AP Calculus AB
Course Overview
The purpose of this course is to (1) prepare students for success in collegiate
mathematics and (2) prepare students for success on the AP Calculus AB
exam. I believe that, as an instructor, if I take care of the former, the latter
will naturally follow.
Each exam in the course is cumulative. Over the course of a school year,
students will take 7 cumulative exams and a “mock AP exam” final. Exams in
the first semester consist of instructor written questions and multiple choice
items from past AP exams. Exams in the second semester consist of former
AP multiple choice and free response questions. The free response rubrics
are used in the assessment process.
Students are encouraged to write throughout the course (“where symbols
fail, words prevail”). Emphasis is placed upon making connections between
concepts and representations.
Text
Larson, Hostetler and Edwards. Calculus of a Single Variable: Eighth Edition.
Boston: Houghton Mifflin Company, 2006.
Supplementary Materials used in class
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Our course website: http://apcalculus.scottsbluff.schoolfusion.us
AP Calculus Multiple Choice Questions, 1969-1997
AP Calculus Free Response Questions, 1969-1978; 1979-1988; 19891998
AB Free Response Questions posted on apcentral (1999-2006)
APCD for Calculus AB
TI-SmartView software (a computer emulator of the TI-84)
TI-84 overhead calculator
Maple (computer algebra system by MapleSoft; single license for
presentation only)
Handouts created on comparing magnitudes of function growth,
conics, polar/parametric coordinate systems
www.calcchat.com Textbook site where students can check odd
numbered homework problems
Supplementary readings from various internet sources
Class Frequency
Our school conducts class in a traditional block schedule (8 total classes, 4
95 minute periods per day, class meets every other day). One month is
approximately 10 class meetings. Each day consists of one 95 minute block.
Graphing Calculators
Each student in class has their own graphing calculator, either a TI-83 or a
TI-84. Our school will receive grant money this summer to purchase
classroom sets of TI-84 calculators. In class lecture, I will use a TI-84
overhead unit for direct instruction on proper graphing calculator use.
Students are required to have a graphing calculator.
I purposefully do not teach nDeriv() or fnInt() commands until February; I
find this approach successful in that my students never mistake calculator
syntax for acceptable calculus notation. I instead teach the students to
graph the function, then use 2nd – TRACE and select either dy/dx or the
command to find the area under the curve.
Course Outline
[note: the numbers 1-10 are intended to list topics, not represent day 1,
day 2, etc.]
Unit I: Algebra Review and Introduction to Limits (approx. 4 days)
1. Algebraic connections between distance formula, Pythagorean
Theorem, trigonometric identities, and slope of the secant
line
2. Understanding the concept of a limit using tables, graphs, and
algebra
3. Existence of a limit
4. Understanding continuity/discontinuity through graphical and
analytical methods
5. Review properties of functions (domain, range, even, odd,
transformations, periodicity of trigonometric functions,
asymptotes)
6. Direct instruction in finding the zeroes of a function using a
a graphing utility
7. Predicting asymptotes graphically and verifying analytically
8. Review of oblique asymptotes and synthetic division
9. Evaluating limits involving infinity
10. Review of trigonometric applications
11. Establish fact that a graph is evidence rather than a conclusive
proof; analytic methods yield definitive results, particularly when
predicting local/global behavior
Unit II:
1.
2.
3.
4.
Introduction to Derivatives (approx. 10 days)
Finding zeroes of a function analytically and numerically
Connecting continuity to limits (concept of delta-epsilon definition)
Intermediate Value Theorem and its graphical implications
Derivative initially presented as
 Graphically: slope of the tangent line
 Numerically: as f (a ) at a given point
 Analytically: Limit of the difference quotient or as the slope
of the secant line when the two points used to compute
the slope are infinitely close together
 Verbally: instantaneous rate of change, velocity, etc.
5. Direct instruction in graphing a function on a graphing utility and
computing the numeric derivative at a point
6. Review of function composition, fundamental theorem of algebra,
binomial expansion, and factoring
 Derivation of the quadratic formula and its graphical and
analytic implications
 Descartes’ Rule of Signs for determining number of possible
zeroes for a given function
7. Local linearity and tangent line approximation to a function at a
given point
8. Vertical and horizontal tangent lines, as well as examples where
there are no tangents
9. Differentiation techniques: Power Rule, Sum Rule, Product Rule,
Quotient Rule, Chain Rule
10. Concept of dominance in limits involving infinity
Unit III: Applications of Derivatives (approx. 10 days)
1. Derivative as instantaneous rate of change
2. Derivative as the limit of an average rate of change (secant line)
3. Mean Value Theorem and its graphical and analytic implications
4. Relationship between continuity and differentiability
5. Determining rate of change from tables
6. Extreme Value Theorem and its implications
7. Direct instruction in graphing a function in an appropriate window
8. Estimating rate of change from the graph of a function
9. Curve sketching and corresponding characteristics between f and
f  by finding critical values
10. Relationship between monotonicity on an interval
(increasing/decreasing) and the sign of f 
11. Review of polar coordinates
12.
