AP Calculus AB Syllabus
2014 - 2015
GUTHRIE HIGH SCHOOL
2014/2015
Authored by: Tim Flanigan
AP Calculus AB Syllabus
2014 - 2015
Course Objective
This course focuses on developing the students’ understanding of the concepts of Calculus and providing experience
with its methods and applications. It will show students how applications of concepts relate to real world situations in
ways they can see and understand. Technology will be used regularly by students and teachers to reinforce the
relationships among the multiple representations of functions, to confirm written work, to implement
experimentation, and to assist in interpreting results. With these core concepts and the actual rigor of the course
work, students will be equipped with the skills necessary for success in post-secondary courses and a potential career
in the field of mathematics.
Student Goals
• Students should be able to work with functions represented in a variety of ways: graphical, numerical, analytical or
verbal. They should understand the connections among these representations.
• Students should 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.
• Students should 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.
• Students should understand the relationship between the derivative and the definite integral as expressed in both
parts of the Fundamental Theorem of Calculus.
• Students should be able to communicate mathematics and explain solutions to problems both verbally and in written
sentences.
• Students should be able to model a written description of a physical situation with a function, a differential equation
or an integral.
• Students should be able to use technology to help solve problems, experiment, interpret results and support
conclusions.
• Students should be able to determine the reasonableness of solutions, including sign, size, relative accuracy and
units of measurement.
• Students should develop an appreciation of calculus as a coherent body of knowledge and as a human
accomplishment.
Course Materials
Students are required to keep a 3-ring binder which will be used as a notebook holding organized notes, assignments,
and tests. Students will also be required to be prepared every day with a pencil and paper. It is also highly suggested
each students has his or her own graphing calculator. The TI-84 is recommended, but either the TI-83 or TI-89 are
acceptable alternatives. These calculators are still allowed on both the ACT and AP Calculus exams.
Textbook
Smith and Minton, CALCULUS – Early Transcendental Functions, 3rd Edition, McGraw-Hill 2007
1
Course Outline
Chapter 0 – Preliminaries for Calculus
2 weeks
Polynomials and Rational Functions
 Study of parent functions, transformations and compositions of functions, and other types of functions we
will be using throughout the course.
Graphing Calculators
 Evaluate a sketch in order to find extrema, intercepts, asymptotes and other relationships between functions
and their graphs.
Inverse Functions
 Review of inverses and evaluating them graphically, algebraically and by using composites.
Trigonometric and Inverse Trigonometric Functions
 Revisit the unit circle and how to use it as well as all 6 trigonometric functions, their graphs, properties and
main identities.
Exponential and Logarithmic Functions
 Examine what exponential functions are, how to recognize them, how to solve exponential equations and
properties of logarithms.
Chapter 0 Sample Activity: Students are given a lab ranging from “Path of a Baseball” to “Coin Toss” where they
must collect data. Then they must make predictions on what type of function the data models and sketch a graph
using graphing calculators. Students conclude by answering free response questions describing what the graph
represents and explaining the real-world meaning of certain values.
2 weeks
Tangent Lines and Length of Curves
 Review of tangent and secant lines for the sake of using those lines to estimate the length of a curve.
Concepts and Computation of Limits
 Define and analyze limits visually, then solve them by using a calculator or analytic calculations. We graph
functions on graphing calculators that have holes and/or asymptotes. We then use the ‘table of values’ utility
on the calculators to find values of points extremely close to non-existent values.
Continuity and Its Consequences
 Define continuity and different types of discontinuity, understanding how it affects functions and their
derivatives.
AP Calculus AB Syllabus | 2014/2015
Chapter 1 – Limits and Continuity
2
Limits Involving Infinity
 Graph various types of functions and use the zoom feature to better describe what graphs look like and
discuss different characteristics observable for each function. The students solve limits graphically or by
analytic calculations.
Formal Definition of a Limit
 Discover how limits can be used to find the derivative of a function at a point, which is also the slope of the
tangent line at the same point.
