Download AP Calculus AB Course Outline

AP® Calculus AB – Mr. Andrew J. Byrum
Course Philosophy
Students will rise to the occasion, given the opportunity. For most students, Calculus is the
culmination of all those years studying mathematics in school. It becomes the “tie that
binds.” The course is designed to be challenging, but not overbearing; with an emphasis
using the multifaceted approach of expressing concepts algebraically, graphically,
numerically, and verbally. My students will explore the two major branches of Calculus:
differential and integral calculus.
Teaching Strategies
Within the first few days of school, each student is issued a Texas Instruments TI-89
graphing calculator. Most students are already familiar with using a graphing calculator,
since I also teach the Pre-Calculus course, in which each student is issued a Texas
Instruments TI-83 or TI-84 graphing calculator. The first week is designed to familiarize the
students with the particular nuances of using the TI-89, with continual emphasis on the fact
that the AP Exam is only 50% calculator based. As content progresses, more tidbits are
revealed about the TI-89’s capabilities; and each student is offered a manual should
individualized exploration be desired.
Throughout the course, students collaborate together, usually informally, on various
classwork activities. The classroom seating structure is that of tables, which make group
work more conducive for cooperative learning. Often, students do problems on the two
marker boards located on the wall at the head of the class; giving verbal explanations for
their work. Marker boards are employed so that students can visualize Calculus through
the use of color. Students are expected to use proper vocabulary and terminology when
giving verbal explanations to their fellow classmates, as well as to the instructor.
Technology is also available through the use of a Texas Instruments TI-89 graphing
calculator that can be viewed via an overhead projector.
I want to complete the teaching of the course by the end of the third marking period, so that
test preparation can commence and continue until the AP exam is administered during the
second week in May. A binder with all of the previous free response problems from the AP
Exams, starting with 2010 and dating to 1998 are provided. These have been downloaded
from
the
CollegeBoard
AP
Central
website
at
http://apcentral.collegeboard.com/apc/members/exam/exam_questions/1997.html,
and
photocopied on colored paper. The Scoring Guidelines and Scoring Commentaries are
also photocopied on the same color paper, and provided in the same order, to be kept in
the back of the binder behind the tests.
1
AP Calculus AB Course Ouline
Unit 1: Limits and Continuity (4 weeks)
A.
Rates of Change and Limits
1.
2.
3.
4.
5.
Average and Instantaneous Speed
Definition of Limit
Properties of Limits (Theorem 2)
One-sided and Two-sided Limits (Theorem 3)
Sandwich Theorem (Theorem 4)
Assignment: Textbook Section 2.1 pp. 62 – 64 # 1 – 32, 53 – 56
Resource: How to Ace Calculus: The Streetwise Guide
Chapter 8: Limits: You Gotta Have Them pp. 46 – 56
B.
Limits Involving Infinity
1.
2.
3.
4.
5.
Finite Limits as x →  ∞
Sandwich Theorem Revisited (Theorem 5)
Infinite Limits as x →a
End Behavior Models
“Seeing” Limits as x →  ∞
Assignment: Textbook Section 2.2 pp. 71 – 72 # 1 – 22
Quiz: Limits (No Calculator)
C.
Continuity
1.
2.
3.
4.
5.
Continuity at a Point
Continuous Functions
Algebraic Combinations (Theorem 6)
Composite Functions (Theorem 7)
Intermediate Value Theorem for Continuous Functions (Theorem 8)
Assignment: Textbook Section 2.3 pp. 80, 81 # 1 – 12, 19 – 24
Resource: How to Ace Calculus: The Streetwise Guide
Chapter 9: Continuity, or Why You Shouldn’t Ski Down Discontinuous Slopes
pp. 57 – 62
D.
Rates of Change and Tangent Lines
1.
2.
3.
4.
5.
Average Rates of Change
The Tangent to a Curve
The Slope of a Curve
The Normal to a Curve
Instantaneous Rate of Change
Assignment: Textbook Section 2.4 pp. 87 – 89 # 1 – 6, 9 – 14, 23 – 28
Test: Limits and Rates of Change – Part: A (No Calculator)
Limits and Rates of Change – Part: B (Calculator Based)
2
Unit 2: Derivatives (6 weeks)
A.
Derivative of a Function
1.
2.
3.
Definition of Derivative: Limit of the Difference Quotient
d
Various Notations Used for Derivative: f (x), y, dy
dx , dx f(x)
One-sided Derivatives
Assignment: Textbook Section 3.1 p. 101 – 103 # 1 – 10, 18
Resource: How to Ace Calculus: The Streetwise Guide
Chapter 11: Limit Definition of the Derivative: Finding Derivatives the Hard Way
pp. 68 – 76
Quiz: Limits Using the Difference Quotient (No Calculator)
B.
Differentiability
1.
2.
3.
4.
