AP Calculus AB Course Description: Advanced Placement Calculus AB is primarily concerned with developing the students’ understanding of the concepts of calculus and providing experiences with its methods and applications. The four major topics of this course are limits, differential calculus, integral calculus, and their applications. All students will be required to analyze problems graphically, numerically, analytically and verbally in this course. The course content will follow the outlines set forth by the College Board and the state. All students will be required to use a TI-89 series graphing calculator for this course. Course Prerequisites: All students enrolling in AP Calculus AB should successfully complete Pre-Calculus. By successfully completing Pre-Calculus all students should have a working understanding of linear, polynomial, rational, exponential, logarithmic, trigonometric, inverse trigonometric, and piecewise defined functions. Students must also be familiar with the properties of functions, the algebra of functions, the graphs of functions, the language of functions (domain, range, odd, even, periodic, symmetry, zeros, intercepts, etc.) and understand the concept of the unit circle. Textbook: Larson, Hostetler, Edwards. Calculus, 6th Edition, Houghton Mifflin, © 1998 AP Review Supplemental Workbook: Lederman, David. Multiple-Choice & Free-Response Questions in Preparation for the AP Calculus (AB) Examination, 8th Edition, D&S Marketing Systems, © 2003 Student Evaluation: 1st, 2nd & 3rd Nine Weeks:

Exams Quizzes Daily Warm-ups Assignments

65% 15% 10% 10%

4th Nine Weeks

Exam AP Free-Response AP Multiple Choice Assignments

40% 25% 25% 10%

AP Calculus AB

Page 1 of 14

Student Assessments: Exams will be given at the completion of a unit (typically at the end of a chapter). Most exams will span over a 57 minute class period. The majority of the exam will be without the use of a calculator and the remainder of the exam will require the use of a TI-89 series graphing calculator. Quizzes will be given during the coverage of the chapter, usually covering 2 – 4 sections of textbook material. These quizzes will be completed during one class period and some will be with the use of a graphing calculator and others will be without. Daily warm-ups are calculus problems given at the beginning of each class to assess the students’ understanding of concepts taught a few days earlier. Daily warm-up problems are collected and graded each day. Assignments are graded works which assess the students understanding of the basic calculus concepts. Assignments are given daily. AP Review Problems will be assigned daily beginning with the fourth nine weeks. These problems are past AP Free Response and AP Multiple Choice questions. All AP free response questions will be assigned to students to complete from the past five examination cycles. AP Multiple Choice will be review assignments from the D&S Marketing review workbook. Students will practice the multiple choice questions in these workbooks to help them prepare for the AP examination. Old AP questions will be given throughout the year as appropriate on quizzes and exams, as well as homework assignments.

AP Calculus AB

Page 2 of 14

AP Calculus AB: Course Outline The 2007-2008 school year begins on August 20, 2007 in Orange County Florida. The AP Calculus Exam will be administered on May 7, 2008. This gives Orange County students 158 in session school days to learn calculus and prepare for the AP Calculus exam. All of the days listed below are an approximation. This is a typical schedule, but it does change slightly from year to year with the number of days spent on each topic. First Semester: Chapter 1:

Limits and Their Properties

15 days

Section 1.1:

A Preview of Calculus

1 day

Section 1.2:

Finding Limits Graphically and Numerically

1 day

Section 1.3:

Evaluating Limits Analytically

3 days

Section 1.4:

Continuity and One-Sided Limits

3 days

Review (Sections 1.1 – 1.4)

1 day

Quiz (Sections 1.1 – 1.4)

1 day

Section 1.5:

2 days

Infinite Limits

Section 3.5: Limits at Infinity

1 day

Review (Chapter 1)

1 day

Exam (Chapter 1)

1 day

Chapter 2:

Differentiation

29 days

Section 2.1:

The Derivative and the Tangent Line Problem

2 days

Section 2.2:

Basic Differentiation Rules and Rates of Change

2 days

Section 2.3:

The Product and Quotient Rules and Higher-Order Derivatives

2 days

Section 2.4:

The Chain Rule

3 days

Review (Sections 2.1 – 2.4)

1 day

Quiz (Sections 2.1 – 2.4)

1 day

Section 2.5:

3 days

Implicit Differentiation

Section 2.A: Supplemental Unit: Particle Motion AP Calculus AB

5 days Page 3 of 14

Review (Section 2.5 & Particle Motion)

1 day

Quiz (Section 2.5 & Particle Motion)

1 day

Section 2.6:

Related Rates

5 days

Supplemental Worksheet on Related Rates

1 day

Review (Chapter 2)