Chapter 1 Sample Activity: Students are given several functions to graph on their graphing calculators. After
sketching the graphs and writing a description of the function behavior near a given value, they find limits
analytically and use the ‘table’ to verify their conclusions.
Chapter 2 – Differentiation
5 weeks
Tangent Lines and Velocity
 Evaluate lines tangent to a curve and how that relates to instantaneous velocity at the point.
Derivatives
 Learn the definition for a derivative and use limits to find the derivative function. We will discuss the
relationship between a function and its derivative function (i.e. when a function is decreasing the derivative
is negative).
The Power Rule
 Practice differentiating by using the Power Rule on specific problems.
The Product and Quotient Rule
 Use the Product or Quotient Rules to differentiate and know the difference between the two.
The Chain Rule
 Use candy to introduce this complex technique as we must differentiate each layer of candy.
Trigonometric Functions
 Study the derivatives of transcendental functions and practice problems involving them.
Exponential and Logarithmic Functions
 Study the derivatives of transcendental functions and practice problems involving them.
Implicit Differentiation and Inverse Trigonometric Functions
 Study the derivatives of transcendental functions and practice problems involving them.
3
The Mean Value Theorem
 Study what the Mean Value Theorem is, how to use it, and how to relate it to real world occurrences.
Chapter 2 Sample Activity: As an introduction to the derivative, students will sketch various functions in the
standard window and then zoom in at a given point to discover the behavior of the function at that point. If there
is local linearity, then they will estimate the slope of the line and write an equation for it. In this chapter, we use
this to connect the idea of derivative to the slope of the tangent line and velocity.
Chapter 3 – Applications of Differentiation
6 weeks
Maximum and Minimum Values
 Find local and absolute maximum and minimum values through analytical use of calculus and a graphing
calculator.
Increasing and Decreasing Functions
 Study trends of derivatives for increasing and decreasing functions.
Concavity and Second Derivative Test
 How we can determine the concavity of a function based on the second derivative of the function and any
extrema.
Curve Sketching and Optimization
 Learn how to use Algebra, Calculus and graphing calculators to model real world situations and to optimize
those situations.
 Follow the 3 step process. First, draw a picture and label variables and constants. Identify the given rate of
change, rate to be found and when to find it. Second, choose an equation that relates the variables. Eliminate
variables that don’t have matching rates. Finally, differentiate both sides with respect to time. Substitute and
solve, but don’t forget correct units!
Rates of Change in Economics and the Sciences
 Answer free response questions from past AP exams.
Chapter 3 Sample Activity: Students review the relationships between f, f’ and f” by answering questions such as
“if f’ is positive, then f is ____________”. They use the relationships to match graphs of several functions to the
graphs of their first and second derivatives. After comparing the various graphs, students sketch possible graphs
of f and f” given a graph of f’.
AP Calculus AB Syllabus | 2014/2015
Related Rates
4
Chapter 4 – Integration
4 weeks
Antiderivatives
 Introduce integration by studying antiderivatives of given functions.
Sums and Sigma Notation
 Graphical representations of antiderivatives and development of Mathematical Induction.
Area
 Discuss how to find area under curves by using Geometric figures and Riemann Sums.
The Definite Integral
 Explore what a definite integral is, what it can be applied to and how it was derived from infinite Riemann
Sums, plus how to solve problems involving a definite integral.
Fundamental Theorem
 Prove the Fundamental Theorem through induction.
Substitution and Numerical Integration
 Study of ‘u-substitution’, ‘Midpoint Rule’ or ‘Trapezoidal Rule’ so that we may evaluate integrals for which
no elementary antiderivative exists.
Natural Logarithms
 Examine Natural Logarithms as integrals, whereby ln x can be thought of in terms of area.