Differentiability at a Point
Differentiability Implies Local Linearity
Differentiability Implies Continuity (Theorem 1)
Intermediate Value Theorem for Derivatives (Theorem 2)
Assignment: Textbook Section 3.2 pp. 111 # 1 – 22
C.
Rules for Differentiation
1.
2.
3.
Power Rule
Product Rule
Quotient Rule
Assignment: Textbook Section 3.3 pp. 120, 121 # 1 – 19, 24, 26, 27, 28, 38
Supplemental: Introductory Mathematical Analysis p. 280 # 10 – 48 even
Resource: How to Ace Calculus: The Streetwise Guide
Chapter 12: Limit Definition of the Derivative: Finding Derivatives the Easy Way
pp. 77 – 83
D.
Velocity and Other Rates of Change
1.
2.
3.
4.
Instantaneous Rates of Change
Displacement – Motion along a Line
1st Derivative – Velocity
2nd Derivative – Acceleration
Assignment: Textbook Section 3.4 pp. 129 # 2 – 4, 6, 8, 10 – 13
Resource: How to Ace Calculus: The Streetwise Guide
Chapter 13: Velocity: Put the Pedal to the Metal pp. 84 – 87
Quiz: Derivatives: Power Rule, Product Rule, Quotient Rule (No Calculator)
E.
Derivatives of Trigonometric Functions
1.
2.
3.
Derivative of the Sine Function
Derivative of the Cosine Function
Derivative of the remaining Trigonometric Functions
Assignment: Textbook Section 3.5 pp. 140 # 1 – 13
3
F.
Chain Rule
1.
2.
3.
The Derivative of Composite Functions
The “Outside-Inside” Rule
Repeated use of the Chain Rule
Assignment: Textbook Section 3.6 pp. 146 # 1 – 20, 29 – 32
Supplemental: Introductory Mathematical Analysis p. 289, 290 # 10 – 44 even
Resource: How to Ace Calculus: The Streetwise Guide
Chapter 14: Chain Rule: S & M Made Easy pp. 88 – 92
Quiz: Trigonometric Functions and the Chain Rule (No Calculator)
G.
Derivatives of Exponential and Logarithmic Functions
1.
2.
3.
4.
Derivative of ex
Derivative of ax
Derivative of ln x
Derivative of logax
Assignment: Textbook Section 3.9 pp. 170 # 2 – 40 even
Supplemental: Introductory Mathematical Analysis p. 296 # 2 – 28 even
Supplemental: Introductory Mathematical Analysis p. 300 # 2 – 20 even
Resource: How to Ace Calculus: The Streetwise Guide
Chapter 24: Exponents and Logarithms: A Review of All That “e” Hoopla pp. 169 – 174
Chapter 25: Doing That Calc Thing to Exponents and Logs pp. 175 – 180
H.
Implicit Differentiation
1.
2.
Implicitly Defined Functions
Higher Order Derivatives
Assignment: Textbook Section 3.7 pp. 155 # 2 – 20, 24 – 36 even
Supplemental: Introductory Mathematical Analysis p. 307 # 1 – 21 odd
Resource: How to Ace Calculus: The Streetwise Guide
Chapter 17: Implicit Differentiation: Let’s Be Oblique pp. 114 – 116
Quiz: Implicit Differentiation( Part: A – No Calculator, Part: B – Calculator Based)
I.
Derivatives of Inverse Trigonometric Functions
1.
2.
3.
4.
5.
Derivatives of Inverse Functions
Derivative of sin-1x (Arcsine)
Derivative of tan-1x (Arctangent)
Derivative of sec-1x (Arcsecant)
Derivatives of the Other Three
Assignment: Textbook Section 3.8 pp. 162 # 2 – 16 even
Test: Derivatives – Part: A (No Calculator)
Derivatives – Part: B (Calculator Based)
Unit 3: Applications of the Derivative (6 weeks)
A.
Extreme Values of Functions
1.
2.
3.
Absolute (Global) Extreme Values
Local (Relative) Extreme Values
Definition of a Critical Point
Assignment: Textbook Section 4.1 pp. 184 # 1 – 10, 19 – 30
4
B.
Mean Value Theorem
1.
2.
Mean Value Theorem for Derivatives (Theorem 3)
Physical Interpretation for the Mean Value Theorem
Assignment: Textbook Section 4.2 pp. 192 # 2 – 18 even
C.
Connections between f(x), f′(x), and f′′(x)
1.
2.
3.
4.
First Derivative Test for Local Extrema
Concavity
Points of Inflection
Second Derivative Test for Local Extrema
Assignment: Textbook Section 4.3 pp. 203, 204 # 1 – 12, 14 – 28 even
Quiz: 1st & 2nd Derivatives, Minimums, Maximums, Points of Inflection, Concavity (No
Calculator)
Cooperative Activity: Box Making from a Single Sheet of Paper (12 x 18)
D.
