AP Calculus AB



AP Calculus AB

Mr. Wren’s Syllabus

Course Overview

It is the intent of this course to present calculus as a singular body of knowledge. James Stewart in his text, Calculus Concepts and Contexts, 2nd edition, remarks “the two branches of calculus and their chief problems, the area problem and the tangent problem appear to be very different, but it turns out that there is a very close connection between them.”(4) Showing the connection between these two branches of calculus is a chief aim of this course.

In order to help students see and appreciate the connections between both integral and differential calculus I employ a “multi-representational” model of instruction and practice. Students encounter on a daily basis functions that are represented verbally, visually (graph form), analytically (algebraic functions), and numerically (in table form). By representing functions in any one of these four forms students learn to apply calculus concepts to solve the problem at hand.

The TI-83 and TI- 89 graphing calculators facilitate the exploration of

functions and enable students to give meaningful approximate answers

to problems when an analytic solution is found wanting. The calculator

allows for a fluid movement between the various ways in which functions are represented. Students for instance can employ the calculator to create

a regression equation from a table of values and then integrate the function

using known techniques of integration (or simply use the calculator to

integrate the function for them). Similarly, students can take data from

a graph measuring the velocity of a particle and use the calculator to find an approximation of the distance the particle traveled using the trapezoid rule.

The course covers topics presented by the College Board in the

Teacher’s Guide – AP Calculus. The tangent problem and differentiation

are considered primarily in the first semester of the course and the area

problem and integration are taught primarily in the second semester

of the course. How the Fundamental Theorem of Calculus relates the

integral to the derivative is a major idea in the second semester.

AP Calculus AB Course Outline

Unit 1 Precalculus Review: Functions and Models (2-3 weeks)

This unit prepares the way for calculus by reviewing the basic concepts of functions learned in precalculus. Students are introduced to the four ways of representing a function. The graphing calculator as a tool for approximating values is discussed

A. Representing Functions

1. Four ways to represent a function

2. Piecewise Functions

3. Symmetry; increasing and decreasing functions

B. Mathematical Models

1. The function as a model for understanding data

2. A review of algebraic and transcendental families

of functions

3. Using the calculator to create regression equations

to model data

C. Transformations and Compositions of Functions

D. Using the graphing calculator

E. Exponential and Logarithmic Functions

1. Exponential growth and decay

2. Inverse functions

3. Logarithmic functions and the properties of logarithms

F. Trigonometric Functions

1. Graphs of basic trigonometric functions

and their inverses

2. Transformations of functions

3. Applications of trigonometric functions

G. Review and testing

Unit 2 Limits and Derivatives (4 weeks)

This unit explores the concept of the “limit” and the properties

of limits. Of particular interest is the use of the “limit” in

developing the definition of the derivative . Derivatives as rates

of change in different situations are explored . Students also learn

how to use the derivative as a function to predict how the original

function behaves.

A. The tangent line problem: an introduction to a “limit”

1. Tangent line viewed as the limit of secant lines

2. Concepts of average versus instantaneous velocity

described numerically, graphically, and in physical

terms.

3. Local linearity: “Zooming” in on a smooth function

B . Limits at a point

1. Informal definition of the limit of a function

2. The definition applied when examining a table

of points and when examining a graph

3. One sided limits

4. Pitfalls of using the graphing calculator to determine

a limit from a table of points or from a graph

E. Calculating limits using algebra

1. Properties of Limits

2. Direct Substitution

3. Rationalization technique

4. Squeeze Theorem

F. Continuity

1. Geometric and mathematical definitions of

continuity

2. Discontinuous functions

a. removable discontinuity

b. jump discontinuity

c. infinite discontinuity

3. Determining the continuity of composite functions

4. The Intermediate Value theorem

G. Limits involving infinity

1. Infinite limits and vertical asymptotes:

2. Limits at infinity and horizontal asymptotes

3. Dangers of using calculators to check limits

(numerically and graphically)

H. Instantaneous Rates of Change

Unit 3 The Derivative (5 weeks)

A. The definition of the derivative as a limit of a difference

quotient

1. The derivative notation

2. Interpreting the derivative as the slope of a tangent

line

3. The derivative as an approximate rate of change

when working with discrete data

B. The Derivative as a function

1. The concept of a differentiable function, interpreted

graphically, numerically, and descriptively

2. Differentiability and Continuity: How can a

function fail to be differentiable?

