AP CALCULUS AB

COURSE SYLLABUS

The AP Calculus AB course follows the AP Calculus AB topic outline from AP

Central. The textbook used is Calculus, 8th edition by Larson Hostetler

Edwards. The course is designed to be a rigorous course equivalent to a

college experience in Calculus I. The goals are for the students to pass the

AP Exam, receive college credit, and have the knowledge to succeed in higher

level mathematics courses. The students enrolled in this AP Calculus AB

course will receive concurrent credit for calculus from a local university by

maintaining an average of at least a C.

Before taking AP Calculus AB, all students should have completed an

equivalence of 4 years of mathematics. These 4 years must include 2 years of

algebra, a year of geometry, trigonometry, and an extensive course on

functions. All students that meet the prerequisites and have a desire to

attend this course are admitted.

The following outline is based on 33 weeks prior to the AP Examination. The

number of days includes days for testing and review. The students currently

attend class 45 minutes a day, 5 days a week. Starting in the 2007-2008

school year, the students will attend class 55 minutes a day, 5 days a week.

AP Calculus AB Course Outline

Preparation for Calculus (One week)

• Solving equations

• Rationalizing

• Logarithms and exponentials

• Graphing

• Functions

• Formulas and rules

• Trigonometry

Unit 1: Limits and Continuity (3 weeks)

• Finding limits numerically

• Finding limits graphically

• Limits that do not exist

• Properties of limits

• Evaluating limts analytically (including rationalizing)

• Using the Squeeze Theorem

• Continuity at a point and on an open interval

• One-sided limits

• Properties of continuity

• Intermediate Value Theorem

• Infinite limits

• Vertical asymptotes and their behavior in terms of limits

• Test (2 days)

� Lab on Limits – See Student Activity 1

Unit 2: Differentiation (6 weeks)

• Graphical and numerical representation of derivatives

• Slope of a tangent line at a point

• Writing equations of tangent lines

• Limit definition

• Differentiability vs. continuity

• Constant rule

• Power rule

• Constant multiple rule

• Sum and difference rules

• Derivatives of sine and cosine

• Rates of change

• Average rate vs. instantaneous rate

• Product rule

• Quotient rule

• Derivatives of trig functions

• Higher order derivatives

• Chain rule

• General power rule

• Implicit differentiation

• Related rates

• Test (2 days)

� Related Rates Activity – See Student Activity 2

Unit 3: Applications and Differentiation (5 weeks)

• Extrema on an interval

• Extreme Value Theorem

• Rolle’s Theorem

• Mean Value Theorem

• Increasing and decreasing intervals

• Fist derivative test

• Concavity intervals

• Points of inflection

• Second derivative test

• Limits at infinity

• Horizontal asymptotes and their behavior in terms of limits

• Curve sketching

• Optimization

• Test (2 days)

� Curve Sketching – See Student Activity 3

Midterm Exam

Unit 4: Integration (5 weeks)

• General solutions and particular solutions of a differential equation

• Indefinite integral notation

• Basic integration rules

• Find antiderivatives

• Sigma notation

• Finding area using upper and lower sums

• Riemann sums

• Evaluating a definite integral using limits

• Evaluate using properties of definite integrals

• Fundamental Theorem of Calculus

• Mean Value Theorem for Integrals

• Average value on a closed interval

• Second Fundamental Theorem of Calculus

• Change of variables in integration

• General Power Rule for integration

• Trapezoidal Rule

• Test (2 days)

� Discovering the Fundamental Theorem of Calculus – See Student

Activity 4

Unit 5: Logarithmic, Exponential, and Other Transcendental

Functions (5 weeks)

• Properties of natural logs

• Derivative of natural log functions

• Log Rule for Integration

• Integrate trig functions

• Inverse functions

• Derivatives of inverse functions

• Properties of natural exponential function

• Differentiate natural exponential functions

• Differentiate exponential functions with bases other than e

• Integrate exponential functions with bases other than e

• Compound interest and exponential growth

• Properties on inverse trig functions

• Differentiate inverse trig functions

• Integration of inverse trig functions

• Test (2 days)

Unit 6: Slope Fields (3 weeks)

• Find particular solutions

• Use slope fields

• Growth and decay

• Separation of variables

• First-order linear differential equations

• Test (2 days)

� Slope Fields in Groups – See Student Activity 5

Unit 7: Applications of Integration (3 weeks)

• Area between 2 curves

• Volume using disk method

• Volume using washer method

• Volume of solid with known cross sections

• Volume using shell method

• Test (2 days)

� How much candy does it take? – See Student Activity 6

Review (2 weeks)

AP Exam

Additional Requirements:

In addition to this outline, the students are responsible for a weekly journal.

Every Monday they are given anywhere from

8–12 released AP questions over material already covered in class. This

journal is due on Friday of the week given.

Teaching Strategies:

On the first day of school, the expectations are explained and the tone

is set for the school year. We spend one class period in open discussion on a

topic and an assignment is made. The homework is to be attempted at home

that night. The next class period is spent working in groups collaboratively

on the assignment with me acting as their coach. Every topic is demonstrated, using multi-representational approaches, i.e. graphically,

numerically, analytically, and verbally. I want all of my students to

understand the reason behind each procedure. Defining all terminology used

helps teach my students to communicate the concepts mathematically. All

students are required to communicate their solutions both verbally and in

writing.

