TEXT:

 

Calculus Graphical, Numerical, Algebraic, 3rd edition;  Finney, Demana, Waits, Kennedy

 

 

REQUIREMENTS:

 

3-ring binder, Pencil, Notebook, and Graphing Calculator (TI-83+, TI-84, TI-84 Silver, TI-89, or TI-Nspire )

 

 

PREREQUISITES:

Before studying calculus, all students should complete four courses of secondary mathematics designed for college-bound students: courses in which they study algebra, geometry, trigonometry, analytic geometry, and elementary functions. These functions include those that are linear, polynomial, rational, exponential, logarithmic, trigonometric, inverse trigonometric, and piecewise defined. 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 of the numbers 0, pi/6, pi/4, pi/3, pi/2, and their multiples.

 

DESCRIPTION:

Advanced Placement (AP) Calculus BC is a college-level mathematics course for students that have previously demonstrated mastery of' algebra, geometry, coordinate geometry, trigonometry, and AP Calculus AB. This course will develop the students' understanding of the concepts of calculus and provide experience with its methods and applications. The course will emphasize a multi-representational approach to calculus, with concepts, results, and problems being expressed geometrically, numerically, analytically, and verbally. The connections among these representations also are important. Technology is used regularly to reinforce the relationships among the multiple representations of functions, to confirm written work, to implement experimentation, and to assist in interpreting results.

COURSE 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.

• understand the meaning of the derivative in terms of a rate of change and local linear approximation and they should be able to use derivatives to solve a variety of problems.

• 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.

• understand the relationship between the derivative and the definite integral as expressed in both parts of the Fundamental Theorem of Calculus.

• communicate mathematics both orally and in well-written sentences and should be able to explain solutions to problems.

• model a written description of a physical situation with a function, a differential equation, or an integral.

• use technology to help solve problems, experiment, interpret results, and verify conclusions.

• determine the reasonableness of solutions, including sign, size, relative accuracy, and units of measurement.

• develop an appreciation of calculus as a coherent body of knowledge and as a human accomplishment.

 

TOPICAL OUTLINE: 

The topic outline for Calculus BC includes all Calculus AB topics. Additional topics are found in paragraphs that are marked with a plus sign (+) or an asterisk (*). The additional topics can be taught anywhere in the course. Some topics will naturally fit immediately after their Calculus AB counterparts. Other topics may fit best after the completion of the Calculus AB topic outline. Although the examination is based on the topics listed here, the course may be enriched with additional topics.

I. Functions, Graphs, and Limits

A. Analysis of Graphs

With the aid of technology, graphs of functions are often easy to produce. The emphasis is on the interplay between the geometric and analytic information and on the use of calculus both to predict and to explain the observed local and global behavior of a function.

B. Limits of Functions (incl. one-sided limits)

• An intuitive understanding of the limiting process.
• Calculating limits using algebra.
• Estimating limits from graphs or tables of data.

C. Asymptotic and Unbounded Behavior

• Understanding asymptotes in terms of graphical behavior.
• Describing asymptotic behavior in terms of limits involving infinity.
• Comparing relative magnitudes of functions and their rates of change. (For example, contrasting exponential growth, polynomial growth, and logarithmic growth.)

D. Continuity as a Property of Functions

• An intuitive understanding of continuity. (The function values can be made as close as desired by taking sufficiently close values of the domain.)
• Understanding continuity in terms of limits.
• Geometric understanding of graphs of continuous functions (Intermediate Value Theorem and Extreme Value Theorem).

E. *Parametric, Polar, and Vector Functions

The analysis of planar curves includes those given in parametric form, polar form, and vector form.

II. Derivatives

A. Concept of the Derivative

• Derivative presented graphically, numerically, and analytically.
• Derivative interpreted as an instantaneous rate of change.
• Derivative defined as the limit of the difference quotient.
• Relationship between differentiability and continuity.

B. Derivative at a Point

• Slope of a curve at a point. Examples are emphasized, including points at which there are vertical tangents and points at which there are no tangents.
• Tangent line to a curve at a point and local linear approximation.
• Instantaneous rate of change as the limit of average rate of change.
• Approximate rate of change from graphs and tables of values.