Chapter 4 Sample Activity: To explore the relationships between f, f’ and f”, we play the “Leap Frog Game”
using 14 laminated cards explaining different behavior for functions (i.e. is increasing, is concave up, has a local
minimum, etc.). Students are read a statement from a Power Point presentation about the behavior of a function,
its derivative or its second derivative. They must use two cards to validate their understanding of the behavior of
the other two functions in order to “leap” others in a race around the class.
Chapter 5 – Applications of the Definite Integral
3 weeks
Area Between Curves
 Demonstrate the versatility of the integral by exploring the area bounded between two curves.
Volume: The Disk, Washer and Shell Method
 Use of integrals and cross sections to compute the volume of a three-dimensional solid.
5
Chapter 5 Sample Activity: Divide students into small groups, each group is given an expandable
holiday decoration. After dividing the flat shape into interval widths, each member of the group
approximates the area of the shape using any summing process they choose. Students then cut their
shape into strips and unfold the strip to make a disk. Students find the volume of their own disk and
then stack the disks to reveal the whole decoration fully expanded.
Chapter 6 – Integration Techniques
5 weeks
Basic Integration Rules
 Examine antiderivatives following directly from derivatives of basic functions as well as by substitution of
variables (including change of limits for definite integrals).
Integration by Parts (Calculus BC)
 Learn the process for integration by parts, where the formula comes from, and when it is necessary or
beneficial to use it.
Trigonometric Integrals
 Review antiderivatives following directly from derivatives of trigonometric functions and learn 9 basic
integration formulas.
Chapter 6 Sample Activity: Students collect data for various decay models. The students graph the data, use their
calculators to find regression models for the data, and then use their models to predict future behavior.
AP Exam Preparation
2 weeks
AP Multiple Choice
AP Free Response Questions
 Review major concepts of Calculus, how to solve problems and, most importantly, how to justify answers.
Post AP Exam
2 weeks
Vectors in a Plane
 Analyze direction and magnitude of a vector along with Theorems associated with real world applications
such as wind tunnels, aviation and physics.
AP Calculus AB Syllabus | 2014/2015
 Review test taking strategies along with how to solve Calculus problems analytically and with and without
the use of a calculator.
6
Modeling Projectile Motion
 Study relationships between position, speed and acceleration within the context of Calculus and Newton’s
Laws of Motion.
Grade Calculation
There will be 1 to 2 quizzes per chapter and a test at the completion of every chapter, along with daily
assignments. Grades will be calculated as follows:
TESTS = 100 points each
FINAL = 200 points (exempt if take AP Test)
QUIZZES AND PROJECTS = 50 points each
BOOK WORK = 10 points each
WORKSHEETS = misc. points
Resources and Supplemental Materials
Numerous resources are used to supplement the textbook and enhance the course. Problems from other calculus
textbooks are added to assignments or used in openers for discussion. Additional resources include items found at the
AP Central website, on the AP electronic discussion group, and materials and ideas obtained at the Advanced
Placement Summer Institute of 2011. Additional projects are as follows:
Rocket Project:
While examining the relationship between position, velocity, and acceleration, we will enrich our understanding of
these concepts by building, launching, timing free-fall, and determining the height and velocity of our rockets at
specific times by using trigonometry or the known constant of gravitational acceleration and both differentiation and
anti-differentiation.
M&M’s and Peanut M&M’s
When introducing the Chain Rule, we will break down the structure of M&M’s and Peanut M&M’s and relate that to
the process of finding the derivative or general anti-derivative of higher order functions.
How to Create the Biggest Box:
When exploring optimization we will create boxes from limited materials. We will then create scatter plots and use
appropriate regression to find the estimated dimensions to create the largest volume a box could have from a set
amount of material. We will then use concepts of finding relative and absolute extrema to determine the exact
dimensions of the box with the maximum volume and compare our previous results.
Car Drive:
Students will be asked to write down the velocity of vehicle during a 20 minute ride, every minute, and write down
the odometer reading before and after the trip, or use the trip odometer. Then they will use concepts of Area and
Riemann sums to estimate total distance traveled and compare it to the odometer readings.
7