3. What the first derivative says about the function “f”

4. Higher order derivatives. What does f′′ say about f′?

C. Linear Approximations

1. Computing linear approximations

2. Using linear approximations to approximate

functions

D. Differentiation of Algebraic and Transcendental

Functions.

In this section the definition of the derivative is used

to derive rules for differentiation. Students are “walked”

through” proofs for the power rule, constant multiple rule

and the product rule. These rules and others not proved

reinforce the idea that calculus is a coherent and

logical body of knowledge.

1. Differentiation of algebraic functions

2. Derivative Rules when combining functions

3. Applications of the derivative to rates of change

in the natural and social sciences.

Group presentations. Students are assigned to work in small

groups to present their solutions to a problem specific to

their field of interest – social science, physical science, etc.

When calculating the derivative students are required to explain the meaning of the derivative as an instantaneous rate of change.

Presentations include a statement of the problem, the data or fun-

tion from which data is derived, method and calculation by which the rate of change is determined (derivative of a known function, limit from a table) and a visual aid to help illustrate the problem.

4. Applications to velocity and acceleration – linear

motion problems

5. Derivative of ex

6. Derivatives of trigonometric functions

7. Chain Rule

8. Implicit Differentiation

9. Derivatives of inverse trigonometric functions

10. Derivative of Logarithmic functions

11. Linear Approximations and Differentials

Unit 4 Applications of Differentiation (6 weeks)

Students are given the opportunity to pursue the applications of differentiation. Students discover the value of determining “extreme values” of a function when faced with solving practical problems that occur in real life. Strategies for solving related rates and optimization problems are discussed

A. Related Rates

Individual Lab Activity: “How Many Licks” –Lab Activity 7

Students determine the rate of change of volume of a

Tootsie Roll Pop as you consume it

(see Kamische, Ellen., A Watched Cup Never Cools 1999. Key

Curriculum Press), pp.23-24)

B. Extreme Values

1. Local extrema and critical points

2. Absolute extrema and the Extreme Value Theorem

C. Derivatives and the shapes of Curves

1. Mean Value Theorem

2. Increasing/Decreasing Test

3. First Derivative Test

4. Concavity

5. Second Derivative Test

6. Curve Sketching

7. Curve Sketching and the use of the graphing

calculator

Students use the their knowledge of calculus to refine the

display of a function graphed on their calculators. Students

discover that for some functions two or three different

viewing windows are necessary to see all the salient features

of the function. Students also use the calculator to determine

approximate values of maxima and minima, inflection points,

and roots for functions that are too difficult to consider without

technology.

D. Optimization Problems

E. Linearization Models – Newton’s Method

F. Antiderivatives

Students are introduced at this point to antiderivatives to

to prepare them for finding the integral of a function in the

unit ahead. Students are introduced to a few “antiderivative”

formulas. It is not enough however to show the students that

the concept of the antiderivative reverses the process of finding

the derivative. What is important is that the students realize

the value in finding the antiderivative. Practice is thus given

to finding a position function from a velocity function first by

a guess and check method, and then learning a few basic anti-

differentiation rules to speed up the process. The idea that a physicist can learn a great deal about the position of a particle at a given time because the velocity of the particle is

known is drilled home with a return to a study of rectilinear

motion. Students are also introduced to slope fields here to

allow students a chance to guess the shape of a family of anti-

derivative curves from the given slope field.

