AP Calculus BC
2006-2007
Primary Text: Anton, Howard, Bivens, I., and Davis, S. Calculus. 8th ed. New York: John
Wiley & Sons, Inc., 2005.
Supplementary Text: Thomas, George B. and Finney, Ross L. Calculus and Analytic Geometry. 8th
ed. Reading, MA: Addison-Wesley Publishing Co., 1992.
(Standards from AP Calculus Acorn Book)
Topics
Understanding Functions
Types: polynomial,
exponential, trig, log and ln,
power, rational; parametric
curves
Concepts: domain, range,
odd and even, inverse, end
behavior, continuity,
intermediate value theorem,
transformations
Limits and Continuity
1. Understanding limits
intuitively
2. Computing limits to a
point (including leftand right-hand limits)
and limits to infinity
3. Understanding end
behavior
4. Understanding
continuity intuitively
5. Definition of
continuous
Standards
Resources
I
Summer Review Packet
Chapter 1: Functions
1.1 Functions
1.2 Graphing Functions Using Calculators
1.3 New Functions from Old
1.4 Families of Functions
1.5 Inverse Functions; Inverse
Trigonometric Functions
1.6 Exponential and Logarithmic
Functions
1.7 Mathematical Models
1.8 Parametric Equations
Chapter 2: Limits and Continuity
2.1 Limits (An Intuitive Approach)
2.2 Computing Limits
2.3 Limits at Infinity; End Behavior of a
Function
2.4 Limits (Discussed More Rigorously)
2.5 Continuity
2.6 Continuity of Trigonometric and
Inverse Functions
Timing
8 classes
9 classes
I, II
Understanding the Derivative
1. Average rate of change vs.
instantaneous rate of
change
2. Velocity vs. speed,
acceleration
3. Derivative at a point
4. Definition of the derivative
5. Understanding
differentiability and points
of non-differentiabilty
(cusp points, endpoints,
vertical tangents,
oscillation)
Finding and using Derivatives
1. Power rule
2. Product and quotient rules
3. Rules for exponential, log.,
ln., trig and inverse trig
functions
4. Chain rule
5. Implicit differentiation
6. Related rate problems
7. Linearization problems
8. L’Hopital’s rule for limits
Applications of Derivatives
1. What the first
derivative tells about a
function (inc. or dec.,
relative extrema)
2. What the second
derivative tells about a
function (concavity,
points of inflection)
3. Finding absolute
extrema
4. Optimization
5. Interpretations of
derivatives to concepts
other than velocity (i.e.
II
Chapter 3: The Derivative
Chapter 4: Derivatives of Logarithmic,
Exponential, and Inverse Trigonometric
Functions
*Aquarium Activity and Bungee Jumper
Investigation (see Student Activity 1,
below)
3.1Tangent Lines, Velocity, and General
Rates of Change
3.2 The Derivative Function
3.3 Techniques of Differentiation
3.4 The Product and Quotient Rules
3.5 Derivatives of Trigonometric
Functions
4.2 Derivatives of Logarithmic Functions
4.3 Derivatives of Exponential and
Inverse Trigonometric Functions
--Derivatives of Parametric Curves
3.6 The Chain Rule
*Derivatives Levels Activity, including
using the graphing calculator to find
numerical derivatives (see Student
Activity 3, below)
4.1 Implicit Differentiation
3.7 Related Rates
*Related Rates game (see Student Activity
4, below)
3.8 Local Linear Approximation
4.4 L’Hopital’s Rule
*Derivative Matching Activity (see
Student Activity 2, below)
AP FRQs (3) on Implicit
Differentiation and Related Rates
Chapter 5: The Derivative in Graphing
and Applications
5.1 Analysis of Functions I: Increase,
Decrease, and Concavity
5.2 Analysis of Functions II: Relative
Extrema; Graphing Polynomials
5.3 More on Curve Sketching: Rational
