AP Calculus AB Syllabus
Camden High School
Course Description
AP courses in calculus consist of a full high school academic year of work and are comparable to calculus
courses in colleges and universities. It is expected that students who take an AP course in calculus will
seek college credit, college placement, or both, from institutions of higher learning. The AP Program
includes specifications for two calculus courses and the exam for each course. The two courses and the
two corresponding exams are designated as Calculus AB and Calculus BC. Calculus AB can be offered as
an AP course by any school that can organize a curriculum for students with mathematical ability.
Calculus AB is designed to be taught over a full high school academic year. It is possible to spend some
time on elementary functions and still teach the Calculus AB curriculum within a year. However, if
students are to be adequately prepared for the Calculus AB Exam, most of the year must be devoted to
the topics in differential and integral calculus. These topics are the focus of the AP Exam questions.
Both the AB and BC concepts will be covered, and students demonstrating proficiency of all material will
take the Calculus BC exam and will receive a BC score and an AB subscore. Calculus AB represents
college-level mathematics for which most colleges grant advanced placement and/or credit. Most
colleges and universities offer a sequence of several courses in calculus, and entering students are
placed within this sequence according to the extent of their preparation, as measured by the results of
an AP Exam or other criteria. Appropriate credit and placement are granted by each institution in
accordance with local policies. The content of Calculus BC is designed to qualify the student for
placement and credit in a course that is one course beyond that granted for Calculus AB. Many colleges
provide statements regarding their AP policies in their catalogs and on their websites.
Success in AP Calculus is closely tied to the preparation students have had in courses leading up to their
AP courses. Students should have demonstrated mastery of material from courses that are the
equivalent of four full years of high school mathematics before attempting calculus. These courses
should include the study of algebra, geometry, coordinate geometry, and trigonometry, with the fourth
year of study including advanced topics in algebra, trigonometry, analytic geometry, and elementary
functions. With a solid foundation in courses taken before AP, students will be prepared to handle the
rigor of a course at this level. Students who take an AP Calculus course should do so with the intention
of placing out of a comparable college calculus course. This may be done through the AP Exam, a college
placement exam, or any other method employed by the college.
Student Goals
Each student who actively participates and fulfills the expectations of the course will be able to:

Work with functions in a variety of ways and understand the connections among these
representations.

Understand the meaning of 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.

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.

Understand the relationship between the derivative and the definite integral as expressed in
both parts of the Fundamental Theorem of Calculus.
August 2009
Lisa Twitty
AP Calculus AB Syllabus
Camden High School

Communicate mathematics and explain solutions both verbally and in written sentences.

Model a written description of a physical situation with a function, a differential equation, or an
integral.

Use technology to help solve problems, experiment, interpret results, and support conclusions.

Determine the reasonableness of solutions, including sign, size, relative accuracy, and units of
measurement.