1. Finding the most general antiderivative of a given function

2. Using slope fields to determine the shape of an unknown function. This exercise anticipates the

lesson on separable equations: solving a differential equation for an unknown function that is a

solution to the differential equation.

Unit 5 The Definite Integral (5 weeks)

The area and distance problems serve as the launching point for introducing the idea of the definite integral. Emphasis is placed on finding the limit of the sum of rectangles of equal width to determine an approximate area under a curve. However, subdividing the area under a curve into unequal partitions is discussed so that taking the limit of the normal to zero can be understood.

Several days are given over to calculating the integral through Riemann sums. A day each is given over to approximating the integral by the midpoint and trapezoidal rules. The proof for the Evaluation Theorem of the Fundamental Theorem is “walked” through, and students are made accountable for reproducing the same on a test. The rationale for this is the inherent value of recognizing the value of applying the Mean Value Theorem within the proof. (Those existence theorems don’t exit for nothing!)

Finally, several days are taken to studying the Fundamental Theorem to discern the subtle idea integration and differentiation are inverse processes.

A. Approximating Areas

1. Riemann sums (left end, right end and midpoint

sums) to approximate the area of a region

2. Calculating approximate distances from a table

or graph of discrete data of varying velocities

3. Evaluating limits of Riemann sums with partitions

of equal width to determine the area of a region

under polynomial functions.

4. The definite integral as a limit of Riemann sums

5.. Properties of the definite integral

B. The Fundamental Theorem of Calculus (part 2)

1. Proof of the Evaluation Theorem

2. Calculating the definite integral using known

formulas of indefinite integrals and the Evaluation

Theorem. Here integrands are chosen to be

velocity functions so that accumulating rates of

change in distance can be seen to accrue to a total

distance

C. The Fundamental Theorem of Calculus (Part 1)

D. Substitution Rule for Integrals

E. Integration by parts ( simple examples)

F. Additional technique of integration

1. Trigonometric substitutions

2. Partial fractions (linear and linear -quadratic

factors

G. Approximating the exact value of a definite integral

using the Trapezoid Rule.

Unit 6 Applications of Integration (5 weeks)

The main idea in this unit is for students to be able to divide a quantity – whether it be an area between two curves or the length of an arc – into small pieces and estimate the answer with Riemann sums, and then recognize the limit of such sums as an integral. Once the integral is represented, students will use the Evaluation theorem or their calculators to evaluate the integral.

A. Area between curves

B. Using areas to determine distances – Students represent

the area between two velocity curves to compute the

the distance between two objects over an interval of time

“t” after starting from the same point.

C. Volumes of Solids of Revolution

1. Disc Method

2. Washer Method

D. Volumes of Solids with known cross sections

E. Average Value of a Function

F. Applications to Physics and Engineering

1. Work – variable force

2. Hydrostatic pressure and force

Individual Writing Project: Tile Project

Each student tries to sell the class on a tile design created in class on the computer or on their graphing calculator. Students use their calculators or classroom computer to create two functions that enclose a region. Each region is “painted” a different color thus creating an aesthetically pleasing pattern for a two-colored ceramic tile. The area between two functions is one color and the area of the remaining part of the tile is a second color. Students integrate the enclosed region (the bounds of which are determined by the intersection of the curves). tudents calculate the ratio of the areas of each colored part of the tile. Students are to duplicate the tiles many times over and arrange the tiles (a minimum of four) in several formations. Students are required to create a business letter introducing their company to the class and point presentation is required for the presentation.

Group Project “As Easy as п” - Lab Activity 15

Students determine the volume of a pan (circular object)

(see Kamische, Ellen., A Watched Cup Never Cools 1999. Key

Curriculum Press), page 57.)

Unit 7 Modeling with Differential Equations (2 weeks)

Students are introduced to differential equations and learn to solve these equations through the technique of separable equations. Students examine data on population growth, radioactive decay, and continuous compounded interest. The unit is introduced with the idea that the growth rate of a population is proportional to the population’s size (for early growth of the population). This example allows the student to see what constitutes a differential equation and what solution is being sought. Direction or slope fields are introduced to show students that the field represents general solutions to the differential equation. Finally the actual technique of solving simple differential equations is given.