Functions; Curves with Cusps and
Vertical Tangent Lines; Using
Technology
*f, f’, and f’’ Matching Exercises (see
Student Activity 5, below)
5.4 Absolute Maxima and Minima
5.5 Applied Maximum and Minimum
Problems
13 classes
9 classes
Economics)
6. Rolle’s Theorem; Mean
Value Theorem
Understanding the Definite
Integral and Understanding
and Using Antiderivatives
1. Summation of area
2. Left, right, midpoint, and
general Riemann sums,
summation notation, and
definite integral
3. Trapezoidal
approximations
4. Application in real life
using summation
techniques
5. Fundamental Theorem
6. Antiderivative Functions
7. Properties of Definite
Integral
8. Average Value Theorem
for Integrals
9. Integral as an Accumulator
10. Antiderivative formulas
a. power
b. log and ln
c. exponential
d. trigonometric
e. inverse trigonometric
f. parametric curves
11. Integration by Substitution
*Thomas & Finney
5.7 Rolle’s Theorem; Mean Value
Theorem
*Venn Diagram Exercise on Theorems
(see Student Activity 6, below)
AP FRQs (3) on Analyzing Graphs
using Derivatives
*Special emphasis on
JUSTIFICATION
III
Chapter 6: Integration
17 classes
6.1 An Overview of the Area Problem
*Using the Graphing Calculator for
definite integrals (see Student Activity 7,
below)
6.4 The Definition of Area as a Limit;
Sigma Notation
6.2 The Indefinite Integral
6.5 The Definite Integral
6.6 The Fundamental Theorem of
Calculus
6.3 Integration by Substitution
6.8 Evaluating Definite Integrals by
Substitution
6.9 Logarithmic Functions from the
Integral Points of View
*Special emphasis on Integral as
Accumulator
*Rule of Four (see Student Activity 8,
below)
AP FRQs (7) on Approximating definite
integrals using Riemann sums,
Evaluating definite integrals on the
calculator, Average value, and
Interpreting definite integrals
Topics
Standards
Resources
Timing
Applications of Antiderivatives
and Integrals
1. Area between Curves
2. Volumes of solids of
revolution and solids
formed by perpendicular
cross-sections
3. Arc length
4. Area of a surface of
revoluation
5. Average Value Theorem
for Integrals
6. Integral as an Accumulator
III
Chapter 7: Applications of the
Definite Integral in Geometry,
Science, and Engineering
7.1 Area Between Two Curves
*Constructing Volumes of Revolution
(see Student Activity 10, below)
7.2 Volumes by Slicing; Disks and
Washers
*Thomas & Finney
7.3 Volumes by Cylindrical Shells
20 classes
*See
MIDTERM EXAM
Midterm Exam, below
7.4 Length of a Plane Curve
7.5 Area of a Surface of Revolution
7.6 Average Value of a Function and
Its Applications
AP FRQs (3) on same materials as
those used in Ch. 6
AP FRQs (4) on Areas and volumes
More Methods of
Antidifferentiation
1. Integration by Parts
2. Trigonometric Integrals
3. Trigonometric
Substitutions
4. Integrating Rational
Functioons by Partial
Fractions (non-repeating
linear factors only)
5. Numerical Integration;
Simpson’s Rule
6. Improper Integrals
III, IV
Chapter 8: Principles of Integral
Evaluation
8.1 An Overview of Integration
Methods
*Integral Levels (see Student Activity
9, below
8.2 Integration by Parts
8.3 Trigonometric Integrals
8.4 Trigonometric Substitutions
8.5 Integrating Rational Functions by
Partial Fractions (*non-repeating
linear factors only)
8.7 Numerical Integration; Simpson’s
Rule
8.8 Improper Integrals
13 classes
Mathematical Modeling with
Differential Equations
1. Solving Separable
Differential Equations
2. Modeling with First-Order
Differential Equations (e.g.