Develop an appreciation of calculus as a coherent body of knowledge and as a human
accomplishment.
Textbook Information
Calculus: Graphical, Numerical, Algebraic. Pearson, 2012 (issued by the school)
Barron’s AP Calculus. Barron’s Educational Series, Inc. (student purchases in the spring semester)
Other Course Materials
Organized notebook (sections to include: notes, homework, activities, reference)
Pencils (standard and colored, or markers if desired)
Graphing Calculator: TI-Nspire CAS handheld strongly recommended
The Role of Technology
The graphing calculator is an integral tool in this course. The TI-83, TI-89, and T-Nspire CAS calculators
are used daily to discover and explore relationships within and between concepts. Students are strongly
encouraged to purchase the TI-Nspire CAS. Concepts will be presented using Nspire software, and
lessons will be taught using the Nspire CAS handhelds if enough students have them. Throughout the
course, graphing calculators will be used regularly in guided activities and spontaneously during class
discussions in a variety of ways to:
 Conduct explorations.
 Graph functions in arbitrary windows.
 Solve equations numerically.
 Analyze and interpret results.
 Justify and explain results of and relationships between graphs and equations.
Topic Outline and Timeline
Unit 1: Limits and Continuity (13 - 17 days, 2 AP Quizzes, 1 Test)
1) Rates of Change and Limits
a) Average Speed
b) Instantaneous Speed
c) Definition of Limit
d) Properties of Limits
e) One and Two-sided Limits
f) Sandwich Theorem
August 2009
Lisa Twitty
AP Calculus AB Syllabus
Camden High School
[Graphing calculators help students to assimilate this idea. Several examples, including finding
𝑠𝑖𝑛𝑥
lim
𝑥→∞ 𝑥
by using the table produced by the graphing calculator to notice that y = 0 appears to
1
be the horizontal asymptote. Students then confirm analytically and then graph 𝑓(𝑥) = 𝑥 and
−1
𝑓(𝑥) = in the same window to connect the analysis to the graph.]
𝑥
2) Limits Involving Infinity
a) Finite Limits as 𝑥 → ∞
b) Infinite Limits as 𝑥 → 𝑎
c) End Behavior Models
3) Continuity
a) Continuity at a Point
b) Continuous Functions
c) Discontinuity (Removable, Jump, Infinite)
[Exploration of discontinuities helps students to appreciate the significance of continuity on
intervals and everywhere continuous functions. In small groups, students complete an activity
in which each group is given a set of five functions with different types of discontinuities. Using
their graphing calculators, students investigate the graphs of the functions to classify the type of
discontinuity of each. Students then compute the limits at each discontinuity for each function.
A culminating class discussion is held in which students discuss the connections between the
limit and type of discontinuity, and how both are revealed in the graph and table for each
function.]
4) Rates of Change and Tangent Lines
a) Average Rate of Change
b) Tangent to a Curve
c) Slope of a Curve
d) Normal to a Curve
e) Instantaneous Rate of Change
Unit 2: The Derivative (20 - 25 days, 3 AP Quizzes, 1 Test)
1) The Derivative of a Function
a) Definition of a Derivative
b) Derivative at a Point
[Students complete an exploration of 𝑓(𝑥) = 𝑥 3 − 4𝑥 2 − 9𝑥 + 46 in which they find the
derivative at x=2 and x=4 using the definition, and then graph the tangent lines at both points.
For both lines, students explain, in writing, how the tangent lines confirm the computed
derivatives.]
c) Relationship Between the Graphs of f and f’
d) Graphing the Derivative from Data
e) One-sided Derivatives
2) Differentiability
a) Where f’(x) Fails to Exist
[Students explore this idea by graphing 𝑦 = |𝑥|, and zooming in at (0, 0); 𝑦 = 1 +
1
2
√𝑥 2 + 0.005, and zooming in to (0,1); and 𝑦 = 𝑥 3 + 4 (𝑥 − 1) ⁄3 and zooming in to (1,1) to
seek linearity in each case. This exploration also helps students realize that the initial
appearance of the calculator graph may or may not be useful.]
b) Local Linearity
August 2009
Lisa Twitty
AP Calculus AB Syllabus
Camden High School
3)
4)
5)
6)
7)
8)
9)
[Class discussion begins with a zoomed-in view of the sine function that appears to be the line
y=x. Once a student guesses this line, I graph it. Of course, it appears to be the same line. By
zooming out once, then twice, then three times, the point is made for the students. In the class