C. Derivative as a Function

• Corresponding characteristics of graphs of 'f and f '.
• Relationship between the increasing and decreasing behavior of f and the sign of f '.
• The Mean Value Theorem and its geometric consequences.
• Equations involving derivatives. Verbal descriptions are translated into equations involving derivatives and vice versa.

D. Second Derivatives

• Corresponding characteristics of the graphs of f, f ', and f ".
• Relationship between the concavity of f and the sign of f ".
• Points of inflection as places where concavity changes.

E. Applications of Derivatives

• Analysis of curves, including the notions of monotonicity and concavity.
• + Analysis of planar curves given in parametric form, polar form, and vector form, including velocity and acceleration.
• Optimization, both absolute (global) and relative (local) extrema.
• Modeling rates of change, including related rates problems.
• Use of implicit differentiation to find the derivative of an inverse function.
• Interpretation of the derivative as a rate of change in varied applied contexts, including velocity, speed, and acceleration.
• Geometric interpretation of differential equations via slope fields and the relationship between slope fields and solution curves for differential equations.
• + Numerical solution of differential equations using Euler's method.
• + L'Hôpital's Rule, including its use in determining limits and convergence of improper integrals and series.

F. Computation of Derivatives

• Knowledge of derivatives of basic functions, including power, exponential, logarithmic, trigonometric, and inverse trigonometric functions.
• Basic rules for the derivative of sums, products, and quotients of functions.
• Chain rule and implicit differentiation.
• + Derivatives of parametric, polar, and vector functions.

III. Integrals

A. Interpretations and Properties of Definite Integrals

• Definite integral as a limit of Riemann sums.
• Definite integral of the rate of change of a quantity over an interval interpreted as the change of the quantity over the interval:

                   

• Basic properties of definite integrals. (Examples include additivity and linearity.)

B. *Applications of Integrals

Appropriate integrals are used in a variety of applications to model physical, social, or economic situations. Although only a sampling of applications can be included in any specific course, students should be able to adapt their knowledge and techniques to solve other similar application problems. Whatever applications are chosen, the emphasis is on using the method of setting up an approximating Riemann sum and representing its limit as a definite integral. To provide a common foundation, specific applications should include finding the area of a region (including a region bounded by polar curves), the volume of a solid with known cross sections, the average value of a function, the distance traveled by a particle along a line, and for BC only the length of a curve (including a curve given in parametric form).

C. Fundamental Theorem of Calculus

• Use of the Fundamental Theorem to evaluate definite integrals.
• Use of the Fundamental Theorem to represent a particular antiderivative, and the analytical and graphical analysis of functions so defined.

D. Techniques of Antidifferentiation

• Antiderivatives following directly from derivatives of basic functions.
• + Antiderivatives by substitution of variables (including change of limits for definite integrals), parts, and simple partial fractions (nonrepeating linear factors only).
• + Improper integrals (as limits of definite integrals).

E. Applications of Antidifferentiation

• Finding specific antiderivatives using initial conditions, including applications to motion along a line.
• Solving separable differential equations and using them in modeling. In particular, studying the equation y ' = ky and exponential growth.
• + Solving logistic differential equations and using them in modeling.

F. Numerical Approximations to Definite Integrals

Use of Riemann sums (using left, right, and midpoint evaluation points) and trapezoidal sums to approximate definite integrals of functions represented algebraically, graphically, and by tables of values.

IV. *Polynomial Approximations and Series

A. *Concept of Series

A series is defined as a sequence of partial sums, and convergence is defined in terms of the limit of the sequence of partial sums. Technology can be used to explore convergence or divergence.

B. *Series of constants

• + Motivating examples, including decimal expansion.
• + Geometric series with applications.
• + The harmonic series.
• + Alternating series with error bound.
• + Terms of series as areas of rectangles and their relationship to improper integrals, including the integral test and its use in testing the convergence of p-series.
• + The ratio test for convergence and divergence.
• + Comparing series to test for convergence or divergence.

C. *Taylor Series

• + Taylor polynomial approximation with graphical demonstration of convergence. (For example, viewing graphs of various Taylor polynomials of the sine function approximating the sine curve.)
• + Maclaurin series and the general Taylor series centered at x = a.
• + Maclaurin series for the functions e^x, sin x, cos x, and 1/(1-x).
• + Formal manipulation of Taylor series and shortcuts to computing Taylor series, including substitution, differentiation, antidifferentiation, and the formation of new series from known series.
• + Functions defined by power series.
• + Radius and interval of convergence of power series.
• + Lagrange error bound for Taylor polynomials.