13.
14.
15.
Review of parametric functions and particle motion
Implicit differentiation; finding the slope of a line tangent to an
implicitly defined function at a given point
Equations containing derivatives and interpreting symbols in the
context of the problem
Related rates problems
Unit IV: Second Derivatives and Function Analysis (approx. 10 days)
1. Concept of concavity of f
2. Relationship between critical points and inflection points (how they
are found, how there is a sign change)
3. Corresponding characteristics of f , f  , and f 
4. Relationship between sign of second derivative and concavity
5. Points of inflection as a location where concavity changes with
many examples and counterexamples (no sign change)
6. Interpretation of velocity, speed, and acceleration in terms of the
graphical behavior of f , f  , and f 
7. Modeling rates of change; more challenging optimization problems
8. Optimization problems
9. Graphing slope fields and specific solution curves
10. Solving differential equations
11. Sketching solutions curves given a slope field
Unit V: In Depth Review of Transcendentals (Approx. 6 days)
1. Inverse Functions
2. Domain, Range, One-to-oneness, and restricting the domain of a
function so that it is one-to-one
3. Implicit differentiation to find the derivative of an inverse function
4. Comparing magnitudes of various functions and their rates of
change
5. Review of logarithmic, exponential functions
6. Derivatives of power, logarithmic, and exponential functions
7. Review of trig functions and special values
8. Review of inverse trigonometric functions and their derivatives
Unit VI: Integration and its Applications (Approx. 30 days)
1. Area under curve from a geometric perspective
2. Definite integral as the limit of a Riemann sum
3. Numerical approximations using Riemann sums: left, right,
midpoint, and trapezoidal rules from a graphical and analytic
perspective and from tables of values
4. Numerically approximate the definite integral using fnInt()
command
5. Definite integral as the change of a quantity over an interval and
evaluated used the Fundamental Theorem of Calculus
6. Basic principles of definite integrals: additivity, linearity, and taking
the opposite when the upper and lower limits are reversed
7. Integral as an accumulated rate of change
8. Relationship between displacement, velocity, and acceleration
functions presented as f , f  , and f  respectively in the context
of integration
9. Average value of a function on an interval
10. Problems involving particle motion along a line
11. Graphical and analytic analysis of functions using the
Fundamental Theorem of Calculus
12. Antiderivatives following directly from derivatives of basic
functions
13. Integration by substitution techniques
14. Integration problems involving change of limits
15. Finding a specific antiderivative given initial condition(s)
16. Solving simple separable differential equations
17. Modeling with differential equations, particularly those of the
18.
19.
20.
form y   ky
Computing the area of a region bounded by two curves
Finding the volume of a solid of revolution using disk, washer,
and cylindrical shells methods
Finding the volume of a solid with known cross sections
VII. Further Topics for Depth of Understanding (4 days)
1. Review of all integration rules
2. Integration by Parts and its connection to the product rule of
differentiation
3. Partial fraction decomposition with linear non-repeated factors,
repeated factors, and irreducible quadratic factors
4. L’Hopital’s Rule and indeterminate forms
VIII. Review for AP Exam (balance of time)
1. Cumulative review of topics and their connections, relationships
2. Shift in focus from periodic MC/FR questions from past AP exams to
a primary focus on MC/FR questions
3. Review of exam format
At the present time, the course is limited to senior students only. Our
seniors complete their last day of school the day before the AP Calculus
exam. Consequently, no activities are required for after the exam.
Student Activity
This problem is taken from the course text and adapted accordingly by the
instructor. The data set actually describes a portion of a logistic curve.
Though the logistic curve is not required in the AB course topics, students
should be aware of its existence, particularly if they plan to take a course on
differential equations in college.
Students are given a set of data describing the number of gypsy moth egg
clusters and the percentage of defoliation of a forest. Students must come
up with several models for the data, including a linear model, a quadratic
model, a cubic model, and an exponential model. If students choose to do
so, they may also compute a logistic model. Students will compare the
various models and verbally justify the model they believe best describes the
data set. In groups, students will try to persuade their classmates their
selection is the most appropriate.
Students will do research on the internet to determine the reasons why
defoliation is a problem for forest rangers and personnel. Based on the
evidence they gather online, students will determine a critical percentage of
defoliation from which the given forest will not recover.
Students will compute derivatives based on the selected model to determine
the rate of change of the number of egg clusters at a given time. Students
will interpret their results in the context of the problem. Based on their
selected figure, students will compute the critical number of egg masses that
will cause defoliation from which the forest cannot recover.
Students will find the horizontal asymptote for their model as the number of
egg masses increases indefinitely. Students will interpret this limit in the
context of the problem, paying particularly close attention to the slowing
rate of increase as the number of egg clusters increases.
This activity provides a great experience for students planning to continue
their mathematics education well beyond calculus.