discussion, I have students come up with a function for them to graph and zoom in at various
points to see the same for themselves.]
c) Derivatives on a Calculator (TI-83, TI-89, TI-Nspire CAS)
d) Differentiability Implies Continuity
e) Intermediate Value Theorem for Derivatives
Rules for Differentiation
a) Constant Multiple, Power, Sum, Difference, Product, Quotient Rules
b) Second and Higher Order Derivatives
Velocity and Other Rates of Change
a) Position, Velocity, and Acceleration
[Students complete an exploration in which they derive velocity and acceleration from position
data (of a bullet fired from a rifle) presented in a table. Students appreciate the value of the
equations derived when asked to compute the position, velocity, and acceleration of the bullet
at t-values not included in the table.]
b) Motion Along a Line
Derivatives of Trigonometric Functions
Chain Rule
Implicit Differentiation
Derivatives of Inverse Trigonometric Functions
Derivatives of Exponential and Logarithmic Functions
Unit 3: Applications of Derivatives (20 - 25 days, 2 AP Quizzes, 1 Test)
1) Extreme Values of Functions
a) Absolute Extreme Values
b) Extreme Value Theorem
c) Local Extreme Values
d) Critical Points
2) Mean Value Theorem
a) Physical Interpretation
b) Increasing and Decreasing Functions
3) Connecting f’ and f” with the Graph of f
a) First Derivative Test for Local Extrema
b) Second Derivative
i) Concavity
ii) Inflection Points
iii) Second Derivative Test for Local Extrema
c) Finding f(x) from f’(x) and f”(x)
[As a culminating activity, students working in pairs are given f(x) cards which they are required to
match with cards containing written descriptions of the graphs and to corresponding f’(x) cards.
Students complete a summary form on which they list the matched sets of three cards and a
written description of how they matched the f’(x) card to its f(x) and description.]
4) Modeling and Optimization
5) Linearization
a) Local Linearization
August 2009
Lisa Twitty
AP Calculus AB Syllabus
Camden High School
b) Differentials
6) Related Rates
Unit 4: The Definite Integral (12 – 16 days, 2 AP Quizzes, 1 Test)
1) Estimating with Finite Sums
a) Distance Traveled
b) Rectangular Approximation Method
i) Left Sum
ii) Right Sum
iii) Midpoint
[Students complete an exploration to discover how the three methods compare. The
𝑥
exploration begins with students graphing two functions, 𝑓(𝑥) = 5 − 4 sin ( ) 𝑎𝑛𝑑 𝑔(𝑥) =
2
2 sin(5𝑥) + 3, on their calculators in the window [0, 3] by [0, 5]. After copying the graph of
each onto graph paper, students draw the rectangles for each method with n = 3 and compute
the area approximations, and finally order them. Once the process is complete for both
functions, students complete a written explanation of function characteristics that determine
the relative sizes of the three approximations.]
c) Volume of a Sphere
2) Definite Integrals
a) Riemann Sums
b) Terminology and Notation
c) Definite Integral and Area
d) Constant Functions
e) Integrals on a Graphing Calculator (NINT)
f) Discontinuous Integrable Functions
[Students complete a quick exploration of 𝑓(𝑥) =
𝑥 2 −4
𝑥−2
in which they first explain the
3 𝑥 2 −4
discontinuity on [0, 3] and then use areas to show that ∫0
5
∫0 𝑖𝑛𝑡(𝑥)𝑑𝑥
𝑥−2
𝑑𝑥 = 10.5. Finishing up by using
areas to show that
= 10.]
3) Definite Integrals and Antiderivatives
a) Properties of Definite Integrals
b) Average Value of a Function
c) Mean Value Theorem for Definite Integrals
d) Connecting Differential and Integral Calculus
4) Fundamental Theorem of Calculus
a) Part 1
𝑥
i) Graphing the Function ∫𝑎 𝑓(𝑡)
[Students complete an exploration in which they graph NDER(NINT(x2, x, 0, x)) and then
NDER(NINT(x2, x, 5, x)) on their calculators. They are then asked to tell the x-intercepts
(without graphing) of NINT(x2, x, 0, x) and NINT(x2, x, 5, x) and explain. Finally, students
𝑥
𝑑
connect these steps to how changing a in the graphs of 𝑦 = 𝑑𝑥 (∫𝑎 𝑓(𝑡)𝑑𝑡) 𝑎𝑛𝑑 𝑦 =
𝑥
∫𝑎 𝑓(𝑡)𝑑𝑡. This exploration helps students to connect their graphical understanding with
their analytic understanding of the FTC.]
b) Part 2
i) Finding Area Analytically
ii) Finding Area Numerically (NINT(|𝑓(𝑥)|, 𝑥, 𝑎, 𝑏))
August 2009
Lisa Twitty
AP Calculus AB Syllabus
Camden High School
5) Trapezoidal Rule
a) Error Bounds
Unit 5: Differential Equations and Mathematical Modeling (16 – 18 days, 2 AP Quizzes, 1 Test)