 

PACING GUIDE:
 

Chapter 1 Prerequisites for Calculus	4 days
Chapter 2 Limits and Continuity	6 days
Chapter 3 Derivatives	22 days
Chapter 4 Applications of Derivatives	14 days
Chapter 5 The Definite Integral	16 days
Chapter 6 Differential Equations and Mathematical Modeling	22 days
Chapter 7 Applications of Definite Integrals	15 days
Chapter 8 L'Hôpital's Rule, Improper Integrals, and Partial Fractions	7 days
Chapter 9 Infinite Series	23 days
Chapter 10 Parametric, Vector, and Polar Functions	16 days

This timeline gives approximately 15 days to “review” the course before the exam.  These times are approximate and subject to change.

 
THE EXAM:

Put your knowledge to the test: The AP Calculus BC Exam assesses your mastery of Calculus BC concepts and techniques. It also gives you the chance to earn college credit while in high school. 

About the Exam

The AP Calculus BC Exam is 3 hours and 15 minutes. The 105-minute, 45-question multiple-choice section tests your proficiency on a wide variety of topics. The 90-minute, six-problem free-response section gives you the chance to demonstrate your ability to solve problems using an extended chain of reasoning.

Section I: Multiple-Choice

The multiple-choice section of the exam has two parts. For Part A, you'll have 55 minutes to complete 28 questions without a calculator. For Part B, you'll have 50 minutes to answer 17 questions using a graphing calculator. For more information, see the calculator policy for the AP Calculus Exams.

Unlike other multiple-choice tests, random guessing can hurt your final score. While you don't lose anything for leaving a question blank, one quarter of a point is subtracted for each incorrect answer on the test. But if you have some knowledge of the question and can eliminate one or more answers, it's usually to your advantage to choose what you believe is the best answer from the remaining choices.

Section II: Free-Response

The free-response section tests your ability to solve problems using an extended chain of reasoning. You'll have 45 minutes for each of the two parts of the free-response section. In Part A, you'll answer three questions using a graphing calculator. In Part B, you'll answer three questions without a calculator. During the second timed portion of the free-response section (Part B), you are permitted to continue work on problems in Part A, but you are not permitted to use a calculator during this time. For more information, see the calculator policy for the AP Calculus Exams.

Scoring the Exam

The multiple-choice and free-response sections each account for one-half of your final exam grade. Since the exams are designed for full coverage of the subject matter, it is not expected that all students will be able to answer all the questions.

Exam Grades

AP Grade Reports are sent in July to the college you designated on your answer sheet, to you, and to your high school. Each report is cumulative and includes grades for all the AP Exams you have ever taken, unless you have requested that one or more grades be withheld from a college or canceled.

 

RESOURCES:

 

AP:Calculus BC	http://www.collegeboard.com/student/testing/ap/sub_calbc.html?calcbc
Ask Dr. Math	http://mathforum.org/dr.math/dr-math.html
Calculus textbook website	http://www.phschool.com/webcodes10/index.cfm?fuseaction=home.gotoWebCode&wcprefix=bkk&wcsuffix=1000
CyberCalc	http://www.npac.syr.edu/REU/reu94/williams/calc-index.html
InterAct Math	http://www.interactmath.com
Karl's Calculus Tutor	http://www.karlscalculus.org/
Mr. Calculus	http://users.adelphia.net/~sismondo/index.html
PowerPoint Lectures	http://online.math.uh.edu/HoustonACT/
Slopefields on TI-89	http://www.jamesrahn.com/Calculator/TI89/ti-893.htm
TI Graphing Calculators	http://education.ti.com/educationportal/
Tutorials for the Calculus Phobe	http://www.calculus-help.com/funstuff/phobe.html
Using the TI-89 Calculator	http://www.prenhall.com/divisions/esm/app/graphing/ti89/index.html
Visual Calculus	http://archives.math.utk.edu/visual.calculus/index.html

 

 

Grade Scale:
A	93 and above
B	85 - 92
C	75 – 84
D	65 - 74
F	64 and below