population growth, halflife, pharamacology,
Newton’s Law of Cooling)
3. Slope Fields
4. Euler’s Method
III
Sequences and Series
1. Idea and notation for
sequences; arithmetic,
geometric, harmonic, and
alternating harmonic
2. Definition and notation of
series; sequence of partial
sums; geometric,
harmonic, alternating
harmonic series
3. Power series; interval and
radius of convergence
4. Taylor series
5. Maclaurin series for e, sin
x, cos x, and 1/1-x
6. Functions defined by series
7. Taylor polynomials
8. Taylor’s Theorem with
Lagrange form of the
remainder
9. Alternating series error
bound
10. Radius of convergence: nth
term test; direct
comparison test; absolute
and conditional
convergence; ratio test
IV
Analytic Geometry in Calculus
1. Polar Coordinates
2. Tangent lines and arc length for
parametric and polar curves
3. Area in Polar Coordinates
Review and preparation for the
AP Exam
Chapter 9: Mathematical Modeling
6 classes
with Differential Equations
9.1 First-Order Differential Equations
*Focus on separation of variables and
use of initial condition(s)
9.3 Modeling with First-Order
Differential Equations
*”Clue” Activity (see Student Activity
11, below)
9.2 Slope Fields; Euler’s Method
AP FRQs (3) on Differential
Equations, Separation of Variables,
Slope Fields
Chapter10: Infinite Series
18-20
10.1 Sequences
classes
10.2 Monotone Sequences
10.3 Infinite Series
10.4 Convergence Tests
10.5 The Comparison, Ratio, and
Root Tests
10.6 Alternating Series; Conditional
Convergence
10.7 Maclaurin and Taylor
Polynomials
10.8 Macluarin and Taylor Series’
Power Series
10.9 Convergence of Taylor Series
Chapter 11: Analytic Geometry in
Calculus
11.1 Polar Coordinates
11.2 Tangent Lines and Arc Length
for Parametric and Polar Curves
11.3 Area in Polar Coordinates
Selected AP FRQs and MCQs
5 days
5-10 days
Teaching Strategies
Students are expected to take responsibility for their own learning, as the course is
designed to be a transition from high school to college curriculum.
A variety of teaching strategies are used in almost every class period (55 minutes), ranging
from direct instruction, to student investigation, to group work, discussion, and presentations, to
journaling. I try to create a community of learners: our classroom is set up in groups of three or four
to facilitate student collaboration; the students choose study groups of three to four people, and a
different group is responsible for presenting and reviewing each homework assignment; take-home
exams are given three times per semester, and students are not only encouraged but expected to
collaborate with their classmates in the Calculus AB class AND their peers who are taking Calculus
BC.
The emphasis in our class is always on UNDERSTANDING: understanding why the limit
definition of the derivative and the definite integral make sense; understanding why the Fundamental
Theorem of Calculus is true and understanding how derivatives and integrals are related; understanding
the multiple representations (graphs, equations, tables of values, and verbal descriptions) of all
functions, including derivative and antiderivative functions; and understanding why rules and theorems
work as they do. The graphing calculator is used to help the students see these multiple representation
and to check their work and verify solutions. Each student has a TI-83 or TI-84 graphing calculator,
we use the calculator at least once a week in class, and each assessment (at least two individual
assessments are given in each chapter) includes both a calculator section and a non-calculator section.
I place strong emphasis on both written and oral explanation and justification: every quiz and test
includes free-response questions, and each student group presents a lesson and homework problems to
the class at least once a month.
Selected Student Activities and Investigations
1. Thinking about Average vs. Instantaneous Rate of Change
a. The students are first given a piece-wise linear graph that measures the depth of the
water in an aquarium over a time interval. They are then asked to write a description of
what might be happening in (or outside of) the aquarium to cause the various changes in
depth. (For example, when the graph has a jump discontinuity, someone may have
dropped a rock into the aquarium, versus when the graph has a constant positive slope
someone is probably pouring water into the aquarium.) Students share and critique their
descriptions.
b. The students are then given a written description of a situation: Think about a person
bungee-jumping off of a bridge. They are then asked to write a description of how
jumper’s height changes over time and sketch a graph modeling this situation. Then I
ask them questions about the average and instantaneous velocity of the jumper at
different times, and we discuss as a class and attempt to sketch a graph of velocity.
