AP Calculus Syllabus 2010-11
Please take the time to thoroughly read this syllabus. All students are responsible for its content.
This syllabus should be kept in your math notebook and periodically reviewed as needed.
Summary of Course:
AP Calculus will introduce and study the topics of functions, graphs, limits, derivatives, and
integrals. We will focus on giving students the opportunity to work with functions represented in
a variety of ways, including graphically, numerically, analytically, and verbally and stress the
connections among these representations. Students will be using TI-83+ graphing calculators and
MacBook laptops to help solve problems, experiment, interpret results, and support conclusions.
A fair amount of instructional time will be devoted to learning how to properly use your
calculator to help accomplish these tasks as they arise. The overall goal of this course is to
master the material outlined above to an extent where the student will be able to successfully pass
the AP Calculus AB test.
Students will work both in groups and individually. Learning will be facilitated through direct
instruction, practicing problems, group discussion, and project based learning to explore
mathematical applications. Expectations are high regarding the quality of student work.
Units of Study:
1. Review Functions, Graphs, Rational and Polynomial Functions:
a. Solving linear, quadratic, polynomial, complex, and rational equations and
inequalities
b. Graphically representing equations
c. Shifting, reflecting and stretching graphs
d. Mathematical modeling
2. Limits and Their Properties:
a. Finding asymptotes graphically and numerically
b. Finding limits graphically and numerically
c. Evaluating limits analytically
d. Continuity and one-sided limits
e. Infinite limits
f. Intermediate Value Theorem and Extreme Value Theorem
3. Differentiation:
a. The derivative and the tangent line problem
b. Basic differentiation rules and rates of change
c. The product and quotient rules and higher-order derivatives
d. The chain rule
e. Implicit differentiation
f. The mean value theorem
4. Applications of Differentiation:
a. Extrema on an interval
b. Rolle’s theorem and the mean value theorem
c. Increasing and decreasing functions and the first derivative test
d. Concavity and the second derivative test
e. Limits at infinity
5.
6.
7.
8.
9.
10.
f. Optimization problems
g. Velocity, speed, acceleration
h. Differentials
i. Slope fields
Integration:
a. Antiderivatives and indefinite integration
b. Area
c. Rieman sums and definite integrals
d. The fundamental theorem of calculus
e. Integration by substitution
f. Numerical integration
Review Exponential and Logarithmic Functions:
a. Exponential functions and their graphs
b. Logarithmic functions and their graphs
c. Using properties of logarithms
d. Exponential and logarithmic equations
e. Exponential and logarithmic models
Differentiation and Integration of Exponential and Logarithmic Functions:
a. Differentiation and integration of exponential functions
b. Differentiation and integration of logarithmic functions
c. Differential equations: growth and decay
Review Trigonometric Functions:
a. Radian and degree measure
b. Trigonometric functions: the unit circle
c. Right triangle trigonometry
d. Trigonometric functions of any angle
e. Graphs of sine and cosine functions
f. Graphs of other trigonometric functions
g. Inverse trigonometric functions
h. Applications and models
i. Solving trigonometric equations
j. Sum and difference formulas
k. Multiple-angle and product-to-sum formulas
Differentiation and Integration of Trigonometric Functions:
a. Limits of trigonometric functions
b. Differentiation of trigonometric functions
c. Integration of trigonometric functions
d. Differentiation of inverse trigonometric functions
e. Integration of inverse trigonometric functions
f. Hyperbolic functions
Applications of Antidifferentiation
a. A variety of projects will be used to further study the various real-world
applications of antidifferentiation. Special attention will be paid to
communicating findings verbally, in writing, and graphically.
Textbooks:
Larson, Hostetler, Edwards: Calculus I with Precalculus: A One-Year Course, Houghton Mifflin
Company, 2006.
Foerster: Calculus: Concepts and Applications, Key Curriculum Press, 2005.
Grading Policy:

Homework- As this is an advanced placement class and is designed to give you college
credit for Calculus, your homework will be graded the same way it would be graded in
college. I will not grade homework and it will not be part of your grade. Homework will
be assigned on a daily basis and you will be expected to do it by the next class period.
We will go over the answers as a class and answer any questions you may have at that
time. Since it is important to understand different approaches to solving problems, we
will spend extra time discussing how to use your calculator to solve and check answers.
Just because homework is not graded does not mean you should feel as though you do not
need to do it. I will keep track of who has their homework completed on time, but again,
it does not factor into your grade.