Assessment:

Every Friday I give a homework quiz over homework problems assigned

that week. This enables me to determine what concepts or topics need to

reviewed or given more attention on a weekly basis. At the end of every unit a

two day unit test is given. The first day of the test is made up of multiple

choice problems, some that require the calculator and some that do not. The

second day of the test is made up of open response questions which are

graded based on a set rubric. Homework is taken up by the week and 4

random problems per assignment are graded for right or wrong. The 4

problems chosen are different than the problems chosen for the weekly quiz.

This allows me to get a closer look into the capabilities of my students at that

time and encourage them to put forth more effort on homework.

Technology and computer software:

All of my students have their own calculators. They range from a TI-82

to a TI-84. There is also a classroom set of TI-84 Plus, Silver Edition

calculators for them to use if need be. The calculators are used on a daily

basis to help solve problems, explore data, and analyze situations.

I have one computer in my room and it is for teacher use. I do have

Calculus in Motion and Geometer’s Sketchpad to use for demonstration

purposes but at this time I have no way to display it other than around my

desk. Symposium boards and projectors are in the plan for the next school

year.

� Student Activities 1. When introducing limits, my students are given a lab assignment in

order for them to see what happens as we approach a certain x-value. I

give them several functions, each with a chart of x-values and ask them

to produce the y-values. The x-values start by approaching with a

whole number interval and then get closer with decimal intervals until

we get within a hundred thousandths from the number we are

approaching. The students must explain their findings in general and

the word limit is them introduced.

2. After we study related rates, my students are all given a worksheet

from the previous year. This worksheet was compiled of written related

rate problems by the students who were in my class the year before.

After completing this worksheet, my current students are then asked to

write a related rates problem of their own and solve it correctly in order

to make the worksheet for next year’s students.

3. When introducing curve sketching, I put a function on the board and a

Cartesian coordinate plane on the overhead projector. My students are

each asked to plot points for the recognizable characteristics of the

function. After we have the intercepts, asymptotes, and such, I

generally have to ask if there are any max or mins. Through this type of

coaching and directing they start differentiating to find the max/mins

and the concavity. I put generic sign charts for the both the first and

second derivatives on the board and someone in the class is asked to

present their findings on the sign chart. Then someone is asked to go

to the overhead projector to plot the new points found. Finally,

someone else is asked to draw the sketch. We repeat this process for

another function before they are assigned several on their own.

4. Before going over the Second Fundamental Theorem of Calculus my

students are given a lab assignment that consists of five different

definite integrals. Each integral has a function of x as the upper limit.

The directions are to integrate and then differentiate. They must

explain their findings per problem. This allows the students to discover

the Second Fundamental Theorem of Calculus before we cover the

material in class.

5. My students are separated into groups of two. Each group is given a differential equation and a coordinate plane and asked to draw a slope

field. They have seen an occasional slop field so I am roaming the

room, leading each group in the right direction. After their slope field is

drawn, they are asked to draw the particular solution using a specific

point. Each group is asked to present their slope field and particular

solution to the class using a blank coordinate plane on the overhead

projector, demonstrating while explaining their solution.

6. We start Volumes of Solids by looking at a set base with certain cross

sections. I assign a project for my students to build a model that is

made up of a base with certain cross sections. They have to label the

base using function notation on a rectangular coordinate plane. They

have to put in at least 10 cross sections in whatever shape they

choose. They have to find the volume of the solid showing all work.

7. When studying the volume of a solid, we start by looking at the function

y = x – 1 on the interval from 0 to 1 and try to picture the solid it forms

if we revolve that around the y-axis. After they are given some time to

tell me what they think it will look like, I pull out one Reese’s peanut

butter cup. After we go over how that is what the solid looks like, we

discuss who this volume might be important to. Then I hand out a

peanut butter cup to everyone BUT THEY CAN’T EAT IT YET. We do the

same for the gum drop and the Hershey’s kiss. Now that they have a

representation of each solid on their desk, we have to decide how to

find the volume. I coach them into noticing that each are made up of a

stack of circles and that if we could find the area of each circle in the

stack and add them up we would have the volume of the solid. They

know from previous work in class that when we add the areas together

we want the depth of each one to approach 0 which means we are

integrating the areas of the circles with respect to the variable. This is

an excellent lead in to the disk method.

Resources:

• Major Text:

Larson, Hostetler, and Edwards. Calculus. 8th ed. Boston, New York:

Houghton Mifflin Company, 2006.

• Supplemental Texts and Resources:

Finney, Demana, Waits, and Kennedy. Calculus Graphical, Numerical,

Algebraic. Massachusettes, New Jersey: Pearson, Prentice Hall, 2003.

Finney, Demana, Waits, and Kennedy. Preparing for the Calculus AP

EXAM with Calculus: Graphical, Numerical, Algebraic. Boston, San

Fransisco, New York: Pearson, Prentice Hall, 2006.

Ostebee, Arnold, and Zorn. Calculus from Graphical, Numerical, and

Symbolic Points of View. 2nd ed. Boston: Houghton Mifflin Company,

2002.

Released AP Exams

AP Central – apcentral.collegeboard.com/calculusab