Course Title: AP Calculus BC
Course Description: Advanced Placement Calculus BC is intended for students who have a thorough knowledge of analytic geometry and elementary functions in addition to college preparatory algebra, geometry and trigonometry. The functions include linear, polynomial, exponential, rational, logarithmic, trigonometric, inverse trigonometric and piecewise defined. 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, symmetric, zeroes, intercepts, and so on.)” (CollegeBoard AP Calculus AB and BC Course Description Acorn Book.)
Upon completion of the course, the student is prepared to take the BC Advanced Placement test in Calculus, which provides eligibility for two semesters of credit in college calculus.
Teaching Strategies
1. Each topic is presented numerically, analytically, graphically,
and verbally as students learn to communicate the connections
among these representations.
2. Justifications of responses and solutions are part of the routine
when solving problems. Students are encouraged to express their
ideas in carefully written sentences that validate their process
and conclusions.
3. Students make extensive use of the TI-84 calculator.
Each student has his or her own calculator.
4. Students use their calculators to:
(a) Analyze graphs for zeroes (solve numerically)
(b) Analyze graphs for points of inflection, relative maxima
and minima
(c) Evaluate derivatives numerically
(d) Perform numerical integration for a definite integral
(e) Compute partial sums
5. From the middle of September throughout the rest of the year,
students are assigned free-response questions from AP
Released Exams. These questions are graded as they would
be at an AP Reading. Students may use a calculator for any
question for which a calculator was allowed when the question
appeared on the exam, and they may not use a calculator for
any question for which a calculator was not allowed when the
question appeared on the exam.
6. Students have reading quizzes concerning examples from the book.
Calculator usage is the same as described above.
7. All tests contain material from previous units. Students are
responsible for all material covered to the date of test.
Many tests are two parts, one with calculator usage
and one without.
8. Students are encouraged to work cooperatively on in-class
worksheets, graded AP problems, and take-home exams.
9. Circular functions, exponential functions, and logarithmic functions
are used throughout the course. Students have previously studied
these functions, so we deal with the derivatives of these functions
early in the course.
AP Calculus BC
Textbook: Larson, Ron, Robert P. Hostetler, and Bruce H. Edwards.
Calculus with Analytic Geometry. 6th ed. Boston:
Houghton Mifflin, 1998.
Chapter 1: Limits and Their Properties (summer work, 1 test)
• An introduction to limits, including an intuitive understanding
of the limit process
• Using graphs and tables of data to determine limits
• Properties of limits
• Algebraic techniques for evaluating limits
• Comparing relative magnitudes of functions
and their rates of change
• Continuity and one-sided limits
• Geometric understanding of the graphs of continuous functions
• Intermediate Value Theorem
• Infinite limits
• Using limits to find the asymptotes of a function
Chapter 2: Differentiation (2-2.5 weeks, 1 test, daily quizzes)
• Understanding of the derivative: graphically,
numerically, and analytically
• Approximating rates of change from graphs
and tables of data
• The derivative as: the limit of the average rate of change,
an instantaneous rate of change, limit of the difference quotient,
and the slope of a curve at a point
• The meaning of the derivative—translating verbal descriptions
into equations and vice versa
• The relationship between differentiability and continuity
• Functions that have a vertical tangent at a point
• Functions that have a point on which there is no tangent
• Differentiation rules for basic functions, including power
functions and trigonometric functions
• Rules of differentiation for sums, differences,
products, and quotients
• The chain rule
• Implicit differentiation
• Related rates
Chapter 3: Applications of Differentiation
(3-3.5 weeks, 1 test, daily quizzes)
• Extrema on an interval and the Extreme Value Theorem
• Rolle’s Theorem and the Mean Value Theorem,
and their geometric consequences
• Increasing and decreasing functions and
the First Derivative Test
• Concavity and its relationship to the first
and second derivatives
• Second Derivative Test
• Limits at infinity
• A summary of curve sketching—using geometric and
analytic information as well as calculus to predict the
behavior of a function
• Relating the graphs of f, f’’ and f’’’
• Optimization including both relative and absolute extrema
• Tangent line to a curve and linear approximations
• Application problems including position, velocity,
acceleration, and rectilinear motion
Chapter 4: Integration
(3-3.5 weeks, 1 test, daily quizzes; cumulative exam)
• Antiderivatives and indefinite integration, including
antiderivatives following directly from derivatives of basic functions
• Basic properties of the definite integral
• Area under a curve
• Meaning of the definite integral