Modeling and Optimization
1.
2.
3.
Examples from Business and Industry
Examples from Mathematics
Examples from Economics
Assignment: Textbook Section 4.4 pp. 214 # 2 – 4, 9 – 11, 13, 18, 19
Resource: How to Ace Calculus: The Streetwise Guide
Chapter 16: Maxima and Minima: The Bread and Butter Section pp. 103 – 113
Quiz: Optimizations (Calculator Based)
E.
Related Rates
1.
2.
Related Rate Equations
Solution Strategy
Assignment: Textbook Section 4.6 pp. 237 – 239 # 1 – 3, 11 – 14, 20, 22
Resource: How to Ace Calculus: The Streetwise Guide
Chapter 18: Related Rates: You Change; I Change pp. 117 – 125
Quiz: Related Rates (Calculator Based)
Test: Applications of the Derivative – Part: A (No Calculator)
Applications of the Derivative – Part: B (Calculator Based)
Unit 4: The Definite Integral (4 weeks)
A.
Estimating with Finite Sums
1.
Rectangular Approximation Method (LRAM, RRAM, MRAM)
Assignment: Textbook Section 5.1 pp. 254, 255 # 1 – 4, 10 – 12, 19
Cooperative Activity: LRAM, RRAM, MRAM of f(x) on [0,4] with n = 4, n = 8, n = 16
B.
Definite Integrals
1.
2.
3.
4.
Riemann Sums
Terminology and Notation of Integration
The Definite Integral and Area
Numerical Integration using a Calculator (nInt)
Assignment: Textbook Section 5.2 pp. 267 # 7 – 22
5
C.
Definite Integrals and Antiderivatives
1.
2.
3.
Properties of Definite Integrals
Average Value of a Function
The Mean Value Theorem for Definite Integrals
avg(f) = f(c) =
b
1
b-a a

f(x)dx
Assignment: Textbook Section 5.3 pp. 274, 275 # 1 – 24
Resource: How to Ace Calculus: The Streetwise Guide
Chapter 20: Intermediate Value Theorem and Mean Value Theorem pp. 130 – 133
D.
Fundamental Theorem of Calculus
1.
Fundamental Theorem, Part 1
F(x) =
2.
x
a
f(t)dt
Fundamental Theorem, Part 2

b
a
3.

f(x)dx = F(b)-F(a)
The Area-Integral Connection
Assignment: Textbook Section 5.4 pp. 286, 287 # 2 – 14 even, 15 – 18, 25 – 28
E.
Trapezoidal Approximations
1.
Trapezoidal Rule

b
a
2.
f(x)dx  T = h2 (y0 + 2y1+ 2y 2 +  + 2yn -1+ yn ), h =
b-a
n
Simpson’s Rule

b
a
f(x)dx  S = h3 (y0 + 4y1+ 2y2 + 4y3 +  + 2yn - 2 + 4yn - 1+ yn ), h =
b-a
n
Assignment: Textbook Section 5.5 pp. 295 # 1 – 8
Resource: How to Ace Calculus: The Streetwise Guide
Chapter 21: Integration: Doing It All Backwards pp. 134 – 145
Test: The Definite Integral – Part: A (No Calculator)
The Definite Integral – Part: B (Calculator Based)
Unit 5: Differential Equations (4 weeks)
A.
Antiderivatives and Slope Fields
1.
2.
3.
4.
5.
Solving Initial Value Problems
Definition of Slope Field
Antiderivatives and Indefinite Integrals
Rules for Integration
Properties of Indefinite Integrals
Assignment: Textbook Section 6.1 pp. 312, 313 # 1 – 6, 8 – 24 even, 32 – 38 even
Supplemental: Introductory Mathematical Analysis p. 394 # 14 – 52 even
Resource: How to Ace Calculus: The Streetwise Guide
Chapter 22: The Definite Integral pp. 146 – 164
Cooperative Activity: Drawing Slope Fields
Quiz: Integration (No Calculator)
6
B.
Integration by Substitution
1.
2.
3.
4.
Power Rule for Integration
Rules for Trigonometric Integration
Substitution in Indefinite Integrals
Substitution in Definite Integrals
Assignment: Textbook Section 6.2 pp. 321, 322 # 1 – 8, 10 – 28 even
Supplemental: Introductory Mathematical Analysis p. 431 # 1 – 14
Quiz: Integration Using Substitution (No Calculator)
C.
Integration by Parts
1.
2.
3.
Product Rule for Integration by Parts
Solving for an Unknown Integral
Tabular Integration
Assignment: Textbook Section 6.3 pp. 328, 329 # 1 – 4, 9 – 14, 16 – 24 even
Supplemental: Introductory Mathematical Analysis p. 432 # 15 – 41 odd
Resource: How to Ace Calculus: The Streetwise Guide
Chapter 28: Fancy-Pants Techniques of Integration pp. 193 – 202
Test: The Indefinite Integral – Part: A (No Calculator)
The Indefinite Integral – Part: B (Calculator Based)
Unit 6: Applications of Definite Integrals (2 weeks)
A.
