AP Calculus AB
SUMMER SYLLABUS
Instructor: Krista Bretz
Email: [email protected]
PREREQUISITES
Before studying calculus, all students should complete four years of secondary
mathematics designed for college-bound students: courses in which they study
algebra, geometry, trigonometry, analytic geometry, and elementary functions. These
functions include linear, polynomial, rational, exponential, logarithmic,
trigonometric, inverse trigonometric, and piecewise-defined functions. In particular,
before studying calculus, students must be familiar with the properties of functions,
the algebra of functions, and the graphs of functions. Students must also understand
the language of functions (domain and range, odd and even, periodic, symmetry,
zeros, intercepts, and so on) and know the values of the trigonometric functions at
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the numbers 0, , , , ,  and their multiples.
6 4 3 2
COURSE DESCRIPTION
Students come to AP Calculus AB with a background of algebra, geometry,
trigonometry, and pre-calculus. They must perform well in those classes to take the
AP Calculus AB course. We do spend a little time reviewing topics from these courses
before beginning any calculus. The students take this class for two semesters, leaving
about two to three weeks for review before the AP exam. I give students a test for
each chapter in which there is a calculator and non-calculator portion. In addition, I
give some short to medium length quizzes throughout each chapter. Practice AP free
response and multiple choice questions are given throughout the course.
Calculus AB is primarily concerned with developing the students’ understanding of
the concepts of calculus and providing experience with its methods and applications.
The courses emphasize a multi-representational approach to calculus, with concepts,
results, and problems being expressed graphically, numerically, analytically, and
verbally. The connections among these representations also are important. First
semester is devoted to limits and derivatives; second semester is devoted to integrals.
The course is the equivalent of 1st semester college calculus but also covers
approximately 60% of 2nd semester college calculus.
The course 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. Many colleges provide statements
regarding their AP policies in their catalogs and on their websites. This website:
https://apstudent.collegeboard.org/creditandplacement/search-credit-policies
provided by AP Central, allows students to enter their prospective college and see the
college’s Advanced Placement policies and scores needed to earn college credit.
SEMESTER ONE AND TWO – UNITS OF STUDY
Primary Textbook
Daily Lessons and Assessments for AP Calculus AB, A Complete Course, Mark
Sparks, 2012.
Summer Pre-Requisites
At Battle Mountain High School, the pre-requisite course for AP Calculus AB is
Pre-Calculus/Trigonometry. I begin the year by reviewing basic pre-calculus facts
and all of the basic types of functions that students need to know by equation, graph,
name, and their characteristics.
It is imperative that students have a solid foundation in functions, their
graphs, and their properties. This includes: ability to sketch a graph of the function,
with transformations, without the use of a calculator; knowing where the holes and
asymptotes of the function exist; knowing end behavior; knowing how to solve the
function for its inverse function; and being able to visualize the graph in order to
graph its derivative.
The pre-requisites of this course are mostly covered in IM1, IM2, and IM3.
However, some students will bypass taking pre-calculus/trigonometry and enter AP
Calculus AB with an advanced Accuplacer score. These students (and others in
pre-calculus if their knowledge is not strong) need to know the UNIT CIRCLE,
coordinates, radians, and degrees of the unit circle, and the graphs and
properties of the sine, cosine, and tangent functions. During your summer,
please, please, please, review these concepts. Watch videos on Khan academy, print
out a unit circle and commit the x and y coordinates to memory for all radians and
degrees of the unit circle. This will make both semesters much easier if you have this
knowledge before starting the school year.
Course Planner
Chapters follow the textbook. We have three 52 minute classes a week and one
95 minute class per week.
Unit 0: Prerequisites for Calculus (14 days)
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Lines
Functions and graphs
Quadratic and Rational Functions
Exponential Functions
Logarithmic Functions
Trigonometric Functions
Using the TI-84 Graphing Calculator
Unit 1: Limits and Continuity (14-20 days)
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Rates of change and limits
Limits involving infinity
Continuity
Intermediate Value Theorem
Unit 2: Understanding the Concept of the Derivative (22 days)
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Derivative of a function
Differentiability
Equation of Tangent and Normal Lines
Graphing Calculator and Derivatives Techniques
Unit 3: Rules of Differentiation (30-40 days)
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Rules for Differentiation
Product, Quotient, and Chain Rules
Velocity and other rates of change
Derivatives of trigonometric and inverse trigonometric functions
Implicit differentiation
Derivatives of exponential and logarithmic functions
Unit 4: Applications of Derivative – Part I (10 days)
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Unit
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Extreme values of functions
Mean Value Theorem
Rolle’s Theorem
L’Hopital’s Rule
Connecting f’ and f’’ with the graph of f
5 – Application of the Derivative – Part II (18 days)
Modeling and optimization
Linearization and Newton’s method
Differentials and change
Related rates
Unit 6: Basic Integration and Applications (15 days)
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Estimating with finite sums (RAM—Rectangular approximation method)
Riemann sums
Definite integrals and area, volume, and average value of a function
Definite integrals and antiderivatives
Fundamental theorem of calculus
Trapezoidal rule and Simpson’s rule
Unit 7A: Differential Equations and Mathematical Modeling (Techniques of
Integration) (15 days)
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Slope fields and Euler’s method
Antiderivatives and the indefinite integral
Antidifferentiation by substitution, trigonometric substitution, and partial
fractions
Separable differential equations
Exponential growth and decay
Logistic growth
Unit 7B: Applications of Definite Integrals (16 days)
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Integral as net change
Areas in the plane
Volumes
Cross Sections
Applications from science and statistics
Review for the AP Exam (10 days)
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We do a full mock exam which will require a commitment of 1 ½ hours outside
of class time to take the mock exam.