• Definite integral as a limit of Riemann sums
• Riemann sums, including left, right, and midpoint sums
• Trapezoidal sums
• Use of Riemann sums and trapezoidal sums to approximate
definite integrals of functions that are represented analytically,
graphically, and by tables of data
• Use of the First Fundamental Theorem to evaluate
definite integrals
• Use of substitution of variables to evaluate definite integrals
• Integration by substitution
• The Second Fundamental Theorem of Calculus and functions
defined by integrals
• The Mean Value Theorem for Integrals and the average
value of a function
Chapter 5: Logarithmic, Exponential, and Other Transcendental
Functions (4-4.5 weeks, 1 test, daily quizzes)
• The natural logarithmic function and differentiation
• The natural logarithmic function and integration
• Inverse functions
• Exponential functions: differentiation and integration
• Bases other than e and applications
• Solving separable differential equations
• Applications of differential equations in modeling,
including exponential growth
• Use of slope fields to interpret a differential
equation geometrically
• Drawing slope fields and solution curves for differential equations
• Euler’s method as a numerical solution of a differential equation
• Inverse trig functions and differentiation
• Inverse trig functions and integration
Chapter 6: Applications of Integration
(4-4.5 weeks, 1 test, daily quizzes; cumulative exam;
semester final)
• The integral as an accumulator of rates of change
• Area of a region between two curves
• Volume of a solid with known cross sections
• Volume of solids of revolution
• Arc length
• Applications of integration in physical, biological,
and economic contexts
• Applications of integration in problems involving a particle
moving along a line, including the use of the definite integral
with an initial condition and using the definite integral to find
the distance traveled by a particle along a line
Chapter 7: Integration Techniques, L’Hopital’s Rule, and Improper
Integrals (2-2.5 weeks, 1 test, daily quizzes)
• Review of basic integration rules
• Integration by parts
• Trigonometric integrals
• Integration by partial fractions
• Solving logistic differential equations and using them in modeling
• L’Hopital’s Rule
• L’Hopital’s Rule and its use in determining limits
• Improper integrals
• Improper integrals and their convergence and divergence,
including the use of L’Hopital’s Rule
Chapter 8: Infinite Series
(4-5 weeks, 2 tests, daily quizzes; cumulative exam)
• Convergence and divergence of sequences
• Definition of a series as a sequence of partial sums
• Convergence of a series defined in terms of the limit
of the sequence of partial sums of a series
• Introduction to convergence and divergence of a series
by using technology on two examples to gain an intuitive
understanding of the meaning of convergence
• Geometric series and applications
• Telescoping series
• The nth-Term Test for Divergence
• The Integral Test and its relationship to improper integrals
and areas of rectangles
• Use of the Integral Test to introduce the test for p-series
• Comparisons of series
• Alternating series and the Alternating Series Remainder
• The Ratio and Root Tests
• Taylor polynomials and approximations: introduction using
the graphing calculator
• Power series and radius and interval of convergence
• Taylor and Maclaurin series for a given function
• Maclaurin series for sin x, cos x, e^x, and 1/(1-x)
• Manipulation of series, including substitution, addition of series,
multiplication of series by a constant and/or a variable,
differentiation of series, integration of series, and forming a
new series from a known series
• Taylor’s Theorem with the Lagrange Form of the
Remainder (Lagrange Error Bound)
Additional Topics: Plane Curves, Parametric Equations,
and Polar Curves (these topics are taught
using supplemental materials*)
(5-6 weeks, 3 tests, daily quizzes)
• Plane curves and parametric equations
• Parametric equations and calculus
• Parametric equations and vectors: motion along a curve,
position, velocity, acceleration, speed, distance traveled
• Analysis of curves given in parametric and vector form
• Polar coordinates and polar graphs
• Analysis of curves given in polar form
• Area of a region bounded by polar curves
Review for Advanced Placement Test
(these topics are taught using supplemental materials*)
(3-4 weeks, 5 practice tests)
*Supplemental materials include, but are not limited to:
• Free response and multiple choice questions from
AP Released Exams
• Lederman, David. Multiple-Choice and Free-Response
Questions in Preparation for the AP Calculus (BC) Examination.
6th ed. New York: D & S Marketing Systems, Inc., 1999
• Kelley, W. Michael and Mark Wilding. Master the AP Calculus
AB & BC Test. 3rd ed. New Jersey: Peterson’s, 2003.
• Hockett, Shirley and David Bock.How to Prepare for the
AP Calculus. 6th ed. New York: Barron’s Educational Services, 1998.
After the Advanced Placement Exam:
Differential Equations (1-2 weeks, 1 test)
• Definitions and basic concepts of differential equations
• First order linear differential equations
• Higher order homogeneous linear equations
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