This activity helps students begin to think more conceptually, helps them distinguish between
average and instantaneous change, and helps emphasize the multiple representations of
functions.
2. Derivative Matching Activity
Students are put into groups of three, and each group is given a packet with 40 cards: 10
graphs of functions, 10 graphs of first derivatives, 10 graphs of second derivatives, and and
10 verbal descriptions of the functions. Working together, the students must match a
function to its first and second derivatives and to its verbal descriptions. We then create a
classroom display of all ten completed matches. The following class, students are assessed
individually by having to write a paragraph describing their reasoning process for one of the
matches.
This activity helps students discover more about the relationships between the key features
(extrema, points of inflection, concavity, etc.) of functions and their derivatives (roots,
extrema, values, etc.), and helps them hone their skills at discussing and writing clear and
concise explanations and justifications.
3. Levels of Derivatives
Students are given packets that consist of 5 different “levels” of derivatives that they need to
work their way up through. Level 1 consists of basic rules, Level 2 problems need to be
simplified before they can be differentiated, Level 3 problems involve products and quotients,
Level 4 is chain rule problems, and Level 5 consists of problems that require two or more rules
(e.g. chain rule within a product). Students work individually, and then we check and discuss
the answers as a class. As a follow-up, students are asked to create five different problems (one
from each level) that involve the derivative at a point, and we use these problems to practice
utilizing the nDeriv and dy/dx functions on the graphing calculators.
This activity helps students gauge their own understanding of different types of derivatives and
increase their proficiency at differentiating all types of functions and using the graphing
calculator to evaluate numerical derivatives.
4. “Wizard of Oz”-themed Related Rates Board Game
The class is divided into groups of three or four students. Each group is assigned a playing
piece (Dorothy, Toto, tornado, etc.), and the game is played similar to “Chutes-n-Ladders.” A
related rates problem is displayed to the class via PowerPoint and the overhead projector; each
team that gets the correct answer in the given amount of time is allowed to roll the dice and
move their game piece.
This activity helps the students work together and gives them extra practice (in a fun manner)
with various related rates problems and implicit differentiation.
5. f, f’, and f’’ Matching Exercises
During our study of Chapter 4 (Using the Derivative), the students start each class with a quick
individual quiz. Each student is given a coordinate plane with four graphs labeled A, B, C, and
D, and they are asked to identify which of these graphs is f, f’, f’’, and which is g, an unrelated
function and they must explain their reasoning. The students then grade each others’ papers
and we discuss as a class the correct answer.
This activity helps reinforce the relationships between the key features (extrema, points of
inflection, concavity, etc.) of functions and their derivatives (roots, extrema, values, etc.), and
helps the students hone their skills at discussing and writing clear and concise explanations and
justifications.
6. Venn Diagram Exercise on Derivatives
Students are given a Venn diagram with 3 circles and are asked to compare “Finding Critical
Points,” “Mean Value Theorem,” and “Rolle’s Theorem.”
This helps the students summarize and compare the processes used for finding and verifying
local extrema and for using the MVT (both satisfying the hypothesis and conclusion).
7. Using the Graphing Calculator to Approximate and Evaluate Definite
Integrals
Students work out a classic Riemann sum problem by hand (usually something like y=x^2 from
x=0 to x=4 with four equal subintervals), and then we use a program (obtained from a weeklong AP institute, that I transfer to all of the students’ TI-83 or TI-84 graphing calculators) to
verify our Riemann sums with four subintervals and then to explore left-, right-, and midpoint
Riemann sums with eight, 16, 32, and more equal subintervals. (Each group of three students is
assigned a particular number of subintervals, and we create a table on the board comparing
number of rectangles to area approximation.)