1 day

Exam (Chapter 2)

1 day

Chapter 3:

Applications of Differentiation

27 days

Section 3.1:

Extrema on an Interval

2 days

Section 3.2:

Rolle’s Theorem and the Mean Value Theorem

2 days

Section 3.3:

Increasing and Decreasing Functions and the First Derivative Test 2 days

Section 3.4:

Concavity and the Second Derivative Test

2 days

Section 3.6:

A Summary of Curve Sketching

2 days

Supplemental worksheets on Sketching Functions

2 days

Review (Sections 3.1 – 3.6)

1 day

Test (Sections 3.1 – 3.6)

1 day

Section 3.7:

Optimization Problems

3 days

Supplemental Worksheet on Optimization Problems

1 day

Section 3.8:

Newton’s Method

2 days

Section 3.9:

Differentials

2 days

Review (Sections 3.7 – 3.9)

1 day

Quiz (Sections 3.7 – 3.9)

1 day

Section 3.10: Business and Economic Applications

1 day

Review (Chapter 3)

1 day

Exam (Chapter 3)

1 day

AP Calculus AB

Page 4 of 14

Second Semester: Chapter 4:

Integration Section 4.1:

Antiderivatives and Indefinite Integration

2 days

Section 4.2:

Area

4 days

Section 4.3:

Riemann Sums and Definite Integrals

2 days

Review (Sections 4.1 – 4.3)

1 day

Quiz (Sections 4.1 – 4.3)

1 day

Section 4.4:

The Fundamental Theorem of Calculus

3 days

Section 4.5:

Integration by Substitution

3 days

Review (Sections 4.4 – 4.5)

1 day

Quiz (Sections 4.4 – 4.5)

1 day

Section 4.6:

Chapter 5:

23 days

Numerical Integration (Trapezoidal Rule and Simpson’s Rule)

2 days

Review (Chapter 4)

2 days

Exam (Chapter 4)

1 days

Logarithmic, Exponential, and Other Transcendental Functions

27 days

Section 5.1:

The Natural Logarithmic Function: Differentiation

2 days

Section 5.2:

The Natural Logarithmic Function: Integration

2 days

Section 5.3:

Inverse Functions

2 day

Section 5.4:

Exponential Functions: Differentiation and Integration

2 days

Section 5.5:

Bases Other than e and Applications

3 days

Review (Sections 5.1 – 5.5)

1 day

Quiz (Sections 5.1 – 5.5)

1 day

Section 5.6:

Differential Equations: Growth and Decay

1 days

Section 5.7:

Differential Equations: Separation of Variables

2 days

AP Calculus AB

Page 5 of 14

Section 5.A

Chapter 6:

Supplemental Unit on Slope Fields

1 day

Supplemental worksheet on Slope Fields & Differential Equations

1 day

Section 5.8:

Inverse Trigonometric Functions: Differentiation

2 days

Section 5.9:

Inverse Trigonometric Functions: Integration

2 days

Review (Sections 5.6 – 5.9, 5.A)

1 day

Quiz (Sections 5.6 – 5.9, 5.A)

1 day

Section 7.1:

1 day

Basic Integration Rules

Review (Chapter 5)

1 day

Exam (Chapter 5 and Section 7.1)

1 day

Applications of Integration

9 days

Section 6.1:

Area of a Region Between Two Curves

1 day

Section 6.2:

Volume: The Disk Method

3 days

Section 6.3:

Volume: Shell Method

2 days

Section 7.7:

Indeterminate Forms and L’Hôpital’s Rule

1 day

Review (Sections 6.1

Review:

6.3, 7.7)

1 day

Exam (Sections 6.1 6.3, 7.7)

1 day

Review for AP Examination

20 days

AP Calculus AB

Page 6 of 14

AP Calculus AB: College Board’s Topic Outline & Correlation to Textbook The sections listed next to each topic, lists a section (or supplemental unit) of the textbook which corresponds to each topic. Although several of these topics appear throughout a calculus course, the section where it is first introduced is included for reference.

TOPIC I.

TEXTBOOK SECTION

Functions, Graphs, and Limits Analysis of Graphs With the aid of technology, graphs of functions are often easy to produce. The emphasis is on the interplay between the geometric and analytic information and on the use of Calculus both to predict and to explain the observed local and global behavior of a function.