A. Modeling with Differential equations

B. Direction Fields

C. Separable Equations

D. Exponential Growth and Decay

Group Project “A Watched Cup Never Cools” - Lab Activity 10

Students determine a mathematical model that describes how a hot cup of water cools.

(See Kamische, Ellen., A Watched Cup Never Cools 1999. Key

Curriculum Press), pp 34-36.)

Review of Specific Concepts (1 week)

A. Logarithmic differentiation

B. Inverse Functions and their derivatives

C. Inverse Trigonometric functions and their derivatives

D. Related Rates

E. Optimization

Preparation for the AP Test (3 weeks)

Students are given over the course of three weeks three complete simulated AP tests from the spiral booklet Preparing for the (AB) AP Calculus Examination 2006 by George Best and J. Lux . Students also use the companion solution manual volume accompanying the spiral text. After school tutoring sessions are arranged for students who have questions surrounding the practice test. Class time is not taken for reviewing the practice tests

Post AP Test (2 weeks)

The remaining days after the AP Test are given over to studying further concepts like arc length and surface area so students can complete one last project called the “Cup”and then prepare for their final test in the class. The project is described below:

Group Project: “The Cup” – Lab Activity 19

Students are given a cup and challenged to find the volume of the cup and the inner surface area of the cup. They are also to find at what rate the water level is rising in the cup when the cup contains half its volume and the water is being poured into the cup at a rate of 2 cubic centimeters per second (see Kamische, Ellen., A Watched Cup Never Cools 1999. Key Curriculum Press), pp. 66-67)

Students must use the regression capabilities of a graphing calculator to find a model curve which they can then integrate on the graphing calculator

Teaching Practices

I encourage my students to be patient with themselves and to enjoy thinking. Math has its own beauty and elegance. By taking time to think students will more readily see the connections present within this discipline we call calculus and find the joy that is the inevitable reward of study.

I begin most class periods with a statement of what we are about to study. I try to reveal the objective for the day in the clearest and simplest language possible. Whenever possible I like to share the purpose for the lesson hoping to instill some excitement that naturally comes when one perceives some relevance to the lesson at hand.

James Stewart’s Instructor’s Guide for Single Variable Calculus, Concepts and Contexts 2001, Brooks Cole publishers has been incredibly helpful for structuring lectures and planning group discovery activities. Steward stresses the need for variety. Some days the lecture approach is most suitable particularly when laying out a proof! I often follow up a lecture with an extended activity where students will work in groups and solve a singular problem posed on a worksheet. On other occasions I introduce a concept by referring the students to their text and posing a thoughtful question related to a major concept presented in the text. “What is the difference between the statements f(a) = L and lim f(x) = L as x approaches ‘a’”? “What does it mean to say “approaches”? “How close can must we get to /a’ to determine its limit?”

Stewart has impressed on me the need keep students from seeing calculus as a string of disparate skills to be learned. Always in the forefront of my approach is to lead the students to understand the concept, the big idea and encourage them to appreciate the power of calculus to make sense of tables of data.

The graphing calculator is a wonderful tool for exploring functions. The calculator is used every day in class. Again, Stewart has much to say about the employing a proper respect for the calculator’s sometimes apparent deficiencies. Knowing where a maximum or minimum value may occur on a graph of a function allows the students to rethink a proper “window” and see the graph as it should appear. Students are often surprised to out how much the calculator “hides” when they are not using the proper viewing window!

Student Evaluation

Assessments are given weekly as quizzes and only once every four to five weeks when the unit of study is complete. Each morning or nearly so a question from a past AP test is put before the students as a morning warm up. I see students gaining confidence throughout the course as they see that they are learning the material and have a fighting chance to pass the AP test. I find that beginning the period with such a question enables me to apprise the class of what is considered a “complete” response – fully answering the question posed and to the correct accuracy.