Homework = 0% of overall grade

Participation/Effort- As in college, you are given no credit for simply showing up and
doing what you are supposed to do. You are expected to be in class on time and prepared
to learn every day. If you are gone a great deal, it will no doubt reflect in your grade
without me calculating it in. What you do receive participation credit for is successfully
completing tasks such as answering questions at the board, teaching a review topic in
preparation for a test, etc. You must become skilled in the art of communicating math,
both verbally and in writing, so you must practice, practice, practice! Special focus will
be placed on showing how you arrive at your answer using both your long-hand math
skills and also your calculator. How do you use the calculator to check your answers?
Participation/Effort = 10% of overall grade

Tests/Quizzes- Short quizzes will be given regularly and usually consist of one or two
questions taken from the AP Calculus practice test question bank. You will not be given
notice and will only be allowed to use materials that you would be allowed to use on the
AP Calculus test. Quizzes will always be given directly after we have gone over
homework and you have asked questions about it, so make sure you understand the
material or ask questions before we move on. Tests will be given at the end of each unit
of study and will encompass all work completed during that unit, and will also include
topics covered in past units. The tests will consist of two parts. The first part will
include you presenting one or two random questions to the class. This part of the test will
be graded on your correctness, but also your ability to communicate mathematically both
verbally and in writing on the board. The second part of the test will be written and will
follow the same structure as the AP Calculus test, including what materials you are
allowed use.
Tests/Quizzes = 60% of overall grade

Projects, Presentations & Class Work- Projects and presentations are an important part
of this class. It is not simply enough to be able to answer questions, you must be able to
apply your knowledge to real world situations and be able to accurately and effectively
present your findings to others in a variety of ways. Specifics on projects will be given
out during the year, but you can expect to average one major project every two units.
Just as in the business world you would not get paid (for very long) if you did not finish
the tasks assigned to you, in this class you will not get a grade on a project until it has
met a high standard of quality. If your project does not meet this standard, it will be
returned to you until it is finished. However, if it is not finished by the deadline, expect
your grade to suffer as a consequence. Refer to the Projects portion on the next page for
more detail.
Projects, Presentations & Class Work = 30% of overall grade
Grades are awarded as follows (round decimals to nearest percent):
92% and above
88-89%
78-79%
68-69%
Below 59%
A
B+
C+
D+
F
90-91%
82-87%
72-77%
60-66%
AB
C
D
80-81%
70-71%
BC-
Required Student Supplies:





A graphing calculator (a TI-83 Silver Edition or TI-84 is strongly recommended)
A spiral bound notebook with pocket (preferred) or section within a three-ring binder
exclusively for homework assignments.
Pencils and erasers
Graph paper
A covered textbook (provided)
Projects:
The following is a sampling of the projects we will complete throughout the year. This list is
subject to change but adequately give the reader an idea of the type of work we will be doing.
These projects were found at:
http://www.mecca.org/~halfacre/MATH/proj.htm
When possible, I will let students search for their own projects. They will have to be checked by
me for appropriateness, but you should spend time working on a topic that you find interesting.
All projects must be completed in written form and/or powerpoint form as you will also have to
report your findings to the class when finished, with special focus on communicating the math
involved in your solution. In each report, aside from reporting your results, I would like you to
describe how you used your calculator to help solve the problems.
OptimizationA Dorm Room's a Dorm Room, No Matter How Small
After months of diligent work, I finally earned a promotion to Vice President for Development
here at Who-U, which shocked quite a few Whos because, as you probably know, I am not more
than two. Oh, they tried to just give me a drink of water and send me to bed, but I worked very
hard on our outreach to other dust specks. I'm very proud of my fund-raising accomplishments,
but sometimes the gifts come with very strict limitations on how they can be used. We just
received such a donation, and when I went looking for help, your enterprising and resourceful
professor naturally referred me to you.
We have a somewhat eccentric alum who has made a major contribution in memory of his
favorite Chia Pet Airplane that recently passed away in a bizarre gardening accident (it's best we
not discuss the details). As a fitting tribute to the dearly departed, the donor has designated that
the funds be used to build a dorm in the shape of an airplane hangar, as shown below. There is an
additional stipulation on the gift: the volume of the dorm must be exactly 225,000 cubic feet,
which is one cubic foot for each sprout on the Chia plane.
We're in the planning stages with the architects now, and we would obviously like to minimize
the cost of the building. This is where I need your help. Currently, the construction costs for the
foundation are $30 per square foot, the sides cost $20 per square foot to construct, and the roofing
costs $15 per square foot. I need your expert advice on what the dimensions of the building
should be to minimize the total cost.