(1.2, 1.3)

Limits of Functions (Including one-sided limits) An intuitive understanding of the limiting process Calculating limits using algebra Estimating limits from graphs or tables of data

(1.2) (1.3) (1.2)

Asymptotic and Unbounded Behavior Understanding asymptotes in terms of graphical behavior Describing asymptotic behavior in terms of limits involving infinity Comparing relative magnitudes of functions and their rates of change (for example, contrasting exponential growth, polynomial growth, and logarithmic growth)

(1,4, 1.5, 3.5) (3.5) (5.6)

Continuity as a Property of Functions An intuitive understanding of continuity. (The function values can be made as close as desired by taking sufficiently close values of the domain.) Understanding continuity in terms of limits Geometric understanding of graphs of continuous functions (Intermediate Value Theorem and Extreme Value Theorem) II.

(1.4) (1.4) (1.4)

Derivatives Concept of the Derivative Derivative presented graphically, numerically, and analytically Derivative interpreted as an instantaneous rate of change Derivative defined as the limit of the difference quotient Relationship between differentiability and continuity

AP Calculus AB

(2.1, 2.2) (2.2, 2.A) (2.1) (2.1) Page 7 of 14

Derivative at a Point Slope of a curve at a point. Examples are emphasized, including points at which there are vertical tangents and points at which there are no tangents. Tangent line to a curve at a point and local linear approximation Instantaneous rate of change as the limit of average rate of change Approximate rate of change from graphs and tables of values

(2.1)

(2.1) (2.2) (2.2)

Derivative as a Function Corresponding characteristics of graphs of f and f Relationship between the increasing and decreasing behavior of f and the sign of f The Mean Value Theorem and its geometric consequences Equations involving derivatives. Verbal descriptions are translated into equations involving derivatives or vice versa.

(3.3) (3.3) (3.2) (5.6)

Second Derivatives Corresponding characteristics of the graphs of f, f , f Relationship between the concavity of f and the sign of f Points of inflection as places where concavity changes

(3.3, 3.4, 3.6) (3.4, 3.6) (3.4, 3.6)

Applications of Derivatives Analysis of curves, including the notations of monotonicity and concavity Optimization, both absolute (global) and relative (local) extrema Modeling rates of change, including related rates problems Use of implicit differentiation to find the derivative of an inverse function Interpretation of the derivate as a rate of change in varied applied contexts, including velocity, speed, and acceleration Geometric interpretation of differential equations via slope fields and the Relationship between slope fields and solution curves for differential equations

(3.6) (3.1, 3.7) (2.6) (5.3) (2.2, 4.4, 2.A) (5.A)

Computation of Derivatives Knowledge of derivatives of basic functions, including power, exponential logarithmic, trigonometric, and inverse trigonometric functions Basic rules for the derivative of sums, products, and quotients of functions Chain rule and implicit differentiation

AP Calculus AB

(2.2, 2.3, 5.1 5.3, 5.4, 5.8, 5.9) (2.2, 2.3) (2.4, 2.5)

Page 8 of 14

III.

Integrals Interpretation and Properties of Definite Integrals Definite integral as a limit of Riemann sums Definite integral of the rate of change of a quantity over an interval interpreted as the change of the quantity over the interval:

(4.2) (4.4)

b

f x dx

f b

f a

a

Basic properties of definite integrals (examples include additivity and linearity)

(4.4)

Applications of Integrals Appropriate integrals are used in a variety of application to model physical, biological, or economic situations. Although only a sampling of applications can be included in any specific course, students should be able to adapt their knowledge and techniques to solve other similar application problems. Whatever applications are chosen, the emphasis is on using the method of setting up an approximating Riemann sum and representing its limit as a definite integral. To provide a common foundation, specific applications should include using the integral of a rate of change to give accumulated change, finding the area of a region, the volume of solid with known cross sections, the average value of a function, and the distance traveled by a particle along a line.

(3.10, 4.1, 4.2, 4.3, 4.4, 6.1, 6.2, 6.3)

Fundamental Theorem of Calculus Use of the Fundamental Theorem to evaluate definite intergrals Use of the Fundamental Theorem to represent a particular antiderivative, and the analytical and graphical analysis of functions so defined

(4.4) (4.4)

Techniques of Antidifferentiation Antiderivatives following directly from derivatives of basic functions Antiderivatives by substitution of variables (including change of limits for definite integrals)

(4.1) (4.5)

Applications of Antidifferentiation Finding specific antiderivatives using initial conditions, including applications to motion along a line Solving separable differential equations and using them in modeling (in particular, studying the equation y ky and exponential growth)

AP Calculus AB

(4.5) (5.6, 5.7)

Page 9 of 14

Numerical Approximations to Definite Integrals Use of Riemann sums (using left, right, and midpoint evaluation points) and trapezoidal sums to approximate definite integrals of functions represented algebraically, graphically, and by tables of values.