The Integral as a Net Change
1.
2.
3.
Linear Motion Revisited
General Strategy for Modeling with Integrals
Determining Net Change from Data
Assignment: Textbook Section 7.1 pp. 371 # 1 – 8, 17 – 20
B.
Areas in a Plane
1.
2.
3.
4.
The Area between Curves
The Area Enclosed by Intersecting Curves
Boundaries with Changing Functions
Integration with Respect to y
Assignment: Textbook Section 7.2 pp. 380, 381 # 1 – 10, 12 – 26 even, 36
Supplemental: Introductory Mathematical Analysis p. 446 # 2 – 20 even
Quiz: The Area between Curves – Part: A (No Calculator)
The Area between Curves – Part: B (Calculator Based)
C.
Volumes
1.
2.
3.
4.
Volume as an integral
Square Cross Sections
Circular Cross Sections
Cylindrical Shells
Assignment: Textbook Section 7.3 pp. 392 # 13 – 16, 18 – 26 even, 43, 44
Quiz: Volumes of Revolutions – Part: A (No Calculator)
Volumes of Revolutions – Part: B (Calculator Based)
7
Student Evaluation
Quarterly grades are computed using attendance, activities, homework, quiz results, and
test results. Eighty percent of the grade comes from attendance, activities, quiz results, and
test results, with the remaining twenty percent coming from homework that is kept in a
notebook and due at the end of each quarter. As stated in the outline, quizzes and tests
vary with respect to calculator usage. Upon completion of the course outline, slated for the
end of the third / beginning of the fourth grading period, students are given practice AP
Tests that follow the same format as the AP Exam. These tests are graded using the same
format as the AP Exam, and given a corresponding numerical grade ranging from 55 to
120.
AP
Exam
Score
5
4
3
2
1
Numerical Score
Composite
Practice Practice Practice Practice
Score
Test 1
Test 2
Test 3
Test 4
80 to 108
120
120
120
120
70 to 79
110
111
113
115
60 to 69
100
101
103
105
50 to 59
90
91
93
95
40 to 49
85
84
82
80
30 to 39
80
79
77
75
20 to 29
75
74
72
70
10 to 19
70
69
67
65
5 to 9
65
64
62
60
0 to 4
60
59
57
55
If a student is able to score a 4 or a 5, then reward should come with a numerical grade of
110 or 120 for their achievement. With each test administration the numerical scoring
difficulty is increased at the lower end to push students toward achieving a mark of 3 or
better. For example, a student may receive a numerical grade of 80 while scoring a 2 on
the first practice exam, but only receive a numerical grade of 70 if still scoring a 2 on the
fourth practice exam. Each quarter grade represents one-fourth of the final grade. No Final
exam is given. It is my belief that the AP Exam given in May is the final exam for the
course.
8
Resource Materials
Primary Resource:
Finney, Ross L., Franklin D. Demana, Bert K. Waits, and Daniel Kennedy.
Calculus – Graphical, Numerical, Algebraic. 1st edition. Menlo Park, California:
Scott Foresman Addison Wesley, 1999.
Secondary Resources:
Adams, Colin, Abigail Thompson, and Joel Hass. How to Ace Calculus: The Streetwise Guide.
2nd Printing. New York, New York: W. H. Freeman and Company, 1998.
Barton, Ray, and John Diehl. Advanced Placement Calculus with the TI – 89.
Dallas, Texas: Texas Instruments Incorporated, 1999.
Barton, Ray, John R. Brunsting, John J. Diehl, Greg Hill, Karyl Tyler, and Steven L.
Wilson. Preparing for the Calculus AP* Exam. 1st edition. Pearson Education,
Inc., 2006
Codner, Robert W. Calculus AB Student Workbook. Evanston, Wyoming: Mathematics Criterion
Center.
Haeussler, Jr., Ernest F., and Richard S. Paul. Introductory Mathematical Analysis.
4th edition. Reston, Virginia: Reston Publishing Company, Inc., 1983.
Hopper, Clarence. Visualizing Calculus. Dale Seymour Publications®, 1998.
Purcell, Edwin J. Calculus with Analytic Geometry. 3rd edition. Englewood Cliffs,
New Jersey: Prentice-Hall, Inc., 1978.
Stewart, James. Single Variable Calculus: Concepts & Contexts. 3rd edition. Belmont,
California: Thomson Brooks/Cole, 2005.
Tam, George. Worked Examples and Exercises in Calculus. 1st edition. Richmond Hill,
Ontario: Alice and George Publishing Co. Ltd., 2005.
9