board.

-Handley, PA math department website.

-AP Calculus in a Nutshell website Franklin Road Academy, TN.

-1997, 1998, 2003 Released Exams from The College Board.

-1989-2010 Free Response Question Collection .pdf file from The College Board.

-Teaching AP Calculus by Lin McMullin. D and S Marketing.

-2011 AP Summer Institute Manual by Phyllis Hillis. Oak Ridge High School, Oak Ridge, TN

Expectations: To succeed in this class a student must

1. BE IN CLASS EVERY DAY

2. BE PREPARED FOR CLASS EVERY DAY

3. DO HOMEWORK ASSIGNMENTS

4. PAY ATTENTION IN CLASS

5. ASK QUESTIONS

6. PARTICIPATE IN GROUP WORK/DISCUSSIONS

7. TAKE GOOD NOTES

8. TAKE THE AP CALCULUS EXAM ON May 8, 2013

9. TRY THEIR BEST TO MAKE A 3,4 or 5 ON THE AP EXAM

Topic Timeline:

AB TOPICS

I. Functions, Graphs, and Limits --6 weeks

II. Derivatives and Applications of the Derivative – 8 weeks

III. Integrals and Applications of Integrals -- 8 weeks

Differential Equations and Slope Fields – 6 weeks

V. Review for the AP Exam

AB COURSE OUTLINE

Unit 1 – Limits

Larson 1.2 - tangent line problem/intro to limits – table, graph and algebra

Larson 1.3 – properties of limits

Larson 1.3 – techniques for evaluating limits

Larson 1.4 – continuity and intermediate value theorem

Larson 1.4 – continuity of piecewise functions

Larson 1.5 – infinite limits and vertical asymptotes

Larson 3.5 – limits at infinity and horizontal asymptotes

Unit 2 – Concept of the Derivative

Larson 2.1 – rate of change by equation, graph and table

Larson 2.1 – graphical interpretation of the derivative

Larson 2.1 – difference quotient, derivative at c

Larson 2.1 – derivative at x, properties of derivatives

Various - displacement, velocity and acceleration

Larson 2.4 – chain rule

Unit 3 – Derivative Formulas

Larson 2.3 – product rule

Larson 2.4 – quotient rule

Larson 2.2, 2.3, 2.4 – trigonometric functions

Larson 5.3, 5.8, 5.9 – inverse functions and inverse trig functions

Larson 2.1 – differentiability and continuity of piecewise functions

Larson 2.5 – implicit differentiation

Larson 2.6 – related rates

Unit 4 – Graphical Analysis

Larson 3.1 – extrema and extreme value theorem

Larson 3.2 – Rolle’s Theorem, Mean Value Theorem

Larson 3.3 – increasing/decreasing, first derivative test

Larson 3.4 – concavity, point of inflection, second derivative test

Larson 3.6 – curve-sketching

Larson 3.7 – optimization

Unit 5 – Integrals

Larson 4.1 – intro to definite integrals

Larson 4.1 – antiderivatives and indefinite integrals

Larson 3.9 – linear approximations and differentials

Larson 4.5 – antiderivatives and indefinite integrals, u-substitution

Larson 4.6 – trapezoid rule (including unequal subdivisions)

Larson 4.4 – mean value theorem for integrals

Larson 4.4 - Fundamental Theorem of Calculus

Larson 4.3 – properties of definite integrals

Various -- functions defined by integrals, second fundamental theorem, accumulation function

Unit 6 – Exponential and Logarithmic Equations

Larson 5.1 – natural logarithmic function and definition of e

Larson 5.1 – derivative of natural log function, logarithmic differentiation

Larson 5.2 – antiderivatives of reciprocal functions and trig functions

Larson 5.3 – inverse functions

Larson 5.4 – derivative of natural exponential function, antiderivatives of natural exponential functions

Larson 5.5 – exponential functions with other bases

Unit 7 – Applications of Integrals

Larson 6.1 – area between two curves

Larson 6.2 – volume by disks, washers and known cross-sections

Larson 4.4 – average value

Various -- distance vs. displacement, velocity vs. speed

Unit 8 – Differential Equations

Differential equations and slope fields will be covered using other books in my arsenal.

Topics Covered:

Solutions (general and particular) to separable differential equations

Slope Fields

Exponential Growth and Decay

Unit 9 – AP EXAM REVIEW

We will spend a couple of days reviewing each unit above. We will cover many released free response and multiple choice questions. We will also fill out study cards with all formulas that will need to be memorized. Students will come up with sample problems that will be worked on in group setting and as a class. The students will sit for at one least full length practice exam. We will cover the testing procedure and how their score will be determined. We will review all major Theorems and definitions and how they are usually presented on the AP exam.

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