AP Calculus AB

(4.2, 4.6)

Page 10 of 14

AP Calculus AB: Evidence of Curricular Requirements Curricular Requirement #1: The course teaches all topics associated with Functions, Graphs, and Limits; Derivatives; and Integrals as delineated in the Calculus AB Topic Outline in the AP Calculus Course Description. This curricular requirement is met and on the four previous pages the entire Calculus AB topic outline was reproduced and the section or supplemental unit where it is covered is denoted. In addition, some topics that are not in the Calculus AB Topic Outline are also covered in this course. Examples of these topics include Simpson’s Rule, L’Hôpital’s Rule, and Volume: Shell Method. Curricular Requirement #2 This course provides students with the opportunity to work with functions represented in a variety of ways – graphically, numerically, analytically, and verbally – and emphasizes the connections among these representations. Example #1: Students are first introduced to limits numerically, then graphically, then analytically. Students will construct tables of values, then graph the function to see if the two answers agree. During the next class the analytical method is introduced and the three methods are compared. Students are encouraged to verbalize their explanations and answers in class on a daily basis. Example #2: To find the area under the curve the students are exposed to several approximation techniques prior to learning how to evaluate the definite integral algebraically. Examples of this would include problems of the following nature: Problem #1: (Numerical) Suppose a volcano is erupting and readings of the rate r t at which solid materials are spewed into the atmosphere are given in the table. The time t is measured in seconds and the units for r t are tones (metric tons) per second.

t

r t

0 2

1 10

2 24

3 36

4 46

5 54

6 60 6

Use the Trapezoidal Rule with 3 subdivisions of equal length to approximate

r t dt . 0

AP Calculus AB

Page 11 of 14

Problem #2: (Graphical) Use the midpoint rule to approximate the area of the region bounded by the graph of the function and the x-axis over the indicated interval. Use n 4 subintervals.

f x

x2 3 ; 0, 2 y

8

6

4

2

x 0.5

1

1.5

2

2.5

Problem #3: (Analytical) A particle moves along the x-axis so that its acceleration at any time t is given by a t time t

0 the velocity of the particle v 0

24 , and at time t 1 its position is x 1

6t 18 . At

20 .

(A) (B) (C)

Write an expression for the velocity v t of the particle at any time t. For what value(s) of t is the particle at rest? For what value(s) of t is the particle moving to the right on the time interval 0,8 ?

(D)

Write an expression for the position x t of the particle at any time t.

At the conclusion of this students can present their finding to class by writing their solutions on the board and verbally explaining the steps to their classmates.

AP Calculus AB

Page 12 of 14

Curricular Requirement #3 The course teaches students how to communicate mathematics and explain solutions to problems both verbally and in written sentences. Students are encouraged to verbally express their suggestions on solving problems in class on a daily basis. Students will also come to the board and demonstrate their solutions to the class. Examples of questions that have been used on examinations for students to express the mathematics in written sentences appear below. Example #1: 120

If oil leaks from a tank at a rate of r t gallons per minute at time t, what does

r t dt represent? 0

Example #2: During the first 40 seconds of a flight, the rocket is propelled straight up so that in t seconds it reaches a height of s 5t 3 feet. It can be shown that

Interpret

ds dt

15t 2 .

ds in the context of the problem. dt

AP Calculus AB

Page 13 of 14

Curricular Requirement #4 The course teaches students how to use graphing calculators to help solve problems, experiment, interpret results, and support conclusions. Example for Solving Problems: Each student enrolled in this course will have a TI-89 series graphing calculator to use. The students are expected to bring their graphing calculators to class with them each day. The calculators are used frequently in class to explore, discover and reinforce the concepts that we are learning. Example for Experiment: When introducing the limit definition of a derivative, the students will sketch a function using their graphing calculator and graph various secant lines to the curve which pass through a specific point x, f x on the curve. Students will calculate the slope of their secant lines. To graph the other secant lines students will choose a second point on the curve closer to fixed point of x, f x . They will continue this pattern until they pick a point very close to the fixed point x, f x . Example of Interpretation of Results: Using the example above, students will notice that our secant line is very close to slope of the curve at the fixed point x, f x . They will explore this further by “zooming in” on their original function close to the fixed point and noticing that it appears almost linear over a small fixed interval. Example of Supporting Conclusions: The first function students experiment with is usually a quadratic. I continue this experimentation with polynomial, exponential and logarithmic functions.

AP Calculus AB

Page 14 of 14