We work many problems and do practice exams from the Princeton Review
book (Kahn, David S., Cracking the AP Calculus AB & BC Exams, 2015 Edition,
C. 2014 by The Princeton Review, Inc., Framingham, MA)
Teaching Methods
We cover everything on the AP Calculus AB syllabus in this course.
I prefer to work all the problem sets by hand, then write out my notes based on
the textbook and problems I want the students to be able to do. I post all my notes
which are written on the Smartboard on Schoology. Every day notes are exported as
pdf files and posted so any student missing class should look here first. These
problems are the backbone of the class…students learn not only how to work out the
mathematics but also how to communicate by writing out their justifications using
complete sentences.
I believe that these kids, on the cusp of being college students, have the
maturity to study on their own with little help from me. I can trust them to do their
work and ask questions when they get stuck…in fact, the onus is on them to do so at
this point in their lives. However, they do benefit from having much more direct
contact with me than they are likely to get in college. When time allows, I give the
students class time to work on their assignments, and they do an excellent job of
forming groups or working individually as they prefer.
Writing notes on the whiteboard while explaining them verbally allows my
students to digest the material at their own pace, whether immediately during lecture
or later when re-reading their notes. I try to teach in a way that appears/feels
friendly and informal but which pays serious attention to detail and uses highly
formal language and algorithms. I also tend to start class with the new lesson and
answer questions on previous lessons later in class as time allows.
Technology Use
I use Geometer’s Sketchpad occasionally to demonstrate a concept such as
slope of a tangent line, local linearity, or other graphical analysis. I also use the
Internet to occasionally find and show a concept demonstration someone else has
published. I make frequent use of the TI-84 graphing calculators when appropriate
but encourage non-calculator solutions whenever possible. The calculator is used as
a tool to experiment, solve problems, and justify conclusions, and I make the
students strongly aware of its limitations. It is an excellent tool for numerically or
graphically finding limits and for exploring continuity or for doing simple
programming which is useful for Newton’s and Euler’s methods. Students become
capable of graphing all the familiar functions (including absolute value, radical,
polynomial, exponential, logarithmic, step, trigonometric, etc.) both by hand when
they are simple or on the calculator when they are not. We never stray far from the
methods used for centuries by mathematicians, such as estimation, table
construction and analysis, Riemann sums, and hand calculation of areas under
curves.
Throughout the course I point out that there are verbal, graphical, analytical,
and numerical ways to represent and solve functions. For some of these the
calculator is invaluable. Students use the integral, derivative, graphing, and table
functions on the TI-84 throughout the course. These all provide a shortcut to finding
information but are often the only way to solve a particular problem.
I encourage my students to experiment with their calculators. Several
examples: they may start with an unfamiliar function and find the window for its
graph using the table feature. They could estimate the value of a definite integral or
area under a curve using the numerical integration feature. They might zoom in or
out to see local linearity or other relevant facts (extrema, limits) about the graph. We
might use the various programs available for download (such as Riemann sums or
slope fields) on the TI-84 to see visual depictions of important concepts. They could
verify results (such as whether they properly performed a derivative or definite
integral) and explore by getting numerical results for difficult expressions or using
the trace function to find what y value a function approaches as x approaches +/infinity or a particular value of x from the left or the right. All of these methods are
also useful for interpreting results and supporting conclusions. For example, a graph
often shows a great deal of information about a function. Pointing out its limits,
areas under curves, and extrema often allows students to interpret what the function
is doing and thereby support their conclusions.
I expect students to be able to justify their work using verbal, graphical,
numerical, or analytical arguments. I constantly model the proper use of
mathematical symbols and written sentences in explaining/justifying my work. If a
student uses a graph from a calculator, I require them to sketch the graph and label
its window when making their justification. I also take pains to work on their use of
language so they can be perfectly clear about what they mean (such as “f is
increasing on the interval” as opposed to “it goes up”).
Student Assessment
I give chapter tests and quizzes (including a quiz at the beginning of the year
over memorized trigonometric basic facts from pre-calculus), plus a midterm (final
exam) in December (a partial AP Calculus AB exam) and a final exam (a complete AP
Calculus AB exam) in the spring.
Tests are written to mirror the difficulty, grading style, and types of problems
on the AP exam at the end of the year. Specifically, I try to give 2 free response
questions (one calculator active, one not), multiple choice non-calculator questions,
and multiple-choice calculator active questions. This adds up to approximately one
hour of expected exam time, which is near the length of our class period, and is
similar to the pace of the AP exam. When possible, I use old exam questions on all
parts. I attempt to mirror AP grading on free response questions which is easiest
when using old exams which have rubrics provided. Using this method, students
gain invaluable experience with time management, how to use English to justify
conclusions on free response questions, how to address multiple-choice questions,
and how AP grading works in practice.
Web Resources
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Kahoot.it
Princeton Review website
Naviance – AP Prep
AP Central (apcentral.collegeboard.com)
Khan Academy
Wolframalpha.com
Any website that explains Calculus concepts in laymen’s terms.