AP CALCULUS BC
DESCRIPTION OF COURSE
This math course is not separate from AP Calculus AB nor is it a brief extension of the AB
course. It is a continuation of the analysis of function behavior using limits, derivatives and
integrals; investigating function behavior based on an instant, an interval, and infinitely; and
using like behaved functions to approximate and determine the level of accuracy of the
approximation. This course in conjunction with the AB course will develop a solid and complete
understanding of single-variable calculus concepts through numeric, algebraic, tabular and
graphical exercises and explorations. The calculator will be an integral part of discovering and
verifying the information generated by calculus procedures and tests. Each student will have a
TI-83or84 graphing calculator to use at school and at home.
DESCRITION OF COURSE TIMELINE
This BC Calculus Course is designed to occur the second semester of the school year in addition to an
ongoing full year AB Calculus Course. The AB Course has been Audit Approved by the AP College Board.
The AB Course begins in August and will cover:
Functions, families of functions, composites and transformations, graphing with the calculator
The tangent curve problem
The limit process using algebra, graphs, and tables, limit laws and calculations, formal definitions,
Continuity (formal and intuitive), velocity and rates of change, the definition of the derivative, graphical
relationships between the derivative and its function, differentiation methods and formulas,
Applications of the derivative, rate of change
Trigonometric derivatives, chain rule, implicit differentiation, higher order derivatives
Graphical relationships between the function, 1st derivative and 2nd derivative
Related rates
Differentials and linear approximations
Maximum and minimum values
Rolle’s Theorem and The Mean Value Theorem
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Using derivatives to shape the graph
Limits as x approaches infinity
Horizontal, vertical and slant asymptotes
Optimization problems including business and economics
Newton’s method
Antiderivatives and indefinite integrals
Riemann sums and area
APPROXIMATE BEGINNING OF THE BC CALCULUS COURSE (January)
Definite integrals
The Fundamental Theorem of Calculus
Properties of the definite integral and the substitution method
Area between curves, volumes of known cross-sections, volumes of revolution by the disk, washer and
shell method
Work problems
Average value
Inverse functions interpreted graphically and algebraically
Natural and general, logarithmic and exponential functions
Exponential growth and decay
Inverse trigonometric functions
Indeterminate forms and L’Hospital’s Rule
Integration by parts
Numerical integration using trapezoidal rule and Simpson’s rule
Slope fields and differential equations
AP CALCULUS BC COURSE TOPICAL OUTLINE (occurs in addition to 2nd semester
AP CALCULUS AB Course)
Unit 1 -5: Quick Review (6 days)
Limits and properties, differentiation, applications of differentiation, and integration
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Sample Activities: An exploration of limits involving infinity that uses tables, graphs and then
functions; A lab using “zooming in” to approximate the derivative at a point; Free Response
Question BC 5 Implicit Differentiation; Free Response BC 4 Written Interpretation of 1st and 2nd
derivative; Free Response BC 3 Trapezoidal Approximation from a table
Sample questions after an exploration or lab: In your own words, what is meant by “limit”?
When can you find the limit by substituting a value for x? Under what conditions does a limit
not exist? How does a limit relate to calculating a derivative, an area bounded by a curve and a
volume of revolution?
Unit 12: Infinite Sequences and Series Part I (10 days)
Sequences, convergence, properties, Squeeze Theorem
Defining an infinite series as a sequence of partial sums
Properties of geometric, harmonic, P-series, and alternating series
Writing a power series expanded about x = 0, x = a
Term-by-term differentiation and integration to find a power series of new functions
Taylor Series and Maclaurin Series (and polynomials)
Sample Activities: An exploration of a geometric series using the sum and sequence commands
on the calculator and the discovery of convergence or divergence; An exploration that uses a
power series for a definite integral; A lab using the calculator to see the interval of convergence
Sample questions after an exploration or lab: Using the graphing feature on the calculator,
what conjecture can be made about the elementary function that fits the series for P(x) when x
is not too far from zero? Provide numerical and graphical evidence that your conjecture is
correct. After writing the fourth-degree Taylor Polynomial, P(x), for g(x) = f(x + 3) about x = 0.
Use P to explain why g must have a relative minimum at x = 0.
Unit 5: Integrals (6 days)
Numerical approximations of a definite integral
-Riemann sums- left, right and midpoint
-Trapezoid rule
-How each technique improves if the number of subdivisions is doubled, tripled, or
multiplied by a factor of k
Sample Activities: A lab comparing the rectangular and trapezoidal methods with the definite
integral while increasing the number of subdivisions; Free Response AB 3 Midpoint
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Approximation from graph and table; Free Response BC 4 Form B Trapezoidal approximation
from a graph; Free Response BC 5 Right Riemann Sum from table
Sample questions after an exploration or lab: Explain why the Riemann sum is between the
corresponding lower sum and upper sum. Why can you conclude that the limit of the Riemann
sum is exactly equal to the integral?
Unit 7: Inverse Functions, Indeterminate Forms and L’Hospital’s Rule (5 days)
Differentiation and integration of inverse functions
Using L’Hospital’s Rule
Sample Activities: Since this is a review for the AB students, the activities will be skill based
reviews of trigonometric, exponential, and logarithmic functions and their inverses along with
differentiation and integration; Free Response AB 3 Particle Motion involving inverse tangent
and an exponential; Free Response BC 1 Revolution problem involving an exponential and
cosine; An exploration using parametric equations and the calculator to verify L’Hospital’s Rule
Sample questions after an exploration or lab: Tell why the parametric chain rule fails to give a
value for dy/dx at this point. What does dy/dx seem to be approaching as t approaches /3?