From this exploration, the students are able to come up with the definition of a definite integral
as the limit of the sum of rectangles. After producing this definition, we verify using the fnInt
and Sf(x) features on the calculator, and then the students have time to explore a few problems
on their own with their calculators.
8. “Rule of Four” for Definite Integrals Exercises
Throughout our study of Chapter 5 (Understanding the Definite Integral), students start each
class with a quick individual quiz. Each student is given a worksheet divided into four boxes:
one box has a verbal description of a problem that involves a rate and asks about a quantity; the
other three boxes ask the students to 1) sketch a graph of the rate function, 2) approximate the
change in quantity using a Riemann sum, and 3) set up and evaluate a definite integral
involving the rate to find the exact quantity.
This helps the students understand the relationship between the area under a rate curve and a
total change in quantity, the different methods of approximating areas, and the set-up and
evaluation of definite integrals.
9. Levels of Integrals
Students are given packets that consist of 8 different “levels” of antiderivatives that they need
to work their way up through. Level 1 consists of basic rules, Level 2 problems need to be
simplified before they can be anti-differentiated, Level 3 problems involve un-doing the chain
rule using “simple” u-substitutions, Level 4 is “harder” u-substitution problems, and Level 5
consists of problems that require using integration by parts to un-do the product rule. Level 6
problems involve trigonometric integrals and substitutions, Level 7 involves partial fractions,
and Level 8 problems are all improper integrals. Students work individually, and then we
check and discuss the answers as a class. As a follow-up, students are asked to create five
different problems (one from each level) that involve definite integrals, and we use these
problems to practice utilizing the fnInt and Sf(x) functions on the graphing calculators.
This activity helps students gauge their own understanding of different types of anti-derivatives
and increase their proficiency at anti-differentiating all types of functions and using the
graphing calculator to evaluate definite integrals.
10.
Constructing Volumes of Revolution
Students are given a 3-dimensional x- and y-axis (made of thin wooden rods) and a party
decoration. They lay the axes flat on a piece of graph paper and then lay the flat party
decoration (usually a sphere when unfolded, thus a semi-circle when flat) with one corner on
the origin and the straight side on the x-axis. They trace the outline of the flattened shape onto
their coordinate plane, and using a few coordinate points and the regression capability on their
graphing calculators, they find a function the gives the outside shape of the decoration. They
estimate the area of one side of the decoration using the boxes on their graph paper, and then
calculate the exact area using a definite integral and, if necessary, their graphing calculators.
Then they stand their axes upright, affix the decoration firmly to the x-axis, and open/ unfold
the decoration by revolving the area about the x-axis. This gives a great visual for students for
volumes of revolution; then we talk as a class about how to calculate the volumes using
cylindrical disks (or washers).
11.
“Clue” Differential Equations Activity
Students are given a “Clue”-style murder-mystery problem to solve, in which they (the
“investigators”) know the temperature of the body and need to use differential equations and
Newton’s Law of Cooling to figure out the time of death and narrow the list of suspects.
Students work cooperatively, and also use the graphing calculator to help solve this problem.
MIDTERM EXAM
Students take a 90-minute midterm exam consisting of a “Calculator” section with 10 multiplechoice questions and 1 FRQ, and a “NO Calculator” section with 14 multiple-choice questions and 2
FRQs. All questions are taken directly from old AP exams, and questions are scored using the AP
scoring rubrics (to the best of my ability). This allows the students to get used to the type of questions,
the timing, and the grading used on the actual AP test.
Final Exam
The students take a full AP exam (made from a compilation of previous AP problems)
over the course of three class periods (3 hours). The exam consists of 54 multiple choice and 6
FRQs, and the AP grading rubric and multiple-choice scoring is used to the best of my ability.
We then have a few days to discuss all of the problems on the exam.