While the cost of the flooring and siding has been fairly stable, a further complicating factor is
that the cost of roofing material has been fluctuating dramatically for as long as I can remember
(at least two months). In addition to your recommendation for the price of $15 per square foot, I
also need a recommendation on the dimensions of the dorm if the roofing costs $R per square
foot.
We are meeting with the architects to discuss plans before Thanksgiving, so I would appreciate
your report by November 11.
It's nap time now,
Cindy Lou
Riemann SumsDear Calculus Students:
I recently became CEO of Spacely Sprockets after a somewhat messy hostile take-over, and I do
not yet trust anyone at the company to help me with the critical decisions that will affect the
success of the corporation. When I went looking for help, your enterprising and resourceful
professor naturally referred me to you.
I am remodeling my (nearly) oval office and replacing the horrid orange carpeting left by my
predecessor. With the Board and shareholders watching my every move, I want to be careful not
to appear profligate. I have already decided on a tasteful aqua shag, but the carpeting comes on
rolls in three different widths: 2.5 yards wide, 1.5 yards wide, and 1 yard wide.
I will use the same width for the entire office, but I'm not sure which one to choose. For each of
the three widths, I need an analysis of how many square feet of carpeting I must buy and an
estimate of the amount of carpeting that will be trimmed and wasted. I also have the option to
special order the carpeting in any width. To impress the board with my thoroughness and allaround competence, I would like to know what width I should order so that no more than 15
square feet of carpeting are wasted.
I am including a sketch of my office, and I would appreciate an answer by October 1 since I need
my office to be ready for an important meeting later in the month with AOL-Time-Warner to
discuss the launching of the Sprockets Channel (``All Sprockets. All The Time''TM).
Exponentials/Newton’s Law of Cooling-
Dear Calculus Student:
I have a difficult problem for which no one in my factory can find a solution, and I hope that you
can help me. Your intrepid and enterprising professor, Dr. Crannell, referred me to you.
At the Jumping Gelatin Factory, we make 42 different flavors of gelatin and provide all the
gelatin for all the local day-care centers and nursing homes, as well as for numerous fraternity
and sorority parties. Because of this, turnaround time is extremely important to us, and lately we
have become increasingly worried about the amount of time it takes to mix and cool our food
product.
As you may know from making gelatin at home, the preparation process is this: The gelatin
powder is added to a quantity of boiling water and stirred at the same temperature until it
dissolves. We have experimented with the time needed at this stage and have discovered that 2
minutes is the shortest amount of time we can take and still maintain the high quality of our
dessert.
The next stage is to remove the gelatin from the burner. At this point we add ice water (the same
amount as the boiling water) and wait for the dessert to gel. It is at this stage that we are hoping to
improve our efficiency. There are some people here who believe that the ice water should be
added immediately; others believe that it should be added bit-by-bit; and still others believe we
should wait an hour and then add the ice water.
No matter which of these procedures we use, national gelatin guidelines proscribe that one hour
after the gelatin is dissolved, we must place the entire mixture into a refrigerator and remove it
only again when gelled. Of course, the cooler it is by this point, the faster it gels. The
refrigeration temperature is also mandated, and so that can not be changed.
The question I have for you is: Given the procedure described above, in what manner should we
add the cold water to cool the mixture the fastest? I would appreciate an answer as soon as
possible, but hopefully no later than September 30, (our budget is due the day after).
Distance, Velocity, and AccelerationDear Calculus Student:
I am so incredibly sorry to be writing to you again; I know that this is a really busy time for you
but I hope that you can somehow find the time to help me once more. As you might have guessed
from the official letterhead, I'm in trouble with the law once again. I'm writing you from the
BCPD jail, where I'm being (yet again) held on suspicion of murder.
Here's how I got myself locked up this time: I went to visit the Absolutely Gorges, which have
long been famous for being such a beautiful natural wonder, and which in the last few years have
been famous because of the death of Bobo the Clown, who was shot out of a cannon and into the
canyon nearly two years ago. At first it was believed that his death was suicide, but then some
calculus students proved that the circus owner, Rick Rasterdly, murdered poor Bobo.
At any rate, the site of Bobo's death has become quite a tourist attraction, and so I took a Saturday
to go visit the place. If I'd known that my infamous ex, Jeremy, and his sleazy pal Sheriff Gocher
were going to be there, too, I never would have gone. Gocher says that I followed Jeremy there,
but I swear that's not true. In fact, I hardly saw him the whole time I was there, except for a few
seconds it's those few seconds that are causing all the ruckus.