1) Slope Fields and Euler’s Method
a) Differential Equations
b) Constructing Slope Fields
c) Euler’s Method
2) Antidifferentiation by Substitution
a) Properties of Indefinite Integrals
b) Power, Trigonometric, Exponential, and Logarithmic Formulas
3) Antidifferentiation by Parts
a) Formula and Protocol for Selecting u and dv
b) Inverse Trigonometric and Logarithmic Functions
4) Exponential Growth and Decay
a) Separable Differential Equations
b) Law of Exponential Change
c) Continuously Compounded Interest
d) Radioactivity
e) Modeling Growth with Other Bases
f) Newton’s Law of Cooling
5) Logistic Growth
a) How Populations Grow
b) Partial Fractions
c) The Logistic Differential Equation
d) Logistic Growth Models
Unit 6: Applications of Definite Integrals (13 – 17 days, 1 AP Quiz, 1 Test)
1) Integral As Net Change
a) Linear Particle Motion Revisited
b) Consumption Over Time
c) Net Change From Data
2) Areas in the Plane
a) Area Between Two Curves (with respect to x and y)
b) Using Geometry Formulas
3) Volumes
a) Volume As an Integral
i) Disk Method
ii) Washer Method
b) Square Cross Sections
c) Circular Cross Sections
d) Other Cross Sections
4) Lengths of Curves
a) A Sine Wave
b) Length of a Smooth Curve
c) Vertical Tangents, Corners, and Cusps
August 2009
Lisa Twitty
AP Calculus AB Syllabus
Camden High School
Unit 7: Review and Exam Preparation (18 – 25 days)
1) Multiple-Choice Practice
Using a variety of practice tests and released items (1969 – 1998), students are given sets of items to
answer in small groups. At least once a week, a miniature (multiple-choice only) AP-exam is
administered (in a calculator/no calculator format) and scored to count as a quiz grade. The day
prior to these quizzes is spent in class discussion of items from packets completed during the week.
Test-taking strategies and embedded concept review are naturally occurring parts of these
discussions.
2) Free-Response Practice
Students work in small groups to answer packets of free-response questions. (Packets generally
include the six questions from a given year; but when it is clear that a particular type of question
needs more attention, topical packets are also used.) Once adequate time to answer all parts of all
questions has been allowed, students are given scoring rubrics to score their responses. In this
collaborative environment, students measure each other and themselves against acceptable
justifications and explanations. Students exchange papers within the group to compare approaches
and to evaluate the clarity and elegance of responses.
3) Whole-Test Practice
A full practice test is administered over four class meetings. On the fifth day, students are placed in
small groups to review their scored exams, after which a class discussion is held about the test.
Once this cycle has been completed, students attend and organized practice test at a local
university.
Contact Information
Phone: (803) 425-8930, Extension 3642
Email: lisa.twitty@kcsdschools.net
Late Work Policy
Homework is assigned daily and is communicated on a weekly assignment sheet and on the eChalk class
page. The expectation is for homework to be turned in on the first class day after the lesson was
covered in class, but assignments may be turned in late for full credit – with two stipulations: a
maximum of two assignments per day will be accepted, and all work for the chapter must be turned in
before the test for the relevant chapter is administered.
Extra Help Policy
Mrs. Twitty will be available for after-school help on Wednesdays from 3:15 – 4:15. Additional assistance
may be arranged by appointment.
August 2009
Lisa Twitty
AP Calculus AB Syllabus
Camden High School
Re-do Policy
The day after a test has been graded, the class will be split into two groups: those that
mastered the tested content (as demonstrated by earning a minimum of 77) and those that did
not. Students who mastered the content will earn bonus points and will serve as peer tutors for
those that did not. The following day students who did not master the content will be retested
for a maximum grade of 77. Any student who scores lower than the first attempt will earn as a
replacement grade the average of the two attempts.
Link for Standards
http://apcentral.collegeboard.com/apc/public/repository/ap-calculus-course-description.pdf
August 2009
Lisa Twitty