Show that your conjecture for the limit of dy/dx agrees with the value found by l’Hospital’s
Rule.
Unit 8: Techniques of Integration (6 days)
Review of integration by parts and substitution
Integration of rational functions by partial fractions
Approximate integration
Improper integrals
Sample Activities: An exploration of distance approaching a limit as an endpoint of a velocity
graph approaches infinity; Free Response BC 4 Interpret Integration given a graph; Free
Response BC 6 Antidifferentiation by Partial Fractions
Sample questions after an exploration or lab: How do you reconcile the fact that this distance
approaches a limit, and the velocity never reaches zero? Which technique is needed to
evaluate the antiderivative?
Unit 10: Differential Equations (10 days)
Slope fields
Separable differential equations
Euler’s Method
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Exponential growth and decay
Logistic growth
Sample Activities: An Exploration of slope fields and Euler’s method using the population of
roadrunners, Generating slope fields manually and on the calculator; Free Response BC Euler’s
method –graph of slope field given; Free Response BC 4 Euler’s Method –includes a 2nd degree
Taylor polynomial
Sample questions after an exploration or lab: For which values of x does the numerical solution
by Euler’s method seem to follow the slope field? For which values of x is the solution clearly
wrong? Explain why Euler’s method gives meaningless answers for larger values of x.
Unit 9 and 11: Further Applications of Integration/Parametric Equations and Polar
Coordinates (12 days)
Arc length
Curves defined by parametric equations
Parametric equations and calculus
Parametric equations and vectors: motion, position, velocity, acceleration, speed, distance
traveled
Analyze curves given in parametric and vector form
Polar coordinates and polar graphs
Analyze curves given in polar form
Area of a region bounded by polar curves
Sample Activities: An exploration calculating the length of a curve using geometry and then
calculus; An exploration finding the area of an ellipse from its polar form; Free Response BC 1
Form B Length of a curve; Free Response BC 3 Find the area of a polar region, particle motion
along that curve and the interpretation of dy/dt;
Sample questions after an exploration or lab: What relationship does the vector have to the
graph? Is the pendulum speeding up or slowing down? How can you tell? Do these two
particles collide and why? Explain the relationship between velocity and speed.
Unit 12: Infinite Sequences and Series Part II (14 Days)
Review of geometric, harmonic, alternating, and power series, Taylor and Maclaurin series
Investigation of divergence, convergence, radius of convergence and interval of convergence
Lagrange form of the remainder
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Tests for convergence/divergence
-nth term test
-geometric series test, ratio test
-P-series test
-alternating series test
-integral test
-comparison test
-limit comparison
Error Analysis for Series
-convergent geometric series
-using integral test
-convergent alternating series
-Lagrange
Sample Activities: Power Series development worksheets designed by Monique Morton; An
exploration of the Ratio Technique; An exploration using improper integrals to test for
convergence; AP Calculus infinite series professional development manual notes, investigations,
and problems will be used; Free Response BC 3 Taylor Series, radius of convergence and error
analysis; Free Response BC 3 like previous problem but from a table; Free Response BC 6 like
previous problem but includes interval of convergence and points of inflection
Sample questions after an exploration or lab: Explain why this sequence of partial sums is
increasing, although the terms themselves are decreasing? Create your own chart of
convergence tests indicating how you will identify which test is required. In your own words,
what is an error bound.
Approximately 10 days to review for AP Calculus BC Exam
After the AP Exam:
Unit 7: Hyperbolic Functions
Unit 11: Conic Sections in Polar Coordinates
Review for Final (only over BC outlined material)
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PRIMARY TEXTBOOK AND OTHER RESOURCES
Stewart, James. Single Variable Calculus, Sixth Edition. Belmont, California: Thompson
Brooks/Cole, 2008.
Foerster, Paul A. 1998. Calculus Explorations. Emeryville, California: Key Curriculum Press.
Materials provided by the AP Calculus BC workshop, Presenter: Monique Morton, Western
Kentucky University, 2010.
CLASSROOM STRATEGIES AND ASSESSMENT
1. Students will work in a cycle of individual investigation and then collaboratively share different
methods and interpretations. (facilitated by the teacher)
2. After discovery and exploration, students will attempt sets of problems that progress through
increasing levels of difficulty, various presentations of the material, and increased amounts of
incorporated material.
3. Frequently, students will be asked to write an explanation of how and when to apply a new
concept or procedure.
4. Assessment will include multiple choice and open response problems over a two day period.
The calculator will be involved on the second day. Some assessments will be taken at home.
All of these will include written explanations and justification and will cover any material already
presented in the AB or BC course.
5. In the case of SNOW DAYS: Students are responsible to complete assignments posted online on
a daily basis till schools resumes.
6. Students will have three opportunities to attend AP study sessions at a local high school.
Attending one of these events is required. Attending all three is highly recommended. Students
will be awarded points for participation.
7. Students can attend before and/or after school “HELP” sessions. A sign-up sheet will be posted
on the door each week. Students can do corrections for improved test scores. Corrections must
include the incorrect problem, a corrected version of the problem showing all work, an
explanation of where the process fell apart and how to perform the necessary procedures.
Students entering Calculus AB and/or BC are about to continue an unending journey in mathematics.
While this is a great accomplishment in high school curriculum, it is only the beginning of the possible
mathematical studies at the college level and beyond. It is with great expectation that I implore each
student to work diligently, think with different analytical perspectives, and seek knowledge and
justification.
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