Students spend a good deal of time working in small groups and regularly are challenged to work their solutions (all groups simultaneously) on the white board. This approach gives me the opportunity to see how the students approach their work and who is really doing the work! On a regular basis students are encouraged to present a solution to the whole class and are rewarded with an extra credit point for their effort. Students are quite quick to rise to the challenge take the “vis-a vis” marker and share their solution on the overhead to that seemingly esoteric problem.

Although time is fleeting in teaching this course, I make time for a few in class projects which I use as an alternative means of assessment. I wish I had more time for more projects and hope in time to learn how to make more use of projects that better embed the objectives of the course. Following the AP Exam est, students complete the course with an end of course final exam.

Teacher Resources

Primary textbook

Stewart, James. Calculus, Concepts and Contexts, Single Variable, 2nd ed. Pacific Grove California: Brooks/Cole Publishing Co. 2001.

Supplementary Resources

Kamische, Ellen. A Watched Cup Never Cools. Emeryville, California: Key Curriculum Press. 1999.

Best, George and Lux, J.Richard. Preparing for the AB Advanced Placement Calculus Examination. Andover MA. Venture Publishing, 2006.

Technology Resources

Students are required to use a TI-83 or TI-83 Plus graphing calculator. Calculators are used on a daily basis both to explore new concepts, solve problems, approximate integrals that cannot be easily integrated, interpret solutions and to write and use simple programs.

Students also make use of several Dell computers in the classroom for creating power-point presentations for classroom projects, searching math websites, and for graphing functions using a TI-Graph Link or an on-line website. Two programs that have proven helpful in illustrating class lectures and in providing ancillary materials to support the class lecture are

Tools for Enriching Learning Calculus, CD-ROM. Harvey Keynes and James Stewart, Brooks/Cole. 2001

AP Calculus Teachers Tools Teacher CD, CD-ROM. Frane Media LLC. 2005. Stewart’s CD covers topics that correlate nicely with the text and provide helpful visuals and discussions. The Frane Media CD-ROM is a collection of “notes” and practice sheets on all the major calculus topics.

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C1 The course provides students with the opportunity to work with functions represented in a variety of ways-algebraically, and verbally- and emphasizes the connections among these representations

C7 The course teaches

Students how to use graphing calculators to help solve problems, experiment, interpret results, and support conclusions

C1 The course provides students with the opportunity to work with functions represented in a variety of ways-algebraically, and verbally- and emphasizes the connections among these representations

C7—The course teaches

students how to use graphing calculators to help solve problems, experiment, interpret results, and support conclusions

C8. Students should be able to determine the reasonableness of solutions, including sign, size, accuracy, and units of measurement

C2 Students should understand the derivative in terms of a rate of change and local linear approximation and should be able to use derivatives to solve a variety of problems

C9 Students should develop an appreciation of calculus as a coherent body of knowledge and as a human accomplishment

C5. Students should be able to communicate mathematics both orally and in well –written sentences and should be able to explain solutions to problems.

C7. Students should be able to use technology to help solve problems, experiment, interpret results, and verify conclusions.

C4. Students should understand the relationship between the derivative and the definite integral in both parts of the Fundamental Theorem of Calculus

C3. Students should understand the meaning of the definite integral both as a limit of Riemann sums and as the net accumulation of change and should be able to use integrals to solve a variety of problems

C3. Students should understand the meaning of the definite integral both as a limit of Riemann sums and as the net accumulation of change and should be able to use integrals to solve a variety of problems

C5 Students should be able to communicate mathematics both orally and in well-written sentences and should be able to explain solutions to problems

C6 Students should be able to model a written description of a physical situation with a function, a differential equation, or an integral

C7. Students should be able to use technology to help solve problems, experiment, interpret results, and verify conclusions.

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