What happened was I walked along the cliff-walk, like everyone else does. The cliff-walk is a
steel walkway that winds along the face of the canyon and goes right by the place where Bobo
met his end. It's pretty shaky and not very wide, and within minutes I was grateful I'm not afraid
of heights, because there's quite a drop. The walk runs more than 1000 feet below the top of the
gorge, but 3,280 feet above the floor! There's a bend in the cliff-wall right where Bobo hit; just
before that, there's a gate where people can, as the brochure says, "punch your ticket", which I
did. (There's a photo-copy of my ticket on the next page). Since the murderer was discovered by a
trick of timing, everyone is asked to synchronize their watches at this gate, but I wasn't wearing a
watch. You can't see around the bend from the gate, and so they use this as a natural place to
charge folks: you have to pay $3 to continue on.
So, I paid my $3, punched my ticket, and walked around the bend. I heard someone screaming,
and thought at first it was some kind of joke, but as I came up on the place where Bobo died, I
saw a person already falling down into the gorge, it was Jeremy, and he was the one screaming. I
didn't even know it was Jeremy at the time, I just stood there, gripping the railing, staring
awestruck. Suddenly Gocher ran toward me from further along the catwalk. He looked over the
edge, saw the body, saw that I was pale as a ghost, and arrested me.
Gocher says that I knew that he and Jeremy and some buddies were going to be there that day,
seeing the sights. He'd been with Jeremy until about a minute before his death, but then left
Jeremy behind in order to he catch up with the rest of the group. By now, he's quite pleased that
he's got me it's revenge for my escaping last time.
Jeremy's watch stopped at 1:54:09; the ticket puncher said that Jeremy had, along with Gocher
and everyone else, set his watch to the official clock. My ticket, as you can see, was punched at
1:53:57. This gives me a whole 12 seconds to have walked from the punch, around the curve, and
pushed Jeremy. At first I thought this proved I didn't do it, but when I counted out 12 seconds to
myself, I realized it's a pretty long time. I just don't know what to do!
I am so sick of this jail cell. Please do help to get me out! My preliminary hearing is November
11 (eleven-eleven: Deputy Dirk would have loved it, rest his soul!). If you could find a way to
clear me before then, I'd be eternally grateful!
VolumeDear Calculus Student:
In these times of great concern over the health of our planet, there are few people who do not
know about the overwhelming problem of waste disposal. We entrepreneurs at Eco-Sludge are
not immune to this concern; in fact, we are directly responsible for disposing of the waste byproducts for the larger part of the state of PU. Unfortunately, our former board of managers,
which has now been forcibly retired by the state's Environmental Agency, spent the greater part
of its energies on improving cost-effectiveness to the grave detriment of safety and health. As a
result, we are now faced with a expensive, and what is worse, dangerous cleaning project.
One of the first tasks facing us is estimating the total damage done. Your keen and piercing
professor, Dr. "Death" Crannell, told me of your resounding success with the Jumping Gelatin
Factory, and so I am being so bold as to ask you for help.
Our sludge is stored in large containers buried beneath the ground. These containers were
manufactured with ease-of-construction in mind, and are really nothing more than parabolic holes
in the ground, lined with concrete and covered with lids. (The problem is that the concrete cracks
because of the changes in temperature due to our erratic climate. The sludge has been seeping out
into the ground and making its way to the water supply. Needless to say, this is a horrendous state
of affairs for our local fishers as well as for any inhabitant wishing to drink water from the
faucet).
There are several measurements concerning the storage containers that we can make with ease.
Obviously, we can determine the diameter and the radius of the vat's opening with a large tape
measure. We have a plumbing rod that we can use to determine the depth of the container and
also the depth of the sludge (this is done much the same way you check the oil in your car: by
pulling the stick out and seeing how much of it is wet). According to specifications, each vat
should be 40 feet across at the top and 40 feet deep. In reality, we have found that the
measurements of the vats vary significantly, although the general parabolic shape does not. And,
to no one's surprise, there is neither rhyme nor reason to the depth of sludge found in a particular
vat.
What we can not easily determine is what we most need to know: the quantity (volume) of the
sludge in any given vat. We are fairly certain that there is, however, a mathematical way of
deriving this formula. We would be most grateful if you could help us with this, either in the
standard (40'x40') case, or even better, in the more general case.
Our report to the trustees must be mailed by November 6; could you please get the answer to us
before that date? We